Ramanan, Global Calculus

Global Calculus

S. Ramanan

Graduate Studiesin MathematicsVolume 65

American Mathematical Society

Global Calculus

Global Calculus

S. Ramanan

Graduate Studiesin Mathematics

Volume 65

American Mathematical SocietyProvidence, Rhode Island

Editorial BoardWalter Craig

Nikolai IvanovSteven G. Krantz

David Saltman (Chair)

2000 Mathematics Subject Classification. Primary 14-01, 32-01, 53-01;Secondary 32Lxx, 32Qxx, 32Wxx, 53Cxx.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-65

Library of Congress Cataloging-in-Publication DataRamanan, S.

Global calculus / S. Ramanan.p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 65)

Includes bibliographical references and index.ISBN 0-8218-3702-8 (alk. paper)1. Geometry, Algebraic. 2. Differential operators. 3. Analytic spaces. 4. Differential geom-

etry. I. Title. II. Series.

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Contents

Preface

Chapter 1. Sheaves and Differential Manifolds:

ix

Definitions and Examples 1

1. Sheaves and Presheaves 2

2. Basic Constructions 9

Differential Manifolds 12

4, Lie Groups; Action on a Manifold 23

Exercises 25

Chapter 2. Differential Operators 27

1. First Order Differential Operators 27

2. Locally Free Sheaves and Vector Bundles 29

3. Flow of a Vector Field 38

4. Theorem of Frobenius 46

5. Tensor Fields; Lie Derivative 50

6. The Exterior Derivative; de Rham Complex 54

7. Differential Operators of Higher Order 61

Exercises 70

Chapter 3. Integration on Differential Manifolds 73

1; Integration on a Manifold 73

2. Sheaf of Densities 79

3. Adjoints of Differential Operators 85

v

vi Contents

Exercises 90

Chapter 4. Cohomology of Sheaves and Applications 93

1. Injective Sheaves 93

2. Sheaf Cohomology 98

3. Cohomology through Other Resolutions 105

4. Singular and Sheaf Cohomologies 107

5. Cech and Sheaf Cohomologies 114

6. Differentiable Simplices; de Rham's Theorem 117

Exercises 122

Chapter 5. Connections on Principal and Vector Bundles; Lifting ofSymbols .125

1. Connections in a Vector Bundle 125

2. The Space of All Connections on a Bundle 131

3. Principal Bundles 135

4. Connections on Principal Bundles 143

5. Curvature 148

6. Chern-Weil Theory `154

7. Holonomy Group; Ambrose-Singer Theorem 164

Exercises 168

Chapter 6. Linear Connections 171

1. Linear Connections 171

2. Lifting of Symbols and Torsion 177

Exercises 182

Chapter 7. Manifolds with Additional Structures, 185

1. Reduct ion of the Structure Group 185

2. Torsio n Free G-Connections 194

3. Compl ex Manifolds 197

4. The O uter Gauge Group 201

5. Riema nnian Geometry 207

6. Riema nnian Curvature Tensor 211

7. Ricci, Scalar and Weyl Curvature Tensors 219

8. Cliffor d Structures and the Dirac Operator 224

Chapter 8 Local Analysis of Elli tic O erators 229.d'Al

p p1. Regularisation 229

Contents vii

2. A Characterisation of Densities 232

3. Schwartz Space of Functions and Densities 233

4. Fourier Transforms 238

5. Distributions 242

6. Theorem of Sobolev 245

7. Interior Regularity of Elliptic Solutions 251

Chapter 9. Vanishing Theorems and Applications 257

1. Elliptic Operators on Differential Manifolds 257

2. Elliptic Complexes 261

3. Composition Formula 270

4. A Vanishing Theorem 276

5. Hodge Decomposition 279

6. Lefschetz Decomposition 281

7. Kodaira's Vanishing Theorem 287

8. The Imbedding Theorem 292

Appendix 301

1. Algebra 301

2. Topology 305

3. Analysis 307

Bibliography 311

Index 313

Preface

This book is intended for postgraduate and ambitious senior undergraduatestudents. The key word is Differential Operators. I have attempted todevelop the calculus of such operators in an uncompromisingly global set-up.

I have tried to make the book as self-contained as possible. I do makeheavy demands on the algebraic side, but at least recall the required resultsin a detailed way in an appendix.

This book has had an unusually long gestation period. It is incrediblefor me to realise that the general tone of the book is very much the sameas that of a course which I gave way back in 1970 at the Tata Institute,which I (hopefully) improved upon a couple of years later, and expandedinto a two quarter graduate course at the University of California, Los An-geles, in 1979-80. The encouragement that the audience gave me, especiallyNagisetty Venkateswara Rao (now at the University of Ohio, Toledo), myfirst student the late Annamalai Ramanathan, and Jost at UCLA, was quiteoverwhelming. The decision to write it all up was triggered by a strong sug-gestion of Stefan Mueller-Stach (now at Mainz University). He drove mefrom Bayreuth to Trieste in Italy, and I used the occasion to clear some ofhis doubts in mathematics. Apparently happy at my effort, he suggestedthat I had a knack for exposition and should write books. I took this formore than ordinary politeness, and embarked on this project nearly tenyears later.

Chanchal Kumar, Amit Hogadi and Chaitanya Guttikar enthusiasticallyread portions of the book and made many constructive suggestions. Mad-havan of the Chennai Mathematical Institute and Nandagopal of the TataInstitute of Fundamental Research helped with the figures. Ms. Natalya

ix

x Preface

Pluzhnikov of the American Mathematical Society took extraordinary carein finalising the copy. All through, and particularly in the last phase, mywife Anu's support, physical and psychological, was invaluable.

I utilised the hospitality of various institutions during the course of writ-ing. Besides the Tata Institute of Fundamental Research, my Alma Mater,I would like to mention specially the Institute of Mathematical Sciences atChennai, the International Centre for Theoretical Physics, Trieste, and atthe final stage, Consejo Superior de Investigaciones Cientificas in Madrid.

I wish to thank all the individuals and institutions for the help received.

I collaborated for long years with M. S. Narasimhan. I wish to takethis opportunity to acknowledge the exciting time that I spent doing math-ematical research with him. Specifically, the idea of what I call the com-position formula in the last chapter of this book, arose in our discussions.Narasimhan also suggested appropriate references for Chapter 8.

I learnt modern Differential Geometry from Jean-Louis Koszul. His lec-tures at the Tata Institute, of which I took notes, were an epitome of clarity.I wish to thank him for choosing to spend time in his youth to educate stu-dents in what must have seemed at that time the outback.

There is a short summary at the start of each chapter explaining thecontents. Here I would like to draw attention to what I think are the newfeatures in my treatment.

In Chapter 1 sheaves make their appearance before differential mani-folds. Everyone knows that this is the `correct' definition but I know of fewbooks that have adopted this point of view. The reason is that generallysheaves are somehow perceived to be more difficult to swallow at the outset.I believe otherwise. In my experience, if sufficient motivation is providedand many illustrative examples given, students take concepts in their stride,and will in fact be all the better equipped in their mathematical life for anearly start.

In Chapter 2 I have introduced the notion of the Connection Algebraand believe most computations can, and ought to be, made in this algebra.

In Chapter 3 the treatment of densities and orientation are, I believe,nonconventional. In particular, the change-of-variable formula is not usedbut is in fact a simple consequence of this approach.

Chapter 4 is a fairly straightforward account of sheaf cohomology.

Connections are treated as tools to lift symbols to differential operators;various tensors connected with connections and linear connections have anatural interpretation from this point of view. These are worked out inChapter 5. The existence of torsion free linear connections compatible with

Preface xi

various structures lead to natural integrability conditions on them. Thistreatment, which I believe is new, is given in Chapter 6.

The additional structures are themselves studied in Chapter 7 with someemphasis on naturally occurring differential operators.

Chapter 8 is an account of the local theory of elliptic operators, whileChapter 9 contains the composition formula I mentioned earlier. A generalvanishing theorem for harmonic sections of an elliptic complex is provedhere under a suitable curvature hypothesis. The Bochner, Lichnerowiczand Kodaira vanishing theorems are derived as special cases, followed by ashort account of how Kodaira's vanishing theorem leads to the imbeddingtheorem.

Mistakes, particularly relating to signs and constants, are a professionalhazard. I would be grateful to any reader who takes trouble to inform meof such and other errors, misleading remarks, obscurities, etc.

Chapter 1

Sheaves andDifferential Manifolds:Definitions andExamples

In Geometry as well as in Physics, one has often to use the tools of differ-ential and integral calculus on topological spaces which are locally like opensubsets of the Euclidean space R', but do not admit coordinates valid every-where. For example, the sphere {(x, y, z) E R3 x2 + y2 + z2 = 1}, or moregenerally, the space {(x1, ... , x,,,) E 1[8n : x? = 1} are clearly of geometricinterest. On the other hand, constrained motion has to do with dynamicson surfaces in R3. In general relativity, one studies `space-time' which com-bines the space on which motion takes place and the time parameter in oneabstract space and allows reformulation of problems of Physics in terms ofa 4-dimensional object. All these necessitate a framework in which one canwork with the tools of analysis, like differentiation, integration, differentialequations and the like, on fairly abstract objects. This would enable one tostudy Differential Geometry in its appropriate setting on the one hand, andto state mathematically the equations of Physics in the required generality,on the other. The basic objects which accomplish this are called differentialmanifolds. These are geometric objects which are locally like domains inthe Euclidean space, so that the classical machinery of calculus, available inILBn, can be transferred, first, to small open sets and then patched together.The main tool in the patching up is the notion of a sheaf. We will first

1

2 1. Sheaves and Differential Manifolds

define sheaves and discuss basic notions related to them, before taking updifferential manifolds.

1. Sheaves and Presheaves

At the outset, it is clear that functions of interest to us belong to a classsuch as continuous functions, infinitely differentiable (or CO°) functions, realanalytic functions, holomorphic functions of complex variables, and so on:Some properties common to all these classes of functions are that

i) they are all continuous, andii) the condition for a continuous function to belong to the class is of a

local nature.

By this we mean that for a continuous function to be differentiable, realanalytic or holomorphic, it is necessary and sufficient for it to be so in theneighbourhood of every point in its domain of definition. We will start withan axiomatisation of the local nature of the classes of functions we seek tostudy.

1.1. Definition. Let X be a topological space. An assignment to everyopen subset U of X, of a set .''(U) and to every pair of open sets U, V withV C U, of a map (to be called restriction map) resuv :.F(U) --* F(V)satisfying

resvw o resuv = resuwfor every triple W C V C U of open sets, is called a presheaf of sets. If.F(U) are all abelian groups, rings, vector spaces, ... and the restrictionmaps are homomorphisms of the respective structures, then we say that Fis a presheaf of abelian groups, rings, vector spaces, ... .

1.2. Definition. A presheaf is said to be a sheaf if it satisfies the followingadditional conditions: Let U = Vi be any open covering of an open setU. Then

S1 : Two elements s, t E F(U) are equal if resUV; s = resUV; t for all i E I.S2 If si E .F(Ui) satisfy resUuudnUj si = resuu nu; sj for all i, j E I, then

there exists an element s E F(U) with resUV; s = si for all i. We willalso assume that .F(O) consists of a single point.

1.3. Examples.1) As indicated above, the concepts of differentiability, real analyticity, .. .

are all local in nature, so that it is no surprise that if X is an opensubspace of R', then the assignment to every open subset U of X, ofthe set A(U) of differentiable functions with the natural restriction of


functions as restriction maps, gives rise to a sheaf. This sheaf will becalled the sheaf of differentiable functions on X. Obviously this sheafis not merely a sheaf of sets, but a sheaf of R-algebras.

2) The assignment of the set of bounded functions to every open set U ofX, and the natural restriction maps define a presheaf on X. However,if we are given as in S2 any compatible set of bounded functions si,then while such a data does define a unique function s on U, there isno guarantee that it will be bounded. Thus this presheaf satisfies S1but not S2 and so is not a sheaf. (Can one modify this in order toobtain a sheaf?)

3) Consider the assignment of a fixed abelian group to every nonemptyopen set U, all restriction maps being identity. We will also assignthe trivial group to the empty set. This defines a presheaf, but is nota sheaf in general. (Why?)

4) The standard n-simplex A,z is defined to be {(xo, xl, ... , xn) E Il8n+1xi = 1, xi > 0} with the induced topology. In algebraic topology,

one way of studying the topology of a space X is to look at continuousmaps of the standard simplices into X and studying their geometry.We will give the basic definitions here. A singular n-simplex in atopological space X is a continuous map of the standard n-simplexAn into X. If A is a fixed abelian group, then an A-valued singularcochain in X is an assignment of an element of A to every singularsimplex. Now, to every open subset U of X, associate the set Sn(U)of all singular cochains in U. If V is an open subset of U, then wehave an obvious inclusion of the set of singular simplices in V into thatin U. Consequently there is also a restriction map Sn(U) -* Sn(V).This makes Sn a presheaf, which we may call the presheaf of singularcochains in X. It is obvious that this does not satisfy the axiom S1for sheaves. For if X = U Ui is a nontrivial open covering, then wecan define a nonconstant cochain which is zero on all simplices whoseimages are contained in some Ui. On the other hand, if a cochain isdefined on simplices with images in some Ui, then one can extend thiscochain to all singular simplices by defining the cochain to be zero onsimplices whose images are not contained in any of the U. Thus thispresheaf satisfies S2.

5) The most characteristic example of a sheaf from our point of view isthe one which associates to any open set U the algebra of continuousfunctions on U. This can be generalised as follows. Let Y be any fixedtopological space and consider the assignment to any open set U, ofthe set of continuous maps from U to Y with the obvious restriction


maps. This defines a sheaf of sets, the sheaf of continuous mapsinto Y.

6) We will generalise this example a little further. Let E be a topologicalspace and 7r : E -+ X a continuous surjective map. Then associate toeach open subset U of X the set of continuous sections of it over U(namely, continuous maps o- : U --> E such that 7r o a = Idu). This,together with the obvious restriction maps, is a sheaf called the sheafof sections of ir. Example 5) is obtained as a particular case on takingE = Y x X and 7r to be the second projection.

We will see below that every sheaf arises in this way, that is to say,to every sheaf F on X, one can associate a space E = E(F) and a mapit : E --> X as above such that the sheaf of sections of it may be identifiedwith F.

1.4. Sheaf associated to a presheaf.In the theory of holomorphic functions, one talks of a germ of a func-

tion at a point, when one wishes to study its properties in an (unspecified)neighbourhood of a point. This means the following. Consider pairs (U, f )consisting of open sets U containing the given point x and holomorphicfunctions f defined on U. Introduce an equivalence relation in this set bydeclaring two such pairs (U, f ), (V, g) to be equivalent if f and g coincide insome neighbourhood of x which is contained in U n V. An equivalence classis called a germ. This procedure can be imitated in the case of a presheafand this leads to the concept of a stalk of a presheaf at a point.

1.5. Definition. Let Y be a presheaf on a topological space X. Then thestalk.F. of Y at a point x E X is the quotient set of the set consisting of allpairs (U, s) where U is an open neighbourhood of x and s is an element ofY(U) under the equivalence relation:

(U, s) is equivalent to (V, t) if and only if there exists an open neighbour-hood W of x contained in U n V such that the restrictions of s and t to Ware the same.

If s E F(X) then, for any x E X, the pair (X, s) has an image in thestalk.F'x, namely the equivalence class containing it. It is called the germ ofs at x. We will denote it by s_,.

Let E = E(.F) be the germs of all elements at all points of X, that is tosay, the disjoint union of all the stalks .F, x E X. What we intend to donow is to provide the set E with a topology such that

a) the map 7r : E ---+ X which maps all of F to x, is continuous;


b) ifs E .F(U) then the section s" of E over U, defined by setting s(x) = sx,is continuous.

We achieve this by associating to each pair (U, s), where U is an opensubset of X and s E F(U), the set ss"(U), and defining a topology on Ewhose open sets are generated by sets of the form s(U). In order to checkthat with this topology, the maps s are continuous, we have only to showthat s-'(t(V)) is open for every open subset V of X and t E F(V). This isequivalent to the following

1.6. Lemma. If s E F(U) and t E .F(V), then the set of points x E U fl vsuch that sx = tx is open in X.

Proof. From the definition of the equivalence relation used to define .fix, wededuce that if sa, = to, for some point a E X, then the restrictions of s and t tosome open neighbourhood N of a are the same. Hence we must have sx = txfor every point x of N as well. Thus N is contained in {x E U : sx = tx},proving the lemma.

Finally if U is an open set in X, then it-1(U) = UxEU.Fx = Us(V)where the latter union is over all open subsets V C U and all s E .F(V).This shows that 7r-1(U) is open in E and hence that it is continuous.

1.7. Definition. The etale space associated to the presheaf F is the setE(F) = UxEX .fix provided with the topology generated by the sets s(U),where U is any open set in X and s E F(U). Moreover, if it is the naturalmap E(.F) -i X which has Fx as fibre over x for all x E X, then the sheafof sections of it is called the sheaf associated to the presheaf F.

1.8. Remarks.1) If we look at the fibre of it over x E X, namely .fix, we see that the

topology induced on it is nothing serious, since s(U) (being a sectionof ir), intersects Fx only in one point, namely sx. In other words, theinduced topology on the fibres is discrete.

2) Consider the constant presheaf A defined by the abelian group A.Clearly the stalk at any point x E X is again A so that E can inthis case be identified with A x X. Any a E A gives rise to an elementof A(U) for every open subset U of X. The corresponding section dover U of 7r : E = A x X --+ X is given simply by x ' (a, x) over U.Hence the image, namely {a} x U is open in E. From this one easilyconcludes that the topology on E = A x X is the product of the dis-crete topology on A and the given topology on X. Hence its sectionsover any open set U are continuous maps of U into the discrete spaceA. This is the same as locally constant maps on U with values in A.


3) Notice that in the construction of the etale space we did not make fulluse of the data of a presheaf. It is enough if we are given F(U) foropen sets U running through a base of open sets.

1.9. Definition. If '1, .72 are presheaves on a topological space X, then ahomomorphism f : F1 --+.F2 is an association to each open subset U of X ofa homomorphism f (U) : 171(U) --+ .7=2 (U) such that whenever U, V are opensets with V C U, we have a commutative diagram

F1(U)f F2(U)

Iresuv resuv

F1(V)f

,7=2(V)

When we consider presheaves of abelian groups, rings, ... , and talk. ofhomomorphisms of such sheaves, we require that all the maps f (U) be ho-momorphisms of these structures.

1.10. Examples.

1) The inclusion of the set of differentiable functions (on a domain in RI)in the set of continuous functions is clearly a homomorphism of thesheaf of differentiable functions into that of continuous functions.

2) Consider the sheaf of differentiable functions in a domain of Rn. Themap f induces a sheaf homomorphism of the sheaf of differen-

kLitiable functions into itself, the homomorphism being one of sheaf ofabelian groups but not of rings.

We will rephrase this as follows. Let V be a (finite-dimensional)vector space over R. Then one has a natural sheaf of differentiablefunctions on V. One can for example choose a linear isomorphism ofV with IR and consider functions of V as functions of the coordinatevariables xl, x2, ... , xn. Then differentiability makes sense, indepen-dent of the isomorphism chosen. For let yi, y2, ... , yn is a set of vari-ables obtained by some other isomorphism of V with Rn, that is tosay,

yi = ai1x1 + ai2x2 + . + ainxn

where (aid) is an invertible matrix. Then f is differentiable withrespect to (xi) if and only if it is so with respect to (yi). Indeed dif-ferentiation can be defined intrinsically as follows. For any v E V,define av f (x) = limt--;o f (x+tv)-f(x) Then f 8of gives rise to atsheaf homomorphism of abelian groups.


3) The inclusion of constant functions in differentiable functions gives a ho-momorphism of the constant sheaf III in the sheaf of differentiable func-tions.

If F is a presheaf and F the sheaf of sections of E(F), then for every openset U in X, we get a natural homomorphism of F(U) into .J'(U), whichmaps any s to the section s. If V C U, then for every x E V, the elementof .cx given by (U, s) is the same as that given by (V, resuv s) by the verydefinition of Fx. This implies that the diagram

.F(U) -->tresuv jresuv

F(V) -- -(V)

is commutative, proving that the natural homomorphisms F(U) , .,'(U)define a homomorphism of presheaves.

1.11. Proposition. A presheaf F satisfies Axiom S1 if and only if theinduced map F(U) --3 is injective, for every open set U of X.

Proof. To say that F(U) -* F(U) is injective is equivalent to saying thatif 51, s2 E F(U) with (sl)x = (S2)x for all x E U, then sl = 82. Butthen the assumption assures us that resuNx sl = resUNN 82 for some openneighbourhood Nx C U of x. Now Axiom S1, applied to the covering U =U Nx says precisely that sl = 52. Conversely, if U = U UZ and s, t areelements of F(U) satisfying resUU, (s) = resUV; (t) for all i, then the sameis true of s" and I. But since satisfies S1, it follows that s = t. Now ifwe assume that F(U) - is injective, it follows that s = t, so that weconclude that F satisfies S1.

1.12. Proposition. A presheaf F is a sheaf if and only if the natural maps.F(U) --> are all isomorphisms.

Proof. In fact, if all these maps are isomorphisms, then .F is isomorphic tothe sheaf F and it follows that F is itself a sheaf. On the other hand, if F isasheaf, then we conclude from the above proposition that the maps F(U) -->F(U) are injective and we need only to verify that they are surjective. Anelement a E is a section over U of the etale space of T. Hence, forevery x E U, there is a neighbourhood Nx and an element s(x) E F(N,)which represents the equivalence class a(x) E F. The section s(x) overNx of the etale space given rise to by s(x), and the section v, coincide atx. It follows that the two sections coincide in a neighbourhood Nx of x,contained in N. This means that there exist an open covering U = U Nxand S(x) E F(Nx) such that (s(x))a = a(a) for all a E Nx. In particular, s(x)


and s() give rise to the same section of J over N n N. But, thanks to theinjectivity of the natural map .F(N, n N,) -j :'(N,,' n N,) we conclude thatthe restrictions of s(x) and s(Y) to NN n Ny are the same. Since F is actuallya sheaf, this implies that there exists s E .7= (U) whose restriction to Nx isslxl for all x E U. Thus we have sx= u(x) for all x c U, or, what is thesame, s = or, proving that F(U) , F(U) is surjective.

Somewhat subtler is the relationship between Axiom S2 and the surjec-tivity of F(U) -* If F satisfies S2, then any section of F gives rise,as above, to an open covering {Nx} and elements s(x) of 1F(Nx). In order topiece all these elements together and obtain an element of F(U) we need tocheck that the restrictions of s(x) and s(y) to Nx n N. coincide, at least afterpassing to a smaller covering. We have the following set-topological lemma.

1.13. Lemma. Let {Ui}iEI be a locally finite open covering of a topologicalspace U and {V }iEI be a shrinking. Then for every x E U, there exists anopen neighbourhood Mx such that Ix = {i E I : Mx n V O} is finite, andif i E Ix then x belongs to Vi and Mx is a subset of Ui. If Mx and M.intersect, then there exists i E I such that Mx U My C U.

Proof. Since {Ui} is locally finite, so is the shrinking, and the existenceof M. such that the corresponding Ix is finite, is trivial. We will now cutdown this neighbourhood further in order to satisfy the other conditions.We intersect Mx with U \Vi for all i E Ix for which x 0 V2. We thus obtainan open neighbourhood of x, and the closures of all V, i E Ix then contain x.It can be further intersected with niElx Ui, and the resulting neighbourhoodsatisfies the first assertion of the lemma. Now if Mx n My 0 0, then for anyz E Mx n My, choose i E I such that z E V. Then Mx intersects V andhence Mx C Ui. Similarly My is also contained in Ui proving the secondassertion.

This lemma can be used to deduce that under a mild topological hy-pothesis, the natural maps .F(U) -+ F(U) are surjective for all open sets U,if the presheaf F satisfies Axiom S2.

1.14. Proposition. If every open subset U of X is paracompact, and thepresheaf _'F satisfies S2, then the map F(U) -+ F(U) is surjective for all U.

Proof. Firstly, given an element o- of F(U), there is a locally finite opencovering {Ui} of U, and elements si E F(Ui) for all i, with the property thatfor all x E Ui, the elements si have the image o-(x) in F. Let {V } be ashrinking of {Ui}. For every x E U, choose M. as in Lemma 1.13. We mayalso assume that the restrictions to Mx of any of the si for which i E Ix, isthe same, say s(x). It follows that the restrictions of s(x) and s(') to MxnMy


are the same as the direct restriction of some si to m, fl my, proving in viewof Axiom S2, that there exists s E F(U) whose restriction to Mx is s(x) forall x E U. This proves that .r(U) --> F(U) is surjective.

1.15. Exercises.

1) Let X be a topological space which is the disjoint union of two properopen sets U1 and U2. Define .F(U) to be (0) whenever U is an opensubset of either U1 or U2. For all other open sets U, define .F(U) = A,where A is a nontrivial abelian group. If U C V and F(U) = A,then define the restriction map to be the identity homomorphism.All other restriction maps are zero. Show that F is a presheaf suchthat .,' = (0).

2) In the above, does F satisfy Axiom S2?

Subsheaves.

1.16. Definition. A sheaf Q is said to be a subsheaf of a sheaf F if we aregiven a homomorphism -> F satisfying either of the following equivalentconditions.

1) Qx Fx is injective for all x E X.2) For any open subset U of X, Q(U) -> F(U) is injective.

To see that the above conditions are equivalent, note that 1) impliesthat E(Q) -+ E(F) is injective and hence the set of sections of Q over anyset is also mapped injectively into the set of sections of F. Conversely,assume 2), and let a, b E Qx have the same image in F. Then there exista neighbourhood U of x and elements s, t E Q(U) such that sx = a, tx = b.Moreover, by replacing U with a smaller neighbourhood we may also assumethat the images of s and t are the same in F(U). This implies by ourassumption that s = t as elements of Q(U), as well. Hence sx = tx in Q.

2. Basic Constructions

When .F is a sheaf, it is legitimate to call elements of F(U) sections of F overan open set U, since they can be identified with sections of the associatedetale space. Continuous sections of the etale space make sense, on the otherhand, over any subspace of X.

2.1. Proposition. If K is a closed subspace of a paracompact topologicalspace X, then any section over K of a sheaf F on X is the restriction to Kof a section of Jc' over a neighbourhood of K.


Proof. In fact, any section s over K is defined by open sets Ui of X thatcover K and elements si E F(UU). Since K is also paracompact, we mayassume (by passing to a refinement, if necessary) that the covering is locallyfinite. Let U = U Ui and {Vi} be a shrinking of this covering. Choose forevery x E U, an open neighbourhood Mx of x as in Lemma 1.13. Denote byIx the set of all i E I such that x E Vi. Then our choice of Mx is equivalentto saying that i belongs to Ix if and only if Mx intersects Vi. Considerthe subset W = {x E U (si)x is independent of i for all i E Ix}. For anyy E Mx, we have Iy C I. For, if i E Iy, then y E Vi and hence Mx which isa neighbourhood of y has nonempty intersection with Vi. This means thati E I. Clearly, the set {y e Mx : (si)y is independent of i E Ix} is an openset containing x and contained in W. Hence W is open. On the other handwe have the inclusion K C W. It is now clear that the si actually give asection of F over W.

2.2. Remark. Let us consider the sheaf of continuous functions on R' forexample. A section of the corresponding etale space over a closed set K isthe same as a continuous function in a neighbourhood of K with the un-derstanding that two such continuous functions are to be considered equiv-alent if they coincide in a neighbourhood of K. Any such `germ' gives, onrestriction to K, a continuous function on K. On the other hand any con-tinuous function on K can be extended to some neighbourhood of K. Thuswe have a surjection of the set of sections over K of the sheaf of contin-uous functions into the set of continuous functions on K. But this is notinjective, even when K consists of a single point. For in this case, a sectionis simply an element of the stalk at the point, which consists of germs ofcontinuous functions at the point.

2.3. Inverse images.We used the construction of E(F) as a means to pass from a presheaf to

a sheaf. However, even when one starts with a sheaf the construction ofthe etale space is useful in some applications. One such is the notion ofthe inverse image of a sheaf. Let f : X --> Y be a continuous map oftopological spaces. If 1 is a sheaf on Y, we seek to define a sheaf f -'.Fon X. Let E(F) be the etale space of T. Then form the fibre product ofthe map f : X --* Y and the map it : E(F) --+ Y. (It is the subspaceof the topological space X x E(F) consisting of points (x, a) such thatf (x) = 7r(a).) This space comes with a natural continuous map into X.The sheaf of continuous sections of this space is called the inverse image of.F by the map f.


In the particular case when X C Y is a subspace, the inverse image isalso called the restriction of F to X. We sometimes use the notation FIXfor this restriction.

2.4. Glueing together.Let (Ui) be an open covering of a space Y and for each i, let Fi be

sheaves on Uj. Then we wish to glue all these together and obtain a sheaf onthe whole of Y. For this we need some glueing data. If the -Fi are allrestrictions to Uj of the same sheaf F on Y, then the restrictions of -Fi and.Fj to Ui n Uj are the same as direct restrictions of F to Ui rl Uj. So, weassume as data, isomorphisms mij : .mij I Ui fl Uj -+ Fi I Ui fl Uj. Actuallywe need more, namely compatibility of these isomorphisms. Consider theset Ui fl Uj fl Uk. We have restrictions to it of the sheaves Fi, Fj and Fk.Besides, the isomorphisms mij, mjk and mik restrict to isomorphisms (twoby two) of these three sheaves on Ui fl U. f1 Uk as well. We will use the samenotation for these restrictions. Then the compatibility condition that wehave in mind is that they should satisfy

mij o mjk = mik

One can easily verify that given such data as above, one can glue thesheaves Fi together and obtain a sheaf .'F on the whole of Y with naturalisomorphisms of FI UU with.Fi.

Actually, in a certain sense, the construction involved in glueing is theconverse of the construction of the inverse image. If X is the topologicalunion of the spaces Ui, then it is clear that the Fi build a sheaf F on it, andwhat is needed is a sheaf on Y whose inverse image under the natural mapX -->YisT.

2.5. Remark. Regarding our glueing construction above, we wish to makethe following remark. Let .77 be a sheaf on X. Let (Ui) be an open coveringand let Gi be subsheaves. of FI Ui, for all i. Then, in order to glue the9i together, we only need to check that gi I Ui n Uj is the same subsheaf ofFl Ui fl Uj as 9j I Ui n Uj. Then we can, not only glue them together, butalso get a homomorphism of 9 into F which makes it a subsheaf.

2.6. Definition. Let F be a sheaf on a space X. Let f : X -+ Y be acontinuous map. Then one can define a sheaf on Y called the direct imageof .F by f as follows. To any open set U in Y, associate F(f -1(U)). It iseasy to check that this defines a sheaf. We will denote this by f" (.F) -

This is related to the inverse image very closely. In fact, let F be asheaf on X in the above situation, and C a sheaf on Y. We may consider onthe one hand, the direct image f*(F) and on the other, the inverse image


f -1(g). Then one can see easily that there is a natural bijection betweenhomomorphisms 9 -> f* (F) and f -'(G) --* F. Indeed, let us start with ahomomorphism T : C -* f*(F). This gives, for every open subset U c Ya homomorphism T(U) : 9(U) -+ (f,.F)(U) = .F(f-1(U)). If x E f-1(U),then we have a natural map F(f -1(U)) --+ .fix. Composing with T (U),we get a homomorphism CP(U) - J1F. It is obvious that as U varies overneighbourhoods of f (x), this is compatible with restrictions, and so inducesa map cf(x) = f _1(c)x -+ F. This is easily checked to be continuous onthe etale space and hence gives a homomorphism of f -1(CJ) into F. It isthis association that gives the bijection, as claimed.

2.7. Modules over a sheaf of algebras.Before we leave this preliminary account of sheaves and move on to dif-

ferential manifolds, we would like to introduce one more notion which is veryuseful in our context. As we have observed, the sheaf of continuous func-tions on a topological space, that of differentiable functions in R', that ofholomorphic functions in a domain in C'y, etc. are all sheaves of algebras.Let then A be a sheaf of algebras over a topological space X. On the otherhand, let M be a sheaf of abelian groups. Then we say that M is a sheaf ofmodules over A, or simply an A- module if for every U, open in X, the abeliangroup M(U) comes provided with a structure of an A(U)-module in sucha way that the restriction maps resuv respect the module structures in theobvious sense, namely

resvv (f s) = resuv (f) resvv (s)

for all f E A(U) and s E M(U).

3. Differential Manifolds

After these preliminaries, we are now ready to define the concept of differ-ential manifolds. These objects provide the proper setting for developingdifferential and integral calculus. In Physics, these are called configurationspaces and may be thought of as the set of all possible states of the systemof which one wishes to study the dynamics.

3.1. Definition. A differential manifold M (of dimension n) consists of

a) a topological space which is Hausdorff and admits a countable base foropen sets, and

b) a sheaf AM = A of subalgebras of the sheaf of continuous functions onM.

These are required to satisfy the following local condition. For any x E M,there is an open neighbourhood U of x and a homeomorphism of U with an

3. Differential Manifolds 13

open set V in IR' such that the restriction of A to U is the inverse image ofthe sheaf of differentiable functions on V.

The homeomorphisms referred to above, defined in a neighbourhood ofany point, are called coordinate charts. This is because composing withthis homeomorphism the coordinate functions in 118, one obtains functionsx1,. .. , xn,. We think of M as a global object on which action takes place,and it is locally described by using coordinates.

3.2. Examples.

1) Since the concept of differential manifolds is based on the notion ofdifferentiability in 118'1, it is clear that 1R'z together with the sheaf ofdifferentiable functions is a differential manifold. Slightly more ab-stractly, any finite-dimensional vector space over II8 is a differentialmanifold.

2) If (M, A) is a differential manifold, and U an open subset of M, thenthe subspace U, together with the restriction Al U of A is a differentialmanifold as well. This will be referred to as an open submanifold of M.

3) Combining these two examples, we see that any open subspace of 118?1(or any finite-dimensional vector space) is a differential manifold. Inparticular, the space GL(n, I18) of invertible (n, n)-matrices, which isactually the open set in the vector space of all (n, n)-matrices givenby the nonvanishing of the determinant, is a manifold of dimensionn2. Incidentally, it is also a group under matrix multiplication and iscalled the General Linear group.

4) Let f be a differentiable function in Jn. Consider the closed subspace(the zero locus of f) of IR' given by

Zf={xEI18n: f(x)=0}.

Then Z f has a natural structure of differential manifold, if at everyx E Z J, at least one of the partial derivatives of f does not vanish.In fact, consider the association to any open set U C Z f, of the set offunctions on U, which can be extended to a differentiable function ona neighbourhood of U in I18n. This gives a presheaf on Zf. (Actu-ally it will turn out that it is a sheaf, but we do not need it here.)Let AZf be the associated sheaf. Then by our assumption, for anyx E Z f, there exists a neighbourhood N of x in RI such that, one ofthe partial derivatives, say -, is nonzero. By the implicit functiontheorem, the projection to R` taking (x1, ... x,1) to (x1, ... , X.-I)is a differentiable isomorphism of N fl Z f with an open set V in Rn-1


(that is to say, a differentiable bijective map from N onto V whose in-verse is also differentiable). It is now clear that this isomorphism givesthe local requirement of the sheaf AZ-f. If we further specialise thefunction to be f (x) _ x? - 1, then Zf is the set of vectors of unitlength in Jn which we call the unit sphere S'-1

On the contrary, if f is taken to be the function xy defined in1182 with coordinates x, y, then clearly it does not satisfy the criteriongiven above. In fact, both the partial derivatives, a and a vanish at(0, 0). So we cannot conclude that Z f is a differential manifold in thiscase. In fact, the topological space Z f cannot have a structure of adifferential manifold, since it is easy to see that no neighbourhood of0 in Z f is homeomorphic to an open interval in R.

5) The above example can be generalised further, by taking, instead ofone function, finitely many functions. So, let f = (fz),1 < i < r, befinitely many differentiable functions in 118n. Then the closed subspace

Zf={xERt: ff(x)=0foralli}

has a natural structure of a differential manifold, if f satisfies thefollowing condition. For every x c Z f, the rank of the (r, n) matrix( ) is r. One may think of f as a function into Rr.

We will now take for f the function on (n, n)-matrices with valuesalso in Rn2 given by A H AA' - 1, where A' denotes the transposeof A. Then one can check that the above criterion is fulfilled andhence the topological space {A E Mn(]l8) : AA= In} is a differentialmanifold. This is actually a subgroup of GL(n, R) as well,' called theorthogonal group and is usually denoted O(n, l18). Similarly take themap f : M(n, C) into itself given by A H AT - In, and get the setf = 0 as a subgroup of GL(n, C). This group is called the unitarygroup and denoted U(n). (Are these manifolds connected?)

6) Consider the real projective space IIBpn defined to be the quotient ofthe unit sphere Sn by the identification of antipodal points x and-x. We may define a sheaf on 118Pn by associating to any open setU C 1E,pn, the algebra of differentiable functions on its inverse imagein Sn which are invariant under the antipodal map. For any pointa E Sn, the open neighbourhood of a consisting of those y E Snwhose distance from a is less than 1 is mapped homeomorphically onan open set in JRpn, and it is easy to see that this takes the sheaf ofdifferentiable functions on Sn isomorphically onto the sheaf definedabove. Thus we see that RPn is a differential manifold in a naturalway.


3.3. Exercises.

1) Show that the topological subspace of the space of all (n, n)-matrices,consisting of those matrices whose determinants are 1, is a differentialmanifold. This is also a subgroup of the group GL(n,R) mentionedabove and is denoted SL(n, JR). It is called the Special Linear group.

2) Is the same true of matrices with determinant 0?

3.4. Glueing up differential manifolds.Often, a structure of a differential manifold is given on a Hausdorff

topological space M with a countable base for open sets, by the followingprocedure. Suppose {Ui} is an open covering, and that each Ui is providedwith a subsheaf Ai of the sheaf of continuous functions making (Ui, Ai)a differential manifold. If we can glue all these sheaves together to get asheaf of algebras on M, then it is clear that it would make M a differentialmanifold. We have already seen (2.5) how we can glue them together. Whatwe need is simply that Ai I Ui n Uj is the same as Aj I Uj n U. In other words,the open submanifold Ui n Uj of Ui is the same as the open submanifoldUinU; of UJ.

In particular, if (Vi, Ai) are open submanifolds of R, then the glueingdata may also be formulated as follows. The space M is covered by opensets U. For each i, one is given a homeomorphism ci of Ui with the opensubset V of R1. If Ui and Uj intersect, then Ui n Uj has as images inV and Vj, two open sets which we may call Vj and Vji. Then cj o c,--1gives a homeomorphism Vii -> Vii. One may use the homeomorphism ci totransport the sheaf of differentiable functions on V to a sheaf of algebras onU. But in order to glue these together, we need to know that its restrictionto Ui n Uj is the same as the restriction of the transported sheaf on Uj.This can be achieved if and only if the above homeomorphism Vi -+ Vii isdifferentiable for every i, j. (Note that the inverse is also differentiable, byreversing the roles of i and j.) This is in fact the traditional definition of adifferential manifold.

V

For example, consider the sphere in R. The map

(xi,...,x,) i--. (xl,...,xi) .... x, )


is a homeomorphism of the open set consisting of points of the sphere forwhich xi > 0 onto the open unit ball in Rn-1. We can therefore providethese open sets with the differential manifold structure of the unit ball. Wecould do the same with the open sets of the sphere in which xi < 0. It isclear that all these open sets, as i varies, cover Sn-1. To glue these up, wehave only to check that the map

(yi,...,yn-1) H (y1,..., 1-W,...4j,...) yn-1),

where the term 1 --1Y11 occurs at the ith place, is a differentiable isomor-phism of the open set of the unit ball onto the open ball.

Another example is provided by the complex projective space. Noticefirst that the real projective space R1Pn may also be defined as the quotientof R'1+1 \ {0} by the equivalence relation: (xo, x1, , xn) - (yo, yl, , yn)if and only if there exists a nonzero real number a such that yi = axi, forall i. An analogous definition makes sense over complex numbers as well.In other words, define the complex projective space C1Pn to be the quotientof Cn+l \ {0} by the equivalence relation:

(zo,...,z,) - (zo,...,zn)

if and only if there exists a E C" such that zi' = azi for all i. Since forany such z = (zo, . . . , zn) at least one zi is nonzero, Cp is covered byopen sets which are images of sets Ui, i = 0, ... , n under the natural mapCC's'+1 \ {0} -> Ian, where Ui = {z =. (zo,... , zn) : zi 0}. It is clear that thesubspace zi = 1 of Ui is mapped homeomorphically onto the image of Ui in(Clan. On the other hand, it is homeomorphic to Cn under the projectionwhich omits the ith coordinate. This can be used as above to put a structureof a differential manifold on it. To check the condition for glueing up, weonly have to check that the map

{(ZO,...,zn),zi = 1, z7 01 H {(ZO,...,zn),zi 7' 0,zj = 11

given by (zo, ... zn) f--, (zo/zj, ... , zn/zj) is a differentiable isomorphism.This is of course obvious.

A little more abstractly, we could have replaced Cn+1 by any com-plex vector space V of dimension n + 1. Then one considers the open setV \ {0} and introduces the equivalence relation: v - v' if and only if thereexists a E Cx such that av = v'. The quotient is defined to be the projectivespace P(V) associated to V. If f is any nonzero linear form on V, the set{v E V : f (v) = 1} is mapped homeomorphically onto an open set Xf ofP(V). What we did above amounts to using this homeomorphism to definea differential structure on X f. We may glue all these structures together toobtain a differential structure on the whole of P(V).


One may also think of the points of P(V) as one-dimensional subspacesof V. Then one may generalise this by considering all r-dimensional vectorsubspaces of V for any fixed r < dim(V). The set thus formed is calledthe Grassmannian of r-dimensional subspaces of V. We will indicate twoways in which one can provide this set with the structure of a differentialmanifold. Firstly, let W be any such subspace. Consider the image of theone-dimensional space A' (W) in Ar (V). Thus to every element of the Grass-mannian we have associated an element of the projective space P(AD(V)).Then one checks the following assertion.

3.5. Lemma. If w is any nonzero element of Ar(V), then the linear mapV -* Ar+1(V) given by v - vnw has kernel of dimension < r. Moreover, thekernel is of dimension r if and only if w E Ar(W) for some r-dimensionalsubspace W of V.

Proof. In fact, it is obvious that if w belongs to Ar(W), then every elementof W is in the kernel of the above map. Let {ei},1 < i < n, be a basis of Vsuch that the first r of these generate W. In other words, el A ... A er canbe taken to be w. Let v = E aiei; then vnw = 0 if and only if ai = 0 forall i > r. Thus the kernel of the map v H v A w is precisely W.

Conversely, if the kernel contains an r-dimensional subspace W, againtake a basis like the one above. Writing out w in terms of a basis asEil<...<i,. ail,...,ireil A ... A ei,., we deduce that if ej Aw = 0, then ail,...,jr = 0whenever j does not belong to the set {il, . . . , i, }. Since we have assumedthat ej A w = 0 for all j < r, we see that the only nonzero coefficient inthe expression for w is al,...,r, that is to say, w E Ar(W). This also shows inparticular that the dimension of the kernel is < r.

The above lemma asserts in fact that the Grassmannian is imbedded inP(A?V) as a closed subset. Using this description, one can check that theGrassmannian is actually a closed submanifold. This imbedding is calledthe Plucker imbedding.

Another way of introducing the structure of differential manifold on theGrassmannian is the following. In order to introduce a coordinate systemin a neighbourhood of any r-dimensional subspace Uo of V, we proceedas follows. Fix an (n - r)-dimensional subspace W of V supplementaryto U0. Consider the set of all r-dimensional subspaces U of V which aresupplementary to W. This set evidently contains Uo. The projection ofV onto W corresponding to the direct sum decomposition V = U ® Wrestricted to U0 gives a linear map AU of Uo into W. When U = Uo thismap is the zero map. In general the linear map determines U as the image ofrl - AU, where i is the inclusion of U0 in V. This sets up a bijection between


Hom(Uo, W) and the above set, thereby coordinatising the Grassmannianin a neighbourhood of Uo.

3.6. Definition. If (M, A) is a differential manifold, then sections of A overan open set U of M are called differentiable functions on U.

Since A is a sheaf, the notion of a differentiable function on M is of alocal nature, so that the following properties are obvious.

1) If f is a nowhere vanishing differentiable function, then so is 11f.

ii) If (U,) is a locally finite covering and (fz) is a family of differen-tiable functions with support in UZ7 then > fz is also a differen-tiable function.

iii) If c o is a differentiable function on Rm and are differen-tiable functions on M, then co(fl, ... , fm) is also a differentiable func-tion.

3.7. Definition. A continuous map f of a differential manifold M intoanother differential manifold N is said to be differentiable if for any x E M,and for every differentiable function co in a neighbourhood U of f (x) in N,the composite co o f is a differentiable function on f -1(U).

From the definition of a differentiable map it follows that there is ahomomorphism of the sheaf AN into f*(AM) and therefore also a homo-morphism of AM into f -1 (AN). This is called the structure homomorphismassociated to f.

3.8. Remark. It is clear from the definition above that if M, N, P aredifferential manifolds, and f : M --i N, g : N --+ P are differentiable maps,then the composite g o f : M -- P is also differentiable. However, if f :M --+ N is a differentiable map which is bijective, then one cannot concludethat the inverse map is also differentiable. The hackneyed counterexampleis the function x H x3 of 1f8 into 118, which is differentiable and bijective, butwhose inverse x H x3 is not differentiable at 0 .

3.9. Definition. A differentiable map f : M -> N of differential manifoldsis said to be a diffeomorphism if there is a differentiable inverse.

A fact, basic to the study of differentiable functions, is that there arelots of them. We will now formulate this precisely.

3. Differential Manifolds

Differentiable partition of unity.

19

3.10. Proposition. The following are equivalent.

i) Given any locally finite open covering (Ui)iEZ of M, there exist differen-tiable functions Wi on M with values in the closed interval [0, 1] suchthat the support of Wi is contained in Ui and > cpi = 1.

ii) Given open sets U, V with V C U, there exists a positive differentiablefunction whose support is contained in U and which does not vanishanywhere in V.

Proof. i) implies ii). Consider the open covering (M \ V, U). By i), thereexist [0,1]-valued functions cp, b with supports respectively in M \ V and Usuch that cp +,0 = 1 everywhere. Then 0 satisfies the requirement in ii).

ii) implies i). Let (V) be a shrinking of (Ui), i.e. an open covering withU C Uj for all i. By Assumption ii), we see that there exist positive differ-entiable functions coj such that supp cpi C Ui and cpi are nonzero everywhereon Vi. Now since the covering is locally finite, the sum cp = E cpz makessense, is differentiable and is nonzero everywhere. The family of functionsf V)i = cpi/cp}, satisfies i).

Of course the point of the above proposition is that the equivalent prop-erties stated there are actually true. We will now prove this fact. Firstly,in order to prove ii), it is enough to do it locally in the following sense.For every m E M there exists a neighbourhood NN such that there is afunction as in ii) with V replaced by V fl Nx and U by U fl N.. Then onereplaces the sets N., by a locally finite refinement and notes that the sumof the corresponding functions fulfils the requirement. Taking Nx to be acoordinate neighbourhood of x, we therefore reduce the problem to provingthe following.

3.11. Proposition. Let S1, S2 be concentric spheres in Rn centered at 0,with Sl C S2. Then there exists a differentiable function which is nonzeroeverywhere inside Sl and has support contained in the interior of S2.

Proof. Clearly it is enough to construct a differentiable function on Rwhich is nonzero everywhere inside the unit ball and zero in the complement.The function x exp(1) for all x inside the unit ball and 0 in thecomplement, is such a function.

3.12. Definition. Let (Ui) be a locally finite open covering of a differentialmanifold M. A family (cpj) of differentiable functions on M with values in[0, 1] is said to be a partition of unity with respect to the covering (Ui), ifthe support of cpi is contained in Ui for every i and E cpi = 1.


We will now derive a simple consequence of the existence of a partitionof unity.

3.13. Proposition. Any section of A over a closed set can be extended toa differentiable function on M. In other words, given a differentiable func-tion co in a neighbourhood of a closed set K, there exists a differentiable func-tion cp on M which coincides with co in a neighbourhood of K.

Proof. In fact, if K C U, and f is a differentiable function on U, considerthe partition of unity with respect to the covering U, M \ V, where V is aneighbourhood of K with V C U. Thus there is a differentiable function cpon M which is 1 on V and with support in U. The function f cp on U hassupport contained in the support of cp. Hence the function f go on U and theconstant function 0 on M \ (supp f) coincide on the intersection, therebygiving rise to a differentiable function on the whole of M as required.

3.14. Proposition. Let M, N be differential manifolds. If f : M -+ Nis a continuous map such that for every differentiable function cp on N thecomposite cp o f is differentiable, then f is differentiable.

Proof. In fact, for any x E M and any differentiable function go in a neigh-bourhood of f (x), we have to show that co o f is differentiable in a neigh-bourhood of x. Let cp be a differentiable function on N coinciding with cp ina neighbourhood of f (x); then we are given that cp o f is differentiable. Butthen cP o f and cp o f coincide in a neighbourhood of x, proving our assertion.

3.15. Product manifolds.Let M and N be differential manifolds. Then one can provide the topo-

logical space M x N with the structure of a differential manifold in thefollowing way. Cover M and N by coordinate charts cj : U2 -* V andc : Uk --> Vk. Then we may cover M x N by Uz x Uk. On each openset UU x Uf one may define a coordinate chart ci x ck onto an open set inR' x R". The compatibility condition that needs to be verified, namelythe differentiability of (ci x ci) o (ci x ck)-1, follows obviously from thoseof cj o cti 1 and c, o cr 1. Thus we have provided the space M x N withthe structure of a differential manifold. Now it is easy to check from thisdefinition, that a mapping of any differential manifold L into M x N is dif-ferentiable if and only if its composites with the projections to M and Nare both differentiable. This manifold is (therefore) called the product of Mand N.


3.16. Submanifolds.We have already defined the notion of an open submanifold. One may

define a closed submanifold as follows. Let M be a closed subset of a differ-ential manifold N with the property that for every m E M, there exists acoordinate chart (U, c) around m in N such that Mn u is the set of commonzeros of some of the coordinates of the chart c.

The rationale of the definition is quite clear. If U is an open submanifoldof Rn, then the set of points (x) in U satisfying xl = X2 = = x,. = 0,should clearly be defined to be a closed submanifold of U. Note that this.setis itself a manifold with the remaining coordinates serving as a coordinatechart.

Combining the notions of an open submanifold and a closed submanifold,we may define a locally closed submanifold, to be a closed submanifold of anopen submanifold.

3.17. Immersed manifolds.There is a more general notion of a subset of a manifold that has the

structure of a manifold, which is formally similar to the above notion, butsubtler. Suppose M and N are differential manifolds. First of all, assumethat we have an injective differentiable map M -> N. Secondly, for any pointm E M, we require that there is a coordinate chart (U, c) in N containingthe image of m and a neighbourhood U' of m in M such that U' maps intoU and its image is the set of common zeros of some of the coordinates inthe chart c. Then we say that M is a manifold immersed in N.

The notion of an immersed manifold is somewhat delicate for the fol-lowing reason. Let us identify M with its image in the following discussion.Notice that we have not required that for every point m of M, there is acoordinate chart (U, c) of N such that u f1 M is defined by the vanishing ofsome of the coordinates of the chart c. The difference is not slight! Indeed,the topology of the immersed manifold M is not necessarily that of its imageinduced from that of N.

Let us consider an example. We know that the real line R and the torusS' x S' are both differential manifolds. Identify Sl with the submanifold ofC consisting of complex numbers of absolute value 1. Let ce E R. Considerthe map f : III --+ Sl x S' given by f (x) = (exp(27rix), exp(27raix)). It isclearly differentiable. It is also injective if a is irrational. For, if f (x) = f (y),then x - y E Z on the one hand and also, a (x - y) E Z on the other. Considerthe maps g : JR -* JR x JR given by x H (x, ax) and h : R x J -> S' x Sl givenby (x, y) H (exp(27rix), exp(27riy)). Clearly we have f = hog. It is obviousthat g imbeds J as a closed submanifold of JR x JR. On the other hand, h isa local isomorphism of differential manifolds and we conclude that f makes


R an immersed manifold in Sl x Sl in our sense. It is easy to see that theimage with the induced topology is not even locally connected. Indeed thisshows that the image is not a subspace at any point.

If a is irrational, it goes round and round infinitely. If it is rational, itrewinds at a finite stage.

We will indicate many simpler examples as well. A figure like 6 (openat the top end) can be realized as a submanifold of R2 by mapping ll8 differ-entiably like

Clearly this figure with the topology induced from that of R2 is not a man-ifold at the nodal point.

Again a figure like 8 can be realised as an immersed manifold of R2 intwo different ways, namely by parametrisii g it as follows.

4. Lie Groups; Action on a Manifold 23

This example also drives home the point that a closed submanifold is notjust an immersed manifold whose point set is closed. It also needs to havethe induced topology.

4. Lie Groups; Action on a Manifold

4.1. Definition. A Lie group G consists of two structures on the sameset G, namely it is a differential manifold and has also a group structure.The two structures are interrelated by the assumptions that the group lawG x G -* G and the group inverse G -- G are both differentiable.

4.2. Examples.

1) Any countable group with the discrete topology is a Lie group in oursense. (Countability is required because in our definition, manifoldsare supposed to have a countable base of open sets.)

2) The real line R is a Lie group under addition since the maps R x IR Rgiven by (x, y) H x+y and I[8 -* JR given by x -- -x are differentiable.

3) It is also clear that ]n is a Lie group under addition.4) The multiplicative group Cx consisting of nonzero complex numbers is

an open submanifold of C and is actually a Lie group under multipli-cation.

5) The groups GL(n, III) or GL(n, C), which are open submanifolds ofJn2 and C'2, are Lie groups. Indeed, the group composition is therestriction of a polynomial map I[8n2 X Rn2 -+ Rn2. If V is any vectorspace of finite-dimension over R or C, then the group GL(V) of linearautomorphisms is a Lie group.

6) The orthogonal group O(n, R) (resp. the unitary group U(n)) is a Liegroup under matrix multiplication.


4.3. Exercise. The quotient of GL(V) by its centre, namely nonzero scalarmatrices (or automorphisms), is called the projective linear group PGL(V).Show that it is a Lie group.

4.4. Definition. A homomorphism G ---> H of Lie groups is a group homo-morphism which is also differentiable. A homomorphism of C into GL(V)is said to be a representation of G in the vector space V.

It is clear that the composite of a homomorphism G1 -> G2 of Lie groupsand another from G2 to G3 is a homomorphism from G1 to G3-

4.5. Definition. A Lie subgroup H of a Lie group G is a Lie group H withan injective homomorphism of H into G.

4.6. Remark. For any g E G, the (right) translation map G -> G given byx H xg is of course differentiable, being the composite of the inclusion G -fG x G given by g H (x, g) and the group operation. Its inverse is translationby g-1. Hence right (and similarly left) translations are diffeomorphisms.

4.7. Definition. Let G be a Lie group and M a differential manifold. Anaction of G on M is a differentiable map G x M -* M denoted (g, m) H gmsuch that gl (g2m) = (9192) (m) for all gl, g2 e G and rn E M and 1.m = mfor all m E M.

In Physics, the role of the Lie group is that of the symmetries of thesystem. Often the most important physical insight turns out to be theintuition for the appropriate group of symmetries.

4.8. Examples.

1) The proper orthogonal group SO(3) acting on 1[83 and the group gen-erated by it and translations (called the Euclidean motion group) arethe group of symmetries in the study of motion of rigid bodies.

2) The group of all linear transformations of R4 which leave the symmetricbilinear form

((xl, x2, x3, x4), (yl, y2, y3, y4)) H -xlyl + x2y2 + x3y3 + x4y4

invariant, is called the homogeneous Lorentz group. If we consider thegroup generated by this group and translations, then it is called theinhomogeneous Lorentz group. Both these groups act on the differen-tial manifold 1184. This is the symmetry group for the theory of specialrelativity.

Exercises 25

In quantum physics, one is often interested in representations of G inthe projective unitary group (namely the group of unitary operators, moduloscalars) of a Hilbert space, but we will not deal with them in this book.

Exercises

1) Which of the following are sheaves ona) For every open set define F(U) to be the space of square integrable

functions on U.b) T(U) is the set of Lebesgue measurable functions on U.c) T(U) consists of continuous functions on U which are restrictions of

continuous functions on I[8n.

2) Show that the stalk at 0 of the sheaf of differentiable functions onRn, n > 1, is an infinite-dimensional vector space over R.

3) Show that the etale space associated to the sheaf of differentiable func-tions on I[8 is not Hausdorff.

4) Show that any section of the sheaf A of continuous functions on a closedset of a normal topological space X can be extended to a section overthe whole of X.

5) Determine for what values of az and c is the intersection of the hyper-plane E azx2 = c with the sphere E x? = 1, a closed submanifold ofRn.

6) Let M be a differential manifold and f a differentiable function onit. Realise the open submanifold of M given by f 0 as a closedsubmanifold of M x R.

7) Consider the map x H x2 of GL(2, IR) (resp. GL(2, C)) into itself andfind its image. Is the image a submanifold?

8) Show that the space of nonzero nilpotent (2, 2) matrices is a closedsubmanifold of the space of nonzero matrices.

9) If G1, G2 are Lie groups, show that the product manifold G1 x G2 withthe direct product structure is also a Lie group.

10) Interpret the Jordan canonical form for matrices, as describing the or-bits under the action of GL(2, C) on the space M(2, C) of all matricesgiven by g.A = gAg-1. Determine which orbits are closed submani-folds.

Chapter 2

Differential Operators

We will now proceed to develop the formal machinery necessary to carrythe notions of differential calculus in the Euclidean space over to arbitrarymanifolds. The first step in this programme is to define differential operatorson manifolds. We deal only with linear differential operators even if we donot say so explicitly each time. We will start with first order operators.

1. First Order Differential Operators

Let M be a differential manifold. We first define homogeneous first orderoperators on the algebra of differentiable functions, taking as characteristicexample, an operator like

Df Of

where coj are some differentiable functions on lR or on an open subset U ofR n. One of the basic properties of such an operator (sometimes called theLeibniz property) is

1.1. D(fg) = (Df)g + f (Dg)

for any two differentiable functions f, g on U. A k-linear homomorphism ofa k-algebra into itself satisfying the Leibniz property, is for this reason calleda derivation. Notice that a consequence of the definition is that D(1) = 0,for D(1) = D(1.1) = D(1).1 + 1.D(1) = 2D(1). It follows that D(A) = 0 forall A E k.

We propose to take the purely algebraic property 1.1 as the definitionof a linear homogeneous differential operator of order 1 on an arbitrary

27

28 2. Differential Operators

manifold. We will now provide the justification for doing so.

Firstly, the algebraic condition implies that the operator D is local inthe following sense.

1.2. Proposition. If D is a linear operator on the Jfk-algebra of differen-tiable functions satisfying the Leibniz property, then the value of D f in anyopen set V depends only on the restriction of f to V.

Proof. In fact, if f = g in an open neighbourhood N of a point x in V,consider a differentiable function cp which is 1 in a smaller neighbourhood ofx and vanishes outside N. Then we clearly have cp(f - g) = 0. ApplyingD and using 1.1 we get (Dcp) (f - g) + cpD(f - g) = 0. In particular, wesee that cpD(f - g) = 0 on N, proving that D(f - g) = D f - Dg = 0 ina neighbourhood of x. Since x is an arbitrary point of V, our assertion isproved.

Consequently, any map A(M) -i A(M) which satisfies the Leibniz prop-erty (in particular, an operator of the form Wi ao) induces a sheaf homo-morphism of A into itself, the homomorphism being one of JR-vector spaces.

1.3. Proposition. Let U be an open submanifold of IR'n. If D : AU --; AUis a sheaf homomorphism of JR-vector spaces, satisfying the Leibniz rule, thenD is an operator of the form f -+ E cpti a for some differentiable functionscpi on U.

Proof. Clearly, if D is to be of the form E cpi, then applying D to thefunctions xi, we see that cpi ought to be Dxi. Replacing D by D-E(Dxi)&we deduce that it is enough to prove the following. If D is a sheaf derivationsuch that Dxi = 0 for all i, then D is the zero homomorphism. Let thena = (al, ... , a,,,) be a point of U and f any differentiable function in aneighbourhood of a. Then f can be written as the sum of the constantfunction f (a) and E(xi - ai)gi in some neighbourhood of a. Let us assumethis for the moment. Now D f = E(xi - ai)Dgi in view of our assumption.But (xi - ai) vanishes at a, implying that (D f) (a) = 0. Since a is any pointin U, it follows that D f = 0, as was to be proved.

It remains to prove our assertion about the decomposition of f. Indeedwe have

if (x) - f (a) = t7-{f (tx + (1 - t)a)}dtf

f axif (tx + (1 - t)a). dt (txi + (1 - t)ai)dt

1 aJ(xi - ai)axZ f (tx + (1 - t)a)dt.f


The above considerations motivate the following definition.

1.4. Definition. A homogeneous first order differential operator on a dif-ferential manifold M is a linear operator on A(M) satisfying the Leibnizrule. Equivalently, it is a sheaf homomorphism A -* A which satisfies theLeibniz rule. A linear differential operator of order at most one on functionsis of the form f H D f + gyp. f where D is a homogeneous operator as above.

From the geometric point of view, a homogeneous first order operator iscalled a vector field or an infinitesimal transformation. We will presentlyexplain this alternate terminology.

2. Locally Free Sheaves and Vector Bundles

The set T(M) of all homogeneous first order operators on M has some nicestructure. Firstly it forms a vector space over IR in an obvious way. In tact,if D1, D2 E T(M) then D1 + D2 is defined by

(DI + D2) (co) = Dlcp + D2W

for all cp E A(M). It is obvious that it is also a derivation. Moreover iff E A(M) then f D defined by

(fD) (co) = fDcois also a derivation, thus making T (M) an A(M)-module. Thirdly, if D1, D2are two such operators, then the operator [Dl, D2] defined by

[D1,D2](co) =

is also one such. For, if cp, 0 E A(M), then we have

[DI, D2](co.b) =Dl((D2W)V) + o(D20)-D2((D1So)' + P(DI'))

= (Dl (D2co))'' + (D2(o) (Dl's)

+(DlW)(D2 ) +cp(Dr(D20))(D2(Dicc)), -(D2(p)(Dlb) - co(D2(Dl,))

= ([DI, D2]co)' + w([DI, D2] )

We will refer to the map (DI, D2) -i [DI, D2] as the bracket operation.Finally, since any D E T (M) may also be described as a sheaf homomor-phism of A into itself satisfying the Leibniz rule, it is clear that if V C U areopen subsets of M, a restriction map T(U) --j T(V) may also be defined,making the assignment to any U of T(U) a sheaf as well. This sheaf will bedenoted T.


2.1. Definition. A Lie algebra over a commutative ring k is a module Vover k with a k-bilinear operation

(X,Y)'-' IX, Y1

which satisfies

i) [X, X] = 0;

ii) (Jacobi's identity) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0,

for all X, Y, Z E V. A homomorphism of a Lie algebra V into another Liealgebra W is a k-linear map f : V --> W which satisfies

f ([X, Y]) _ [f (X) I f (Y)]

for all X, Y E V.

2.2. Examples.

1) Consider the n2-dimensional vector space M(n, k) of (n, n) matriceswith coefficients in a commutative ring k. Define the bilinear map(X, Y) XY - YX to be the bracket operation (using matrix mul-tiplication). Then it is a Lie algebra over k. More generally, any(associative) algebra can be regarded as a Lie algebra by defining[x, y] = xy - yx where on the right side we use the algebra multipli-cation. If V is a vector space of dimension n, the algebra End(V) canbe regarded as a Lie algebra and this is the abstract version of thematrix Lie algebra.

2) We can use various subspaces of End(V) or M(n) which are closedunder the bracket operation and obtain Lie algebras. For example, ifb is a bilinear form on V, we may consider {X E End(V) : b(Xv, w) +b(v, Xw) = 0 for every v, w E V}. It is easy to check that if X and Ysatisfy this condition, XY - YX also does.

3) Any vector space with the bracket operation defined by [v, w] = 0gets a Lie algebra structure. A Lie algebra in which all the bracketoperations are 0 is called an abelian Lie algebra.

4) Let A be any associative algebra over k. Then the space of k-derivationsof A form a Lie algebra over k under the operation [D1.D2] = D1D2 -D2D1i for any two derivations D1, D2.

The relevance of this definition here is of course that the bracket oper-ation we defined above on the space of homogeneous first order operators,endows it with the structure of a Lie algebra over R. Although T(M) is aLie algebra over R and is a module over A(M), it is not a Lie algebra over


A(M) since the bracket operation we defined above is not A(M)-bilinear.In fact, we have

[Di,fD2](co) = (fD2)(Di((p))= D1 f D2cp + f Di (D2co) - f D2(Dico)

= f [D1i D2])(co)

It is obvious that the restriction maps T(U) -+ T(V) are all Lie alge-bra homomorphisms so that the sheaf T is a sheaf of Lie algebras over R.Besides, we now have the following situation. On the one hand, A and Tare sheaves on M and on the other T (U) is an A(U)-module for every opensubset U of M. The module structure is compatible with the restrictionmaps in an obvious sense. We recall the definition of a sheaf of A-modules,already indicated in [Ch. 1, 2.7].

2.3. Definition. i) Let A be a sheaf of rings over a topological space X.Let M be a sheaf of abelian groups over X with a structure of A(U)-moduleon M(U) for every open subset U of X. Then we say that M is a sheaf ofA-modules or that it is an A-module, if

resuv(am) = (resuv a)(resuv(m))

for all a E A(U) and m E M(U) and open sets V C U.ii) An A-homomorphism of an A-module M1 into another A-module

.M2 is a homomorphism f of sheaves. of abelian groups such that the homo-morphisms f(U) : M1(U) -> M2(U) are A(U)-linear, for all open sets Uin X.

2.4. Examples.

1) With this definition, we see that if (M, A) is a differential manifold,then T is an A-module, besides being a sheaf of Lie algebras over R.This sheaf will be called the tangent sheaf.

2) The direct sum A' of A with itself r times (where A is a sheaf of rings)is obviously an A-module.

3) If (M, A) is a differential manifold and Z C M is a closed set, one mayconsider for each open subset U of M, the set of all elements of A(U)which vanish on z fl U. This gives a sheaf ZZ which is clearly anA-module. This is called the ideal sheaf of Z in M.

The local structure of the sheaf T as an A-module is quite simple. Wehave seen that if (U, x) is a coordinate system, the sheaf T j U is actuallyisomorphic to A.n, the map (ai) H ai

z2

providing such an isomorphism.


2.5. Definition. Let A be a sheaf of rings on a topological space X. AnA-module M is said to be locally free of rank r if every x E X has a neigh-bourhood U such that the restriction of M to U is isomorphic to A"I U asan AI U-module.

With this definition, our observation above implies that the tangentsheaf of a differential manifold is locally free as an A-module, where A isthe sheaf of differentiable functions.

2.6. Example. Let N be a closed submanifold of a differential manifold M.We defined in 2.4, Example 3) the sheaf of ideals I by the prescription1(U) _ { f E A(U) : f (x) = 0 for all x c U fl N}. If N has dimension n -1,then this sheaf is a locally free A-module of rank 1. In order to see this,first observe that this being a local statement, we may assume that M isthe unit ball in 118' and that N is the closed submanifold given by xn = 0.If f is a function vanishing on N, then it can be written as xn.g. In fact,we have

10

1f1

f (XI, ... , xn) _ ' f(xl, ... , txn)dt = xn / 9f (xi, ... , txn)dt.

In other words, the ideal of functions vanishing on N is a principal idealgenerated by xn.

Any function f which generates the ideal of functions vanishing on N,can be written as xn.g with g nonvanishing. This implies that at any pointof N, the function - = g + xn a does not vanish at any point of N. Inother words, a function f is a generator of the ideal at 0 if and only if itvanishes on N and at least one of the partial derivatives of f is nonzero at 0.

2.7. Exercise. Show that the R-linear map of the maximal ideal M of)o) induces anfunctions vanishing at 0 into R' taking f to (( )o, ... , (;xL

isomorphism of M/M2 onto R'. Conclude that if r > 1, then the idealsheaf M of 0 is not locally free.

2.8. Definition. Let A be a constant sheaf of rings. Then any locally freesheaf of A-modules is called a local system.

Let (M, A) be a differential manifold. Since sections of A are simplydifferentiable functions on U, it is natural to call sections of the A-moduleA' = = EDT A, systems of functions or vector-valued functions. In the generalcase of a locally free sheaf E, let us discuss whether we can think of sectionsas some sort of functions. Locally it is indeed possible since we have assumedthat £ is locally isomorphic to AT. If x E M, and Mx is the ideal of A.consisting of germs of functions at x which vanish at x, one may consider

2. Locally liee Sheaves and Vector Bundles 33

for any f E Ax its image in AxI.AA,,. The natural evaluation map Ax -> Rtaking any f to f (x), gives an isomorphism of Ax/.M., with R. Therefore onemay think of the evaluation map as taking the image off in Ax/Mx. Guidedby this fact, we may take the 118-vector space £x/.Mx£x to be the space inwhich the sections of £ take values at the point x E M. The (important)difference between the case of A and the general case of a locally free sheaf, isthat the vector space associated to x E M, namely Ex = dependson the point x E M. In other words, there is no natural isomorphism ofthese vector spaces at two different points. Consider the set union E of allthese sets, namely UXEM Ex. This comes with a natural map 7r into M,with 7r-1(x) = E. For any x E M there exists an open neighbourhood Usuch that £ J U is isomorphic to A', so that 7r-1(U) may be identified withU x 1[8' and the map it-1(U) -> U with the projection U x R' --> U. Inparticular, the set 7r-1 (U) can be provided with the differential structure ofthe product. This is obviously independent of the isomorphism £JU -+ A'chosen and hence may be patched together to yield a differential structureon E. It is easy to see that the topology we have introduced is Hausdorff andthat it admits a countable base for open sets. Now we have the followingstructures on E.

a) E is a differential manifold.

b) 7r : E -* M is a differentiable map.

c) For any x E M, there exist an open neighbourhood U of x and adiffeomorphism 7r-1(U) -> U x JR such that it becomes the projectionU x W --* U following this isomorphism.

d) There is a structure of a vector space on each fibre 7r-1(x) for everyx E M, which is compatible with the isomorphism above. This meansthat if we identify 7r-1(x) with 118' using the local isomorphism in c)above, then the identification is a linear isomorphism.

2.9. Definition. A differentiable map 7r : E --; M satisfying a) - d) above,is called a differentiable vector bundle of rank r. A homomorphism of avector bundle E into F is a differentiable map E -* F which makes thediagram

E - F

M

commutative such that the induced maps on fibres are all linear.

The isomorphism of the type described in c) above is referred to as alocal trivialisation of the bundle E.


We have associated, to any locally free sheaf E of A-modules, a differ-entiable vector bundle E -+ M. Conversely, if it : E --- M is a differen-tiable vector bundle, then the sheaf of differentiable sections of it is a locallyfree sheaf of A-modules. Moreover, if F is another locally free sheaf of A-modules with associated vector bundle F, then there is a natural bijectionbetween the set of A-linear sheaf homomorphisms E -* F and the set ofvector bundle homomorphisms E -* F. Thus one may use the notions of`locally free sheaf' and `differentiable vector bundle' interchangeably.

2.10. Remark. One has however to guard against the following possibleconfusion. A subbundle of E is a vector bundle F together with a homomor-phism F -* E which is injective on all fibres. It gives rise to an injectiveA-homomorphism of F into E, namely an A-subsheaf of E. Conversely, theinclusion of an A-module F in E gives rise to a homomorphism F into E,but the latter need not be injective on all fibres.

2.11. Exercise. Consider the inclusion of the ideal sheaf of any point in R

in the sheaf A. Show that the corresponding homomorphism at the vectorbundle level is not injective on all fibres.

If cp : M -+ M' is a differentiable map and E is a differentiable vectorbundle on M', then one can define a differentiable vector bundle cp*E calledthe pull-back of E as follows. Take the subspace of M x E defined by

cp*E = {(m, x) E M x E : cp(m) = -7r(x)l.

It is obvious that it is Hausdorff and admits a countable base for open sets.This subspace comes with two natural maps, namely, ir' : cp*E -i M givenby lr'(m, x) = m and cp : cp*E -+ E given by cp(m, x) = x. If U is an openset over which E is trivial and V = cp-1(U), then 7r'-1(V) can be identifiedwith V x R1 so that cp*E is a differential manifold, with respect to which7r' is differentiable. Clearly the fibre 7ri-1(m) over m E M can be identifiedwith the fibre 1r-1(cp(m)) and hence has a natural vector space structure. Itis now easy to verify that ir' : cp*E -* M satisfies the conditions a) - d) andhence cp*E is a vector bundle.

2.12. Definition. The vector bundle cp*E defined above is called the pull-back of the vector bundle E by the map cp : M ---> M. For every differen-tiable section s of E one can define a section of cp*s of cp*E again called thepull-back of s in such a way that

c (c *s(m)) = s((p(m))forallmEM.

Suppose E is a locally free sheaf of A-modules on M' and cp is a dif-ferentiable map M --p M'. Then the inverse image cp-1AM' as defined in


Chapter 1, 2.3 is a sheaf of algebras over M. The inverse image co ' (£) is nota sheaf of AM-modules but only a locally free sheaf of cp- (AMA)-modules.The companion map of cp, namely the map cp : cp-'(Am') --; AM may beused to define cp*£ as cp-1£ (&

W_J(A11') AM, where AM is considered as asheaf of cp-'AMA-modules via cp. With this definition it is easy to see thatthe vector bundle associated to cp*£ can be canonically identified with thepull-back cp*E of the vector bundle E associated to £.

2.13. Remarks.

1) All the constructions which we have made above are also valid for locallyfree sheaves of AM ® (C-modules. The corresponding vector bundlesare bundles of vector spaces over C. If E is a complex vector bundle,one can define r by setting E = E as a differential manifold butchanging the vector space structure on the fibres of E by redefiningmultiplication by i = as multiplication by -i in E. Given anyreal vector bundle E one can associate to it a complex vector bundleEC by tensoring with G. Of course in such a case, we have a canonicalisomorphism of EC with E.

2) We refer to complex vector bundles of rank 1 as line bundles. Realvector bundles of rank 1 are not that interesting. In fact, if L is onesuch, then consider the relation given by declaring two points v, v' ofL to be equivalent if there exists a positive real number a such thatv' = av. Then the quotient space maps onto M and has precisely twopoints on each fibre. In view of the local triviality of the line bundle,the quotient is easily seen to be a two-sheeted (etale) covering spaceM' of M. If M is simply connected, this covering consists therefore oftwo copies of M. Choosing one of them is equivalent to choosing forall m E M, one of the components of Lm, \ {0}, where Lm, is the fibreof L at in. We may call elements of that component positive vectors.Local trivialisation of L is equivalent to the data consisting of anopen covering {Ui} and sections si on each Uj which are everywherenonzero. We may in the above situation, actually choose positivesections s,, i.e. sections whose values are in the chosen component inthe fibre.

We may then take a partition of unity {cpj} for the covering andget a global everywhere nonzero section E cpisi of L. This shows thatL is globally trivial. This argument actually shows that even if Mwere not simply connected, we can pull back L to the two-sheetedcovering M' and trivialise it on M.

3) If M is a closed submanifold of Rn of dimension n - 1, then we have,remarked already in (2.6) that its ideal sheaf is a locally free A-module


of rank 1. By 2) this sheaf is globally trivial. Therefore there isa section which generates the local ideal everywhere, that is to say,there is a function f which vanishes precisely on M and generates theideal sheaf locally at all points.

Let us get back to our sheaf T, the tangent sheaf of M. Applying theabove considerations to it, we get a vector bundle, which is called the tangentbundle T. For any x E M, the fibre over x is called the tangent space atx. Elements of this space are called tangent vectors at x. A differentiablesection of this vector bundle may therefore be called a vector field and thusthe concept of a homogeneous differential operator of first order and that ofa vector field are essentially equivalent.

The tangent space at a point x E M is thus the quotient space Tx/Mx2.If X is a germ of a vector field at x then the map f i-, X f induces aderivation of the R-algebra Ax. The map f H (Xf)(x) gives rise to a mapt : Ax -* JR which is a derivation in the sense that

t(fg) = (tf)g(x) + f(x)t(g)

for all f, g E Ax. By this correspondence, one easily verifies that the tangentspace at x can be identified with the set of linear maps Ax -j JR satisfyingthe above condition.

In the case when M = IR1, or more abstractly a vector space V ofdimension n, the tangent bundle is trivial. The tangent space at any pointv E V can be identified with the vector space itself. In fact, we associate toany x E V the derivation a, :,4,, -+ R given by f (x) --+ limt_,o f (x+tv)-f (x)

tIf N is a submanifold of V, then the tangent bundle of N is a subbundle ofthe trivial bundle N x V. Thus the tangent space at any point P E N isa subspace of V. We visualise this geometrically as the coset space of thissubspace in V which contains P. In other words, the geometric tangent spaceis then the space parallel to the abstract tangent space, passing through P.

2.14. Exercise. If a submanifold N of lR'z is given by f = 0 where f isa differentiable function with the property that at least one of the partial


derivatives of f is nonzero at every point of N, write down the equation ofthe tangent space at a point n E N.

2.15. Differential of a map.If cp : M --> M' is a differentiable map of a manifold M into M' and t

is a tangent vector at m E M, then one can define a tangent vector W(t)at m' = cp(m), by setting cp(t)(f) = t(f o cp) for all f E A,,,,,. This definesa linear map T, (M) -> T,, (M') called the differential of W. The tangentspace at m is to be thought of as the linear approximation of M near mand the differential of cp as the linear approximation of the map cp. Globallyspeaking, the differential gives a vector bundle homomorphism of TM intocp* (TM,) , or what is the same, a sheaf homomorphism of TM into cp* (TM,) .

This homomorphism is usually denoted by dcp. Occasionally, we may writecp for this differential as well.

If (U, x) (resp. (V, y)) is a coordinate system in M (resp. M'), such thatcp(x) = y and cp(U) C V, then cp is given by functions cpy(x1, ... , x,,). Thenits differential takes to the vector at W(x) which takes the coordinatefunctions yj to axk (cpj (x)). Hence we deduce that the differential of cp maps

to ask (cps (x)) yj . In other words, the matrix of the linear mapT --> Ta(x) with respect to the standard bases, is given by AZj _ (cps ).

This matrix is called the Jacobian of the map W.If f is a differentiable function, namely a differentiable map M --> R,

then df is thus a homomorphism TM --> f*(TR). But TT is a trivial bundlesince the vector field a forms a basis for the tangent spaces at all pointsof R. Hence df may be regarded as a homomorphism of TM into the trivialbundle of rank 1, or what is the same, a section of the dual vector bundle.This may simply be called the differential of f.

2.16. Normal bundle.If f : N --+ M is a differentiable map such that the differential of f is

injective at all points of N, then the differential of f, namely the homomor-phism TN -3 f *TM, is a subbundle inclusion. The cokernel of this is calledthe normal bundle of N in M. Thus we have the exact sequence of vectorbundles:

0 -+ TN - f*(TM) -+Nor(N,M) - 0.

Under this assumption, the implicit function theorem assures us of theexistence of a local coordinate system (U, x) at a point n E N and a localcoordinate system (V, y) at f (n) such that f IU is injective and f (U) C Vis given by the vanishing of some of the coordinates, say yl, ... , y,.. Thetangent space to N is freely generated by aya1 , ... , TY49T.


If in addition f is injective, it follows that N is a manifold immersed inM. In that case, the notion of normal bundle is more geometric. Supposenow that N is a closed submanifold of R. Then the tangent space atP E N is a subspace of Jn, and the normal space is a quotient. But usingthe Euclidean metric on IEBn, we may identify this quotient with a subspace aswell. We may visualise it as the coset of this latter space, passing through P.

2.17. Exercises.1) Let N be a submanifold of R' given by f = 0 as in 2.14. Write down

the equation of the normal space at any point n E N.

2) If M is a closed submanifold of JRn of dimension n - 1, show that itsnormal bundle is trivial.

On the other hand, the differential of f may even be an isomorphism atall points, without the map f being injective. .

2.18. Definition. If the differential of f : N -+ M is an isomorphism at allpoints of N, we call it an etale map.

2.19. Example. The map t -, exp(it) of JR -* S' is an etale map.

If an etale map is also injective, it follows from the definition that it isan inclusion of an open submanifold.

3. Flow of a Vector Field

A geometric way of looking upon differentiation in RI is the following. Sup-pose given a vector a = (al, ... , an). For any t E R, consider the transfor-mation Cot : (x,, ... , x,,,) ' (x, + tat, ... , xn + tan) of RI into itself. This isa homomorphism of the additive group R into the group of diffeomorphismsof Rn. For any differentiable function f on R', define

(Dd(f))(x) = t o f(ct(x)) - f(x)t


This exists and in fact defines the differential operator > az axi . We maygeneralise this idea to an arbitrary differential manifold. Let (cot), t E II8, bea one-parameter group of diffeomorphisms of M in the sense that

i) the map IE8 x M -* M given by (t, x) --+ cot(x) is differentiable.

ii) the map coo : M -- M is the identity.iii) cot o cot' = cot+t' for all t, t' E R.

Then we may define differentiation of functions f with respect to theabove data by setting

(Xwf)(x) = lof(cot(x)) - f(x)

tIt is easy to see that Xw is indeed a homogeneous first order operator (i.e. avector field). In fact, the linearity of X. is obvious, while we have for everyf, g c A(M),

(Xw(fg))(x) = limo f (cot(x))g(wt(x))-g(x)

+ limt-,o g(x) (Wt (At -f x

= f (x)(Xwg)(x) +g(x)(Xwf)(x)-

3.1. Examples.

1) Take M=Randcpt(x)=x+tforallt,xER.2) Take M = R and cot(x) = etx, for all t, x E R.

In these two cases, we see that the associated operators are respec-tively dx and x x .

3) Again take M = R and consider the function cot(x) = itx. It is easy tosee formally that cot o cot' = cot+ti But, for any given x 0, (pt (x) isonly defined for t < 1/x. We are dealing here with a local 1-parametergroup of local automorphisms. In other words, for any x there exist aneighbourhood U and e > 0, such that Sot(y) is defined for all I ti < eand y E U. The group condition iii) above is satisfied to the extentit makes sense. But notice that the associated vector field is stillmeaningful.

3.2. Exercises.1) Compute the vector field given in Example 3) above.

2) Determine the vector field given by the one-parameter group (cot) whoseaction on 1[82 is given by

cot(v, w) = (cos(t)v + sin(t)w, - sin(t)v + cos(t)w).

With this extended notion we have a converse.


3.3. Theorem. Let X be a vector field on a differential manifold M. Thenfor every x E M, there exist an open neighbourhood U of x, c > 0, and mapscpt : U -+ M for all Itl < E, satisfying:

i) the map (-e, E) x U -+ M given by (t, y) --> Wt (y) is differentiable.

ii) the map cpo : U -* M is the inclusion.

iii) (cpto t,) (y) = cot+t, (y) for all ItI, I t'J < e and y E M such that lt+t'l < Eand y, cot' (y) are in U.

iv) X.

Proof. Since X,, and X are both vector fields, the assertion is purely localand so we may replace M by an open set in R' and prove the existence ofmaps cpt such that XWxi = Xxi for all i. This reduces our task to showingthe following. Given differentiable functions ai, we need to find functions

cp(t, x) : (-E, E) X U - 1R

for some neighbourhood U of a given point such that

tcpz(t,) - x2 = ai(x), for all i.lim

a (Pi (t, x) = lim cpi (t' + t, x) - cpi (t, x)t'-+O t'

coi(t', V(t, x)) - Vi (t, x)t'-o t'ai(co(t, x)).

We also have the initial condition cpi (0, x) = xi. So we start with this equa-tion and note that it has a unique solution in a neighbourhood of (0, x) inIl8 x IR". To prove that iii) is satisfied, we use the uniqueness of the solu-tion. In fact, both cp(t+t', x) and cp(t, cp(t', x)) are solutions of the equation

at x) = ai (0 (t, x)) with the initial condition 4'i (0, x) = cpi (t', x). Finallyequation iv) is obvious from the construction.

From this point of view, the term `infinitesimal transformation' is anappropriate alternative to that of a `vector field'.

3.4. Definition. The one-parameter group associated to a vector field iscalled the flow of the vector field.


z

y -x

Ot(X)

3.5. Remarks.

1) Although a vector field gives rise in general only to a local 1-parametergroup, a limited globalization is possible. Indeed, given a compact setK C M, we can define Wt in a neighbourhood of K for all Itl < e. For,by Theorem 3.3 this can be done in a neighbourhood of every point ofK. Since K can be covered by finitely many of these neighbourhoods,cpt(y) are defined in the same open set Its < e for small enough e, andfor y in a neighbourhood of K. In particular, if M is itself compact,then cot is defined as an automorphism of M for all Itl < e, and henceby iteration, we get in this case, a global flow.

2) It is obvious that if X depends differentiably on some parameters s, thenthe one-parameter group is defined for small values of the parametersand depends differentiably on them.

3.6. Definition. A vector field which gives rise to a global flow is said tobe complete.

We have seen above that any vector field on a compact manifold iscomplete. It is easily seen that the vector field x2 d.- on R is not complete.

3.7. Exercise. Let M be a compact manifold and X a vector field. Ifm E M, determine when the restriction of X to the open submanifoldM \ {m} is complete.

3.8. Definition. If X is a vector field and (cot) the flow corresponding toit, the orbit of a point m E M under cot, namely the map t H cot(m), iscalled an integral curve for X.

3.9. Remark. The integral curve of a vector field X has the property thatthe differential of this map at t maps A to Xt(,,,t). This characterises thecurve. In particular, the curve degenerates to a constant map if and only ifXm, = 0. If X is 0 at a point m, we say that m is a singularity of X. Our


remark amounts to saying that the one-parameter group cot fixes a point mif and only if m is a singularity of X.

Suppose that m is not a singularity. Then by continuity, we see thatXx 0 0 for all x in a neighbourhood of m. Now the integral curve hasinjective differential at all points near 0 and hence it is an immersed manifoldof dimension 1. Actually there is a local coordinate system (U, x) in whichX is given by al . Indeed, suppose X = ai axti , where by our assumptionone of the ai's, say al, is nonzero. Consider the coordinate system given by(yi,... , y,,,) where yi = coi(xi, 0, x2, ... , xn). We now compute the partialderivatives a at m given by xi = 0 for all is

axi = ai(0);yi

= 6 for j > 2.axe

This shows that (yi, ... , yn) is a coordinate system in a neighbourhood ofm. It is easy to see that this coordinate system serves the purpose.

Invariant vector fields.We wish to study now vector fields on a Lie group.

3.10. Definition. Suppose M is differential manifold and a Lie group Gacts on it. Then a vector field X on M is said to be invariant under theaction if the transform of X by any element of G is the same as X, thatis to say, for every m E M and g E G, Xg,,.,, is the image of the vector X,,,,under the differential at m of the map x H gx of M into itself.

From the uniqueness of the flow corresponding to a vector field, wededuce that if the flow of X is cot, then the flow corresponding to gX isgiven by't(m) = gcot(g-im). Therefore, if X is invariant under G, thenthe flows t and cot are the same, so that cot commutes with the action of Gfor all t.

Notice also that from the definition of Lie brackets of vector fields itfollows that if X and Y are G-invariant, then [X, Y] is also invariant. Inparticular, the vector space of invariant vector fields is actually a Lie algebra.We have already remarked that G acts on itself by left translations and soleft invariant vector fields of a Lie group form a Lie algebra over JR.

3.11. Definition. Let G be a connected Lie group. The Lie algebra ofvector f ields which are left invariant (i.e. invariant under left translation byelements of G) is called the Lie algebra of G, and is often denoted by Lie(G)or B.


3.12. Remark. There is a natural linear map of g into the tangent spaceT1(G) at 1, given by X -> X1. It is an isomorphism and the inverse as-sociates to any vector v E T1(G), the vector field X given by X9 = Lg (v),where L9 denotes the differential at 1 of the left translation by g. So g is aLie algebra of dimension n = dim(G).

3.13. Examples.

1) The left invariant vector fields on the additive Lie group R are of theform a dt , a E R. Hence the Lie algebra of R is canonically isomorphicto the abelian Lie algebra R. In the same way, if we take the additiveLie group underlying a vector space V, then its Lie algebra is identifiedwith the abelian Lie algebra V.

2) On the other hand, if we take the multiplicative group R' or its con-nected component IIg+ containing 1, then the invariant vector fieldsare scalar multiples of ti. Thus its Lie algebra is also the abelian Liealgebra R.

3) The Lie algebra of the group GL(n, R) or the connected componentGL(n,1 )+ containing 1, can be identified with M(n,R). It is clearthat the tangent space at 1 of this open submanifold is canonicallyM(n) as a vector space. It only remains to compute the Lie algebrastructure. Denote by xij the function which associates to any matrixA its (i, j)th coefficient. Let A E M(n) and X be the left invariant vec-tor field on GL(n) such that X1(xij) = Aij. For any s E GL(n,R)+,we have X9(xij) = Xl(xij o Ls), where LS is left translation by s.Hence the function Xxij is given by s H >k xik(S)Xlxkj = > xikAkjIf Y is any left invariant vector field with Yl (xij) = Bij, then wehave (XY - YX)1(xij) = Xi(Yxij) - Y1(Xxij) = X1(>xikBkj) -Y1 (E xikAkj) = E AikBkj - E BikAkj = (AB - BA)ij. In otherwords, if we identify left invariant vector fields on GL(n) with M(n),then the Lie bracket is given by the bracket associated with the mul-tiplication in the matrix algebra.

3.14. Exercises.1) Show that the left invariant vector fields on GL(n) are generated by

Ep,q = xi,p and that the right invariant vector fields by Fp,q =i'q

xq,j

2) Deduce that Ep,q and Fr,9 commute. Explain this in terms of theirflows.

The integral curve through 1 of a left invariant vector field, namelyt -- cpt(1), is actually a homomorphism of the group R into G. In fact,


left invariance of X implies that for every g E G, the automorphism x -4gcpt(x) is the same as x cpt(gx). Taking g = cpt'(1), we get cpt,(1)cpt(1) _

cot(cOt'(1)) = cOt+t'(1)

3.15. Definition. A differentiable group homomorphism of the Lie groupJR into a Lie group G is called a one-parameter group .

If p : I[8 --> G is a 1-parameter group, then p(at) gives a vector at 1which in turn defines a left invariant vector field X. It is clear that p givesan integral curve for X.

3.16. Remarks.

1) If the vector field X is 0, the corresponding one-parameter group isthe constant homomorphism t -> 1. Even if X is not 0, the mapt cpt(1) mentioned above, may not be injective. If we take G =Sl = {(x, y) E 1182 : x2 + y2 = 1}, then the left invariant vector fieldsform a 1-dimensional vector space generated by X = x y - ya . Theimage of at under the map t (cos at, sin at) is easily computed tobe aX. Hence it is the one-parameter group of the vector field aX.Its kernel is the subgroup 2a Z of R.

2) Also the image of a one-parameter group is not in general closed. Forexample, consider the case when G = S1 x S' and X = (at , at).Then the induced one-parameter group is given by

t H ((cost, sin t), (cos at, sin at) ).

This is a closed submanifold if and only if a is rational. See the figurein [Ch. 1, 3.17].

3) If g E G, then (the differential of) the inner automorphism x H gxg-1of G takes the vector field X to another left invariant vector field whichwe may denote gXg-1. If t --> c(t) is the flow of X, then the flow ofgXg-1 is given by t H gc(t)g-1.

3.17. Definition. The representation of a Lie group G into GL(g) whichassociates to g E G the automorphism Ad(g) = X gXg-1 is called theadjoint representation of G.

3.18. Remark. The linear automorphism Ad(g) is actually an automor-phism of the Lie algebra g.


3.19. Exercise. Show that if the image of the one-parameter group isclosed, then it it is actually a closed submanifold.

If G and H are Lie groups and T : G -* H is a homomorphism of Liegroups, then the differential at 1 is a linear map Tl (G) -p Tl (H). As suchit gives a linear map t : g -* C) as well, using the isomorphisms Tl (G) -i gand Tl (H) -+ C .

3.20. Proposition. Let X, Y be left invariant vector fields of a Lie groupG. If T : G -i H is a homomorphism of Lie groups, then we have

t[X, Y] = [t(X),t(Y)]

Proof. Let f be a differentiable function in a neighbourhood of 1 in H.Then for any Z E and any x E G, we have (Z f) (Tx) = Zl (LTx f ), since Zis an invariant vector field. Let us take Z = tX, that is to say, Z is the leftinvariant vector field whose value in Tl (H) is the image of Xl E Tl (G) by thedifferential at 1 of T. Then Zi (LTxf) = X1(LTxf o T) = Xl (L. (f o T)) =Xx (f o T) since X is left invariant. In other words, (tX) (f) o T = X (f o T).Hence if Y E g, we may replace f by (tY) f in this equation and obtain(tX)(tY)(f) o T = X((tY) f o T) = XY(f o T). Interchanging X and Yand subtracting, we get [tX, tY] (f) o T = [X, Y] (f o T). Evaluating at 1,we see that the image of [X, Y] under the differential of T at 1 is actually[t(X), t(Y)]. This proves our assertion.

We have remarked that if a vector field depends differentiably on someparameters, then its flow also depends differentiably on the parameters.From this it also follows that the one-parameter group associated to a leftinvariant vector field depends differentiably on it, meaning that there existsa differentiable map p : Tl (G) x ll8 -* G such that for any X E Tl (G) the mapt --> p(X, t) is the corresponding one-parameter group. The restriction of pto Tl (G) x { 11 is a differentiable map g --* G. This is called the exponentialmapping. Let v E T1(G). Then we will evaluate at v, the differential at 0of the exponential. We might as well restrict p to ll8v x R first in order tocompute this differential. In other words, we consider the map (sv, t) -4c(sv, t). But this is the same as c(v, st). Setting t = 1, we need to computethe differential of s c(v, s) at s = 0. By definition it is v. In other words,we have shown

3.21. Proposition. The differential at 0 of the exponential map from gto G is the identity map. In particular, the exponential map gives a dif-feomorphism of a neighbourhood of 0 in g onto a neighbourhood of 1 in G.


It follows from the uniqueness assertion regarding one-parameter groups,that if T : G -> H is a differentiable homomorphism of connected Lie groups,the induced map t commutes with the exponential map in the sense thatexpHot=ToexpG.

If T, T' : G -f H are two homomorphisms of connected Lie groups suchthat the induced homomorphisms t, f are the same, then T = V. Forfrom Proposition 3.21, our assumption implies that T and T' coincide in aneighbourhood of 1 in G. But if G is connected, any neighbourhood U of 1generates G as a group. Hence T, T' coincide on the whole of G.

Moreover, if a homomorphism T : G -* H of connected Lie groups isinjective, then the induced homomorphism t is also injective. For if X is inthe kernel of t, then the image of the one-parameter group t H T(exp(tX))has tangent 0 at 1 and should be the constant homomorphism. By ourassumption, the map t' -+ exp(tX) is also constant, and hence X = 0.

Conversely, if t is injective, the kernel of T cannot intersect the expo-nential neighbourhood, and is therefore a discrete normal subgroup of G.

Finally, if t is an isomorphism of Lie algebras, then T has discrete kernelN and goes down to a homomorphism of the Lie group G/N into H. Thisinduces an isomorphism at the Lie algebra level. Since the differential of Tat 1 and hence at any other point is an isomorphism, it is a diffeomorphismonto an open subgroup of H. Since H is connected, the image coincideswith H. In other words, T : G -* H is an isomorphism.

3.22. Remark. We have associated to every Lie group, a Lie algebra andto every Lie group homomorphism, a Lie algebra homomorphism of thecorresponding Lie algebras. This correspondence helps one to understandan analytical object such as a Lie group, by a purely algebraic object, namelyits Lie algebra.

4. Theorem of Frobenius

Suppose M is a differential manifold and we are given a subbundle E of TM.We seek to find conditions under which one can assert that at every m E M,there exists a local coordinate system (U, x) such that '9

a, ... , , k =rk(E) generate the subbundle E at all points of U. Note that this impliesthat all sections of ,6 over U are of the form f; . Hence if we considertwo sections of E as vector fields and take their bracket, the resulting vectorfield is also a section of E. The theorem of Frobenius asserts that thiscondition is also sufficient.


4.1. Definition. A subbundle E of the tangent bundle of a differentialmanifold M is said to be integrable if for any two differentiable sections Xand Y of E, the bracket [X, Y] is also a section of E.

4.2. Theorem. If E is an integrable subbundle of TM, then at every pointof M, there exists a local coordinate system (U, x) such that EAU is generated(freely) by sections ... , , k = rk(E).

Proof. We will prove this by induction on the rank of E. We have seenin (3.9) that the theorem is true for k = 1. Let m E M. Notice that ifX1, ... , Xk are vector fields which freely generate E in a coordinate neigh-bourhood at m, then all of these vector fields are nonsingular. In particular,Xk is of the form aal in a suitable coordinate system (U, x). Writing out theother Xi's in terms of the basis (a ii ), we see that by subtracting from eachof Xi, i < k, a multiple of Xk, one may assume that they are all of the formEi>2 fi /- Direct expansion of [Xi, Xj], i < j < k, shows that the subbun-dle of E generated by Xi, i < k is also integrable. Our induction assumptionthen implies that there is a coordinate system (V, y) at m such that Xi =

DTi

for all i < k. In this coordinate system, suppose Xk E fib. Again bysubtracting from Xk the linear combination >i<k fig of Xi, i < k, wemay assume that Xk = Ei>k fi Moreover, since Xk is nonsingular, atleast one of the fi's, say fk, is invertible. We may replace Xk by Xk/fk.Then we have Xi = for all i < k and Xk = + >i>k gi . Now

[X5, Xk] = i>k on the one hand, and is a linear combination ofy

Xi, i < k on the other. This implies that a are all zero. In other words,gi are independent of yj, j < k. Let us now take the coordinate neigh-bourhood in the form V1 x V2 with V1 (resp. V2) a domain in Rk-1 (resp.

Ian-k+1) with coordinates yl,... , Yk-1 (resp. yi, i >_ k). Since Xk can beregarded as a nonsingular vector field in V2, we can find a coordinate sys-tem z1i ... , zn_k+1 in which Xk has the expression -. Now it is clear that(yi,1 < i < k - 1, zj, 1 < j < n - k + 1) is a coordinate system with therequired property.

4.3. Definition. An immersed manifold cp : N --+ M is said to be integralfor a subbundle E if the differential of cp at any point p E N maps thetangent space Tp (N) isomorphically onto the fibre

In view of Theorem 4.2, there do exist integral submanifolds for anintegrable subbundle. In fact, take a coordinate system (U, x) as in Theorem4.2 (where for convenience of notation we will assume that U is an open cubein RI in the coordinate system). Then the closed submanifolds S(a) = {(x) :


xi = ai for all i > k}, are obviously all integral manifolds. An integralmanifold obtained in this way is called a slice.

Suppose co : V -+ U is any connected immersed manifold which is inte-gral for E. Then the functions xi o cp satisfy a = 0 for all i > k and j < k.By our assumption, this implies that vcpi = 0 for all i > k and all tangentvectors v at any point of V. Hence cpi is a constant on V. The manifold Vis therefore contained in the slice S(a), and indeed as an open submanifold.

4.4. Remark. If co : M -3 N is an integral manifold, then for every p E N,there exists a neighbourhood U such that W(m) f1 U is a countable unionof closed submanifolds of U. We may of course also assume that these areconnected components of W(M) fl U.

4.5. Exercise. Explain in the above light which of the examples of im-mersed submanifolds given in [Ch. 1, 3.17] are integral curves and whichnot, for a suitable line subbundle of the tangent bundle.

We will now show that immersed integral manifolds do not admit pathol-ogies of the kind we pointed out in [Ch. 1, 3.17].

4.6. Proposition. If go : N -> M is any integral manifold and L is anyother manifold with a map f : L --3 N, then f is differentiable if and only ifcp o f is.

Proof. If f is differentiable, the composite coo f is of course differentiable. Inproving the converse, the key point is that the differentiability of coo f impliesthe continuity of f . Let I E L and (U, x) be an open cube containing coo f (l)as in Theorem 4.2. The open submanifold (g)-1(U) of N is a countableunion of open connected manifolds, each of which is an open submanifold ofa slice. The map co o f maps a connected open neighbourhood W of l intothe union of these slices. But the image in 1[872-k is countable and connectedand hence consists of a single point. In other words, the map co o f maps Winto a locally closed submanifold of M. Hence it is differentiable as a mapinto the submanifold as well.

4.7. Corollary. If cp : N --} M and co' : N' --+ M are two connected integralmanifolds for E with the same image, then they are diffeomorphic.

Proof. Indeed the natural maps N --> N' and N' -+ N are both differen-tiable, by Proposition 4.6.

We can now take the set of all (connected) integral manifolds for a givenintegrable subbundle E (identifying them with their images) and partiallyorder them by inclusion. Clearly it is an inductive family and therefore thereexists a maximal element. These are called maximal integral manifolds.


We will now give an application of the Frobenius theorem to Lie groups.

4.8. Corollary. Let G be a connected Lie group and rj a Lie subalgebra ofthe Lie algebra g of G. Then there is a Lie subgroup H of G with Cl as itsLie algebra.

Proof. Consider the subbundle E which is left G invariant such that El =the subspace Cl of T1 = g. Its sections are generated over A by a basis(Xi) of the vector space lj. In other words, all sections are of the formE fXi. If X = E fiXi and Y = E giXi, then [X, Y] = E fi(Xigj)X3 -7 gi (Xi fj)Xj +E figj [Xi, Xj] is again a section of E. Hence E is integrable.Let H be a maximal integral submanifold for E, containing 1. For anyh E H, the left translation by h-1 of H gives another maximal integralsubmanifold for the same integrable subbundle, since E is invariant underleft translations. But since h E H, this translate contains 1 and so coincideswith H. Hence h-1 E H and H is closed under multiplication. In orderto show that H is a Lie group, we have to verify for example that the mapH x H -* H given by (h1i h2) e h1h21 is differentiable. By Proposition4.6, it is enough to check that this map, considered as one from H x H intoG, is differentiable. But this latter map is the composite of the inclusion ofH x H in G x G and the corresponding group multiplication map of G x Ginto G. This completes the proof.

4.9. Remarks.1) Even in the case of a Lie subgroup H the topology of H may not

coincide with that of the image, as the illustration in [Ch. 1, 3.17]shows. However we have the following comforting situation in thecase of Lie groups. Since H induces injection on the Lie algebra, it isan integral submanifold of G for the left invariant subbundle of thetangent bundle of G defined by C}. Hence the subgroup H can beprovided with the structure of an immersed manifold.

2) A connected Lie subgroup is a locally closed manifold only if it is actu-ally a closed submanifold. For its closure is a connected subgroup inwhich it is open. But an open subgroup is necessarily closed. In thiscase, let p be a subspace of g, supplementary to Cl. The exponentialimage of this space in the exponential neighbourhood intersects Honly at 1. The coset space G/H is a Hausdorff topological space witha countable base for open sets. Besides there is a neighbourhood ofthe trivial coset which can be provided with a differentiable structure,via the exponential map. From this we easily conclude that G/H isa differential manifold and that the natural map G --+ G/H admits adifferentiable section.


3) One can show that if H is a closed subgroup of G then it is automaticallya Lie subgroup, and therefore the above considerations do apply.

4) It can also be proved that an arcwise connected subgroup of a Lie groupis a Lie subgroup.

Suppose that H and G are connected Lie groups with lj, g as their Liealgebras. Given any Lie algebra homomorphism t of Cj into g, consider thegraph of t, namely the Lie subalgebra of C x g defined by the set of elements(X, tX), X E rj. Let H be the corresponding connected Lie subgroup ofH x G. The projection homomorphism of H into H induces an isomorphismof Lie algebras. Hence there exists a discrete normal subgroup N such thatH/N is mapped isomorphically on H. Thus although the map t may notcome from a homomorphism of H into G, there is an etale covering of Hfrom which there is a homomorphism giving rise to t.

4.10. Remark. We have carried out the correspondence between Lie groupsand Lie algebras except for one particular. It is also true that every Liealgebra is actually the Lie algebra of a Lie group. This would follow forexample, if we can show that there is an injective homomorphism of the Liealgebra into gl(n, C), in view of Corollary 4.8. This latter assertion is knownas Ado's theorem, and we do not prove nor use it in this book.

5. Tensor Fields; Lie Derivative

The usual algebraic operations that one performs on vector spaces may alsobe done on vector bundles. Thus if E and F are differentiable vector bundles,then their tensor product E ® F may be defined as follows. Consider the setE®F = UZEM Ex®Fx. It can be made into a differential manifold by notingthat when x varies on a small enough open set U around any point, this unioncan be identified with U x (Rk (9 RI) by using local trivialisations of E andF. Identifying R ®lR with Rkl, we may introduce a differentiable structureon UXEU E® ® F. It is easy to see that these structures do not dependon the particular local trivialisations one uses, and consequently glue up toprovide a differential structure on E ® F. It is clear how to define a vectorbundle structure on this space. The dual E* as well as the tensor, exteriorand symmetric powers of a vector bundle E may also be defined in a similarway. If £ and.F are the locally free sheaves of A-modules corresponding to Eand F, then the tensor product E ® F is the vector bundle associated to thepresheaf which assigns to any open set U the A(U)-module £(U)®A(U).F(U).Its stalk at any point x E M is easily seen to be £., ®A. .F. .

On a differential manifold M, many interesting geometric objects aredescribed by sections of vector bundles, usually the tensor powers of the


tangent bundle and its dual, the cotangent bundle. We will only deal withdifferentiable sections even if we do not say so explicitly each time.

Sections of the cotangent bundle are called differential forms of degree 1and sections of its pth exterior power AP(T*) are called differential forms ofdegree p. Note that there are no nonzero differential forms of degree greaterthan the dimension of M. A differential form of degree p may also beregarded as an alternating A(M)-multilinear form of degree p on the spaceof vector fields (i.e. takes the value 0, whenever two of the argument vectorfields are equal). With this interpretation, one can see that the exteriorproduct of two differential forms a, /3 of degree p and q is given by

5.1.

(a A/ 3)(X1,...,Xp+q)

E'a(X'(1), ... , XQ(p))/3(XX(p+1), ... , Xv(p+q)),

where a runs through the so-called `shuffle' permutations and EQ is the sig-nature of a. (A shuffle is a permutation which preserves the relative ordersin the two subsets, that is to say, it is a permutation of {1, ... , k + l} suchthat o-(1) < < u(p) and a(p + 1) < < v(p + q).)

In order to develop differential calculus on a differential manifold M,we first need the notion of differentiation of a tensor field. We then seekto define the derivative of a tensor field, namely a section of ®r T ®s T*,with respect to a vector field X. The most natural definition that one mightgive, following the geometric definition of a vector field as an infinitesimaltransformation given above, is the following.

5.2. Definition. If w is a tensor field, then the Lie derivative L(X)(w) ofw with respect to X is

lim(cotww)

t-.o twhere cot is the flow determined by X.

It is not difficult to show that the above limit exists if (as we assume) thevector field X and the tensor field in question are differentiable. Note thatthe automorphisms cot give also isomorphisms cot between tensors at anypoint m E M and those at cot (m). With this understanding, the above def-inition amounts to saying that the value of the tensor field L(X) (w) at anypoint m E M is the tensor l1mtyo is the obvious general-isation of the action of X on functions, that is to say, we have L(X) f = X ffor functions f.

Imitating the proof for the product rule for differentiation, one can showthe following.

MaX Planck- Institut


5.3. Proposition. Assume given an A-bilinear sheaf homomorphism(w, w') B(w, w') which associates to any two tensor fields w, w' of fixedtypes, a third one. Assume that for any diffeomorphism cp of an open sub-manifold U with U', we have B(cp*w, cp*w') = cp*B(w, w'). Then for anyvector field X on M, the following identity holds:

L(X)B(w, w') = B(L(X)w, w') + B(w, L(X)w').

Proof. In fact, by definition, (L(X)B(w,w'))P is

= limcptB(w, w'),Pt(P) - B(w, w')P

t-.o tat(P), cot w

limot(P)) - B(wP, wp)

=t

B (cpt wit (P) - wP cot wit (P)) + B (wP, cPt wlim it (P) - wp)

t- o t

limB(to tWwt(t) +B(wp, mo tW (t)

_W )B(L(X)w, w')P + B(w, L(X)w')P.

5.4. Corollary. i) If w, w' are tensor fields, then

L(X)(w ® w') = L(X)w ® w' + w (D L(X)w'.

ii) If a, ,Q are differential forms, then

L(X)(aA13) = L(X)a A +aAL(X),Q.

iii) If w is a differential form of degree 1 and X, Y are vector fields, then

X(w(Y)) = (L(X)w)(Y) +w(L(X)Y).

Proof. We simply have to take in the above proposition the map B to bei) (w, W') H W (9 w';

ii) (a,,3) H a A 0;iii) (w, Y) H w(Y).

5.5. Remark. From the definition of Lie derivation it is easy to deducethat if L(X)w = 0, then w is left invariant under the flow of X, i.e. cpt takesw to itself. Hence we give the following definition.


5.6. Definition. Let X be a vector field and w a tensor field. Then we saythat w is invariant under X if L(X)w = 0.

5.7. Computation of L(X).In Proposition 5.3, we assumed that B is A-bilinear, but used mainly

that it was R-bilinear. The only point at which we used A-linearity waswhen we took the limit inside the argument in B. Notice that if X, w aredifferentiable (as we always assume), then the limit as t tends to zero, of

namely L(X)(w), exists even uniformly on compact sets. Indeedtthis exists even uniformly for the partial derivatives. We do not wish to gointo this question extensively because it is irrelevant to the present discus-sion. We mention it only to point out that the conclusion of Proposition5.3 is valid even when B is not A-bilinear but satisfies a suitable continuityaxiom. We will apply this to the following examples in which in any casethe identity can be directly checked.

5.8. Examples.1) (Y, f) ra Y f where Y is a vector field and f is a function. The corre-

sponding identity gives X (Y f) = L (X) (Y) + YX f , which computesthe Lie derivatives on vector fields to be

L(X) (Y) = XY - YX.

2) (Y, Z) -* [Y, Z], where Y, Z are vector fields. Then we get, using theabove computation, that

[X, [Y, Z]] = [[X, Y], Z] + [Y, [X, Z]]

which is just the Jacobi identity.3) The above identity can also be rewritten as

L([X, Y]) = L(X)L(Y) - L(Y)L(X)

on vector fields. It is easy to see that this is valid even for derivativesof other tensor fields.

5.9. Remark. From Remark 5.5, we conclude that if X and Y are com-muting vector fields, that is to say, [X, Y] = 0, then the flow cot of X leavesY fixed. If "t is the flow of Y, then it follows that the group cots o It o (cots)-1gives rise to the vector field V(Y) = Y. The uniqueness of the flow thereforeimplies that cots and Ot commute for all t, t'.

Corollary 5.4 also computes L(X) on differential forms. Firstly, on formsof degree 1, we have, by 5.4, iii),

(L(X)w)(Y) = Xw(Y) - w([X,Y]),


while 5.4, ii) gives the computation on differential forms of higher degree.Indeed, we get

p(L(X)a)(Xi, ... , Xp) = Xa(Xl, ... , Xp) - L a(Xl, , [X, Xi], . , Xp)

i=1

While the Lie derivative is a good notion of differentiation of a tensorfield with respect to a vector field, it does not lead to a notion of differen-tiation of a tensor field with respect to a tangent vector at a point. Thereason for this is that the value of the Lie derivative of a tensor field at apoint depends on the value of the vector field not only at that point, but ina neighbourhood. It is easy to see that from the algebraic point of view thisis due to the fact that the map X H L(X) is itself a differential operatorand not an A-linear map. For instance, if f E A(M), and a is a 1-form, wehave

(L(fX)a)(Y) = (fX)a(Y) - a([fX,Y])= f (Xa(Y)) - a(f[X,Y] - (Yf)X)= fX(a(Y)) - fa([X,Y]) + (Yf)a(X)= f (L(X)a)(Y) + (Yf)a(X).

Thus we have

5.10. L(fX)a = f L(X)a + a(X)df.

In order to get a good notion of differentiation on tensor fields, we needsome additional structure on the manifold. We will deal with this in somedetail in Chapter 5.

6. The Exterior Derivative; de Rham Complex

6.1. Definition. Let a be a differential form of degree p. Then we definethe exterior derivative da of a to be the differential form of degree p + 1given by the formula

p+1

(da) (Xl, ... , Xp+1) = E(-1)2+1Xia(X1,... , j, ... , Xp+1)i=1

+ (-1)z+ja([Xi1Xi], X1,...,±i,...,Xj,...IXp+1)1<i<j<p+1

where a hat over a symbol means that the symbol does not occur.


6.2. Remark. Firstly, one easily checks that da is A-linear. To show thatit is alternating, notice that if Xk = X1 = X, k < 1, the terms of the firsttype on the right side of the definition reduce to two terms, namely thosecorresponding to i = k and i = 1, and it is clear that these cancel out. On theother hand, in the second type of terms, we have to consider the summationover terms with i = k, j 1 and i k, j = 1, while the term correspondingto i = k, j = 1 is zero as it involves [Xk, X1] in one of the arguments. Itis easy to see that the above two terms also cancel out, proving that da isindeed alternating.

6.3. Example. If a is a form of degree 0, namely a function f, thenthe definition gives df (X) = X f . If a is of degree 1, then da(X, Y) _Xa(Y) -Ya(X) -a([X,Y]).

6.4. Definition. If X is a vector field and a is a differential form of degreep, we define the inner product txa of a with respect to X to be a form ofdegree p - 1 given by

(txa) (X1, ... , XP-1) = a(X, X1, ... , XP-1)

6.5. Lemma. For forms a, 3 of degree p, q we have

tx(a A,(3) = txa A0 + (-1)Pa A tx(O).

Proof. This is a simple consequence of Definitions 6.1 and 6.4.

6.6. Remarks.1) The space of all differential forms constitutes a graded associative al-

gebra. It has also a Z/2-gradation coming from its gradation. If alinear endomorphism preserves the parity of forms, we say that it isZ/2-homogeneous of even type. Similarly if an endomorphism takesodd forms to even ones, and even to odd, we say it is homogeneousof odd type. A linear endomorphism is said to be a derivation ofeven type if it is homogeneous of even degree and is a derivation inthe usual sense. But if an endomorphism D is homogeneous of odddegree and satisfies D(a A 3) = Da n,3+ (-1)deg'a A D,3, we say thatit is a derivation of odd type. For example, by Lemma 6.5, the innerproduct tx is a derivation of odd type while L(X) is one of even type.If both of them are of even type, we have seen that D1D2 - D2D1 isalso a derivation of even type. If one of them is of odd type and theother even; then D1D2 - D2D1 is still a derivation, but of odd type.


Finally if both of them are odd derivations, then D1D2 + D2D1 is aderivation of even type.

2) In terms of the linear map tx, formula 5.10 can be rewritten as

L(f X) (a) = f L(X) (a) + df A txa

for a 1-form a. Using the fact that L(f X ), f L(X) and the map adf A txa are all derivations, one sees that this formula is valid forforms of arbitrary degree. In particular, if a is of degree n, we have0 = tx (df A a) = tx (df) A a - df A txa = (X f )a - df A 1Xa, so thatL(fX)(a) = fL(X)a+ (X f)(a).

6.7. Proposition. The exterior derivative d has the following properties.

i) It is additive, namely, d(a + 3) = da + d,8.ii) For any vector field X, we have dtx+txd = L(X) on differential forms.iii) If a and 3 are differential forms of degree p and q, respectively, then

d(a A,3) = da A 6 + (-1)pa A do.

In other words, d is a derivation of odd type.iv) dod=0.

Proof. i) is obvious from the definition of d.ii) In fact, we have

(dtx +txd)(a)(X1,...,Xp) = E(-1)i+1Xti(txa)(X1,...,ki,...,Xp)+ E(-1)i+i (txa) ([Xz, Xj],... , Xi,... , X,"... Xp) + (da) (X, XI, XP)= Xa(Xi, ... , XP) + E(-1)ia([X, Xil , Xl...... i, ... , XP)_ (L(X )a) (Xl, ... , Xp)for any form a of degree p and vector fields X1i. . . , X.

iii) is proved by induction on p + q. Our assertion is obvious from thedefinition if one of p, q is 0. So let us assume that p + q > 0. In order toprove the equality of the differential forms on the two sides, it is enough toshow that they are the same after applying tx for any arbitrary X. Now wemay use ii) and the induction assumption to complete the proof.

iv) Since d is a derivation of odd type, we have, by Remark 6.6, 1),that d o d + d o d = 2d2 is an even derivation. In order to show thatit is 0, it is enough to check this on the generators of the R -algebra ofdifferential forms, namely functions and 1-forms. Firstly, (d2 f) (X, Y) =X(df)(Y) - Y(df)(X) - df([X,Y]) = XY(f) - YX(f) - [X,Y](f) = 0.Using the definition we may check similarly that it is 0 also on 1-forms.Alternatively, we note that it is enough to prove that it is 0 on small opensets. In a domain in R', however, it is clear that any 1-form is a linear


combination of forms of the type a = f (x)dx. But then da = df A dx. Itfollows therefore that d2 vanishes on a.

6.8. Local computation.

2 ax"generate the space of all vector fields as an A(U)-module. The dual basisfor differential forms of degree 1 is (dxl,... , dx,,,) where (xl,... , x') arethe coordinate functions and dxi are their exterior derivatives. For, if fis a function and X a vector field, we have by definition, df (X) = X f.In particular, (dxi) a = Sid . Hence every differential 1-form canbe uniquely expressed as a = E fidxi. If f is any function, we have bydefinition, (df) ( ) = a . Hence df = dxi. More generally, any r-adifferential form w can be expressed as w = >il<...<i,. aij,...,irdxi, A Adxir,where aji,...,jr are functions. Using 6.7 we obtain

dw = daii,...,ir A dxi, A ... A dxi,,

and since dail,...,ir has been computed above, this completes the local compu-tation of the exterior derivative. Moreover, this also gives the computationof L(X) for any vector field in local coordinates. One notes that these localexpressions are first order differential operators, in the sense that the effectof d on a involves the derivatives of the coefficient functions.

6.9. A digression on complexes.We recall the notions of complexes, morphisms of complexes, homotopy

between morphisms, ... of A-modules, where A is any ring. Analogousnotions are available for sheaves and will be dealt with in Chapter 4, butwe need the basic definitions here. A complex of A-modules is a sequenceMi of modules and homomorphisms Mi -+ Mi+1 for all i E Z such that thesuccessive composites Mi-1 -> Mi , M'+' axe all 0. The hornomorphismsare called the differentials of the complex, and usually all of them are denotedby d. The complex itself is often denoted by M°.

6.10. Examples.1) Recall that a singular r-cochain of a topological space is simply an as-

signment to every singular r-simplex, of an element of a fixed abeliangroup A. For every 0 < i < r, consider the restriction 8i to the stan-dard r + 1-simplex, of the linear map Rr+1 --} Rr+2 defined on theelements of the standard basis as follows: ai(ej) = ej for 0 < j < iand 9i (ej) = ej+1 for i + 1 < j < r + 1. This gives a map Ar -* Ar+1.The ith face of a singular r + 1-simplex s : Ar+1 -* X is defined tobe the composite s o ei. Then we define the differential of a singulari-cochain c by setting dc(s) _ E(-1)ic(s o a j). Then one can check

, eWe have seen that on a domain U in R, the operators --!2-09X,


that d2 = 0 and the resulting complex is called the singular complexof X, with coefficients in A.

2) We have checked above that the exterior derivative on differential formsgives rise to a complex called the de Rham complex.

If M° is a complex, one defines its cohomology groups Hi to be thegroups given by B= where Zi is the kernel of the map Mi -} Mi+1 and Be isthe image of the map Mi-1 --1 M. Notice that since M° is a complex, wehave the inclusion Bi C Zi. Thus Hi can be taken as a measure of deviationof Be from being Zi. Another way of saying it is that He = 0 if and only if

Mi-1 -* Mi -+ Mi+1 is exact, that is to say, the kernel of Mi --+ M'+' isthe same as the image of Mi-1 Mi.

6.11. Definition. The cohomology of the singular complex with coefficientsin an abelian group A is called the singular cohomology of the space withcoefficients in A.

A morphism f' of a complex M° into another complex N° is a se-quence of homomorphisms f' : M' --p Ni for all i such that the diagrams

MiPNi

are all commutative. The maps (fi) induce in an obvious way homomor-phisms Zi(M°) --> Zi(N°) and Bi(M°) --+ Bi(N°) and consequently a ho-momorphism Hi(M°) --3 Hi(N°) of the cohomology groups.

6.12. Remark. A continuous map f : X -+ Y of topological spaces givesin an obvious way a morphism of the singular complex of Y into that ofX. In fact, any singular r-simplex s : A -* X in X gives rise to one in Y,namely f o s. In particular, it gives rise to a homomorphism of the singularcohomology group of Y into that of X.

Two morphisms f0, g° of complexes from M° into N° are said to behomotopic if there exist homomorphisms ki : Mi -i Ni-1 for all i such that

dki + 0+1 d = fi - gi

If f °, g : M° -+ N° are two morphisms of complexes which are homo-topic, then the induced homomorphisms on the cohomology groups are thesame. This is trivial to verify from the definition of homotopy. In particular,if M° and N° are homotopically equivalent in the sense that there are mor-phisms in both directions such that both composites are homotopic, thentheir cohomology groups are isomorphic.


6.13. Definition. The complex

0-*A->T*-+A2T*->...

with all sheaf homomorphisms given by the exterior derivative, is called thede Rham complex.

6.14. Proposition. The de Rham complex is exact as a sequence of sheavesexcept at the initial stage, and the kernel of A -* T* is the constant sheaf R.

Proof. Clearly it is enough to show that if U is an open cube in Rn, thenany form

w = E ail,...,irdxil A ... A dxiri i <... <ir

on U satisfying dw = 0 can be written as da where a is an (r -1)-form. Wewill generalise this a little in order to set up a suitable induction. The gener-alisation we intend is to let ai1..... ir to be not merely differentiable functionson U, but differentiable functions on V x U where V is a parameter spaceassumed to be an open cube in R. Then we claim that the form a canbe chosen to depend differentiably on the parameter space as well, in thesense that the coefficient functions are also differentiable on V x U. Whenwe take a form in n variables with coefficients which depend on m othervariables as well, the notation d for exterior derivative could be somewhatambiguous. To obviate this, we will denote by do the exterior derivativewhen a is treated as a form on an open set of Rn, perhaps depending onsome parameters. Now we want to use induction on n. Any form as above(abbreviating the parameters to t) is best written as

w (t) = W1 (t, xn) A dxn + w2 (t, xn)

where W1 (t, xn) and w2(t, xn) are uniquely determined forms of degrees r -1 and r, respectively, which depend differentiably on (t, xn) and do notinvolve dxn in the wedge part of their expressions. In other words, they aredifferential forms in n - 1 variables depending on m + 1 parameters. Thenwe get

dnw(t) = dwl(t, xn) A dxn + dnw2(t, xn)

= d w1(t x) A dx d w2(t x) + (-1)r'2wa tx' A dxn-1 n n+n-1 n X,n n

Here when we write -, we mean that we differentiate all the coefficientfunctions in the local expression for w. Thus we deduce that

+dnw(t) = (dn-1w1 (t, xn) 1 -37Xn A dxn + dn-1w2(t, xn)(- )


But we are given that dw(t) = 0, so that we obtain two equations:

dn_1W1(t, X"') + (-1)r 8w2(t,xn)8xn

dn-lw2(t, xn)

The second equation implies (by the induction assumption) that there existsa form a(t, xn) in n - 1 variables such that dn_la(t, xn) = w2(t, xn). Thenthe first equation gives on substitution,

dn-lw1(t, xn) + (-1)r80n do-1a(t, xn) = 0,

or what is the same,

dn-1(w1(t, xn) + (-1)r 80n a(t, xn)) = 0.

Again by the induction assumption, there exists a form 3(t, xn) in n - 1variables such that

dn-1,3(t, xn) = w1(t, xn) + (-1)r 8x a(t, xn)

Now let us set -y(t)

=

/3(t, xn) A dxn + a(t, xn). Then we have

do-Y(t) = do_1 (t, xn) A dxn + do-la(t, xn) + (-1)r-18a5-xgin) A dxn

_ (w1(t,xn)+(-1)r8a()/dxn+w2(t,xn)

TX 11 -1

+ (-1)r-18a t,xn A dxnxn

= w(t),

as was to be proved. In order to start the induction, we have to prove theassertion in the case n = 1. This means that if f is a function of x, say in theinterval (-1, +1), and parameters t, then there exists a function cp(x, t) suchthat e4 xx t) = f (x, t). Indeed the function defined by cp(x, t) = fox f (x, t)dxis clearly differentiable in x and t and has the required property.

Finally it is obvious that on a connected open set of R', if df = 0, i.e.L. = 0 for all i, then f is a constant.

6.15. Remark. While the de Rham complex is exact as a sequence ofsheaves, it is not true (in general) that the sequence

0-rR-+A(M)->T*(M)->...

is exact. In other words, it is not in general true that if a differential formw satisfies dw = 0 (namely, a closed form) then it can be expressed as da(namely, an exact form) on an arbitrary manifold M.


6.16. Example. Let us consider the manifold S1. The differential formw = XdY - YdX where X, Y are restrictions of the coordinate functionson R2 to S1, is clearly closed since there are no nonzero forms of degree 2on S'. But there is no function f on S' such that df = XdY - YdX. For,assume that such a function exists. Locally to S' we can find a function 0such that X = cos 0 and Y = sin 0. Substituting in the defining equationfor w, we see that w = dB. Since by assumption df = w as well, we see thatd(f - 0) = 0 and so f and 0 differ by a constant. In particular the localfunction 0 can be extended to the whole of S1. It is well known and easy toprove that this is not possible.

The above example has to do with the fact that S' is not simply con-nected. This suggests that the problem of the exactness of the global deRham complex is related to the topology of the manifold in question. Wewill take up this question for detailed discussion in Chapter 4. For now, wesimply wish to point to the phenomenon of an exact sequence of sheaves

0-, 1,,not giving rise to an exact sequence

0 , F'(M) , F(M) -+ F"(M) -+ 0.The study of this question leads to the concept of cohomology of sheaveswhich will be studied in Chapter 4.

7. Differential Operators of Higher Order

We will now turn to differential operators of higher order and set up themachinery of a symbol calculus. This will be systematically used in thesubsequent chapters.

We have remarked that interesting operators on a differential mani-fold are often defined as maps of a tensor bundle into another, which are,in terms of their local expressions, differential operators in the usual sense.Many such operators are of first order, but there are also interesting opera-tors of higher orders, for example the Laplacian on a Riemannian manifold.We will therefore proceed to define higher order operators, first on functionsand later, from one vector bundle to another.

7.1. Definition. A (linear) differential operator (of finite order) on func-tions is an ]R-linear sheaf homomorphism AM -+ AM which is in the al-gebra generated by differential operators of order < 1. It is said to be oforder < k if it can be expressed as a linear combination of composites ofk operators of order < 1. The set of all such operators will be denoted byDiffM or sometimes for shortness D.


7.2. Remarks.

1) Since we have seen that differential operators of order < 1 have localexpressions of the form E fz + cp, it is clear that a differential op-erator of order < k is locally of the form Ela1<k fa aaa .

2) If M is not compact, it is possible to construct differential operators of`infinite order' by constructing operators on compact sets with ordersthat increase at infinity. For example, consider the open covering ofR+ by open intervals Un = (n, n + 2), n E Z. If {con} is a partitionof unity with respect to Un, then W,Z' is such an operator. Butthis phenomenon cannot happen if M is compact. In any case, wewill only consider operators of finite order.

3) Any sheaf homomorphism of A into itself can be shown to be locally adifferential operator. This is a theorem of Peetre.

Structure of the algebra of differential operators.

We have seen that the set of homogeneous first order operators form aLie algebra, besides being a module over the ring of functions. We wouldlike now to construct an analogue of the universal enveloping algebra inthis case, although it is not a Lie algebra over functions. Recall first thedefinition of the enveloping algebra.

7.3. Definition. Let g be a Lie algebra over a field k. Then the envelop-ing algebra of g is the quotient of the tensor algebra of the vector space g bythe 2-sided ideal generated by elements of the form X ® Y - Y ® X - [X, Y],with X, Y E g.

In order to imitate this construction, we would like first to construct theanalogue of the tensor algebra. As a starting point, it is better to take D1than T, since the former is well adapted to the construction of the tensoralgebra. We will now explain why.

As a sheaf of IR-vector spaces, V1 is simply A®T. For any f E A(M) itis convenient to denote the corresponding element of D1 (M) by m(f) ratherthan f itself (although when we judge that no confusion is likely, we mayuse f instead of m(f)). Then D1 is an A-bimodule. In fact, if f EA(U),and D = m(g) + X E Dl (U), then we may define

f o D = m(f) o (m(g) + X) = m(fg) + fX,

D o f = (m(g) + X) o m(f) =m(fg+Xf)+ fX.


The reason for the latter definition is that

(D o m(f))(,p) = (m(g) +X)(fco)= g.fco+(Xf)M +fXco= m(gf + X f )cp + fX co.

It is easy to check therefore that this definition makes Dl an A-bimodule.It will turn out that the structure of the algebra of differential operators

can be determined from the structure of D' as an A-bimodule on the onehand and as a Lie algebra over II8 on the other. Indeed, the construction isakin to that of the enveloping algebra of a Lie algebra.

In the case of a Lie algebra g over a field k, the situation is like startingwith g ® k. Let us denote inclusion of k in g ® k by A i-* m(A). We leavethe following as a simple exercise to the reader.

7.4. Exercise. Consider the tensor algebra of W = V ® k. Pass to the quo-tient by the 2-sided ideal generated by m(1) - 1. Show that it is canonicallyisomorphic to the tensor algebra of V.

Consider Dl ® D' where the tensor product is with respect to the rightA-module structure of the first factor and the left one of the second factor.Also using the bimodule structure of both we see that the tensor product isalso a bimodule. We can iterate this construction to obtain a tensor algebra.Now we will take the quotient of this ring by the two-sided ideal spannedby m(1) - 1 to obtain an R-algebra. This is the algebra which we sought toconstruct. We will denote it by C(M).

It is easy to describe this algebra by a universal property.

7.5. Proposition. Let B be any ]R-algebra containing A. Then B has anatural bimodule structure over A. Any bimodule homomorphism P of D1into B which is the identity on A gives rise to a unique R-algebra homo-morphism of C into B which is the identity on A and coincides with P onDl.

Proof. In fact, we will define inductively, for each r, a bimodule homo-morphism p(r) from ®'(V') into B. Taking P(°) to be the identity mapon A, and assuming that PH has been defined, we notice that the mapDl x ®r D1 -+ B given by (Y, Z) H p(y)p(r) (Z) is balanced. Hence itgoes down to a map P(r+1)

; ®r+1 Dl - B as claimed. It is evidently ahomomorphism from the JR-algebra T(D1) into B. From the assumption,we see that m(1) - 1 is in the kernel of this map and so induces a map of Cinto B. It is unique since V1 and A together generate T(D1).


7.6. Remark. We could have restated the assumption on P by saying thatit is a left A-linear map from T into B which satisfies

P(X) f = X f + f P(X).

Thus we also have a characterisation of C-modules. It is an A-moduleE together with an 1R-bilinear map T(M) x E -+ E denoted (X, s) --+ Vxmsuch that

7.7. 7fX(s) = fVxs, and Vx(fs) _ (Xf).s+ fVxs.

We have constructed, for any open set U, such an algebra C(U) (althoughfor notational convenience we dropped the parenthetical U). It is easy todefine compatible restriction maps and obtain a presheaf C. We will denotethe image of any vector field X in C by Vx and the image of any function fby m(f).

7.8. Definition. We call C the connection algebra of M.

Notice that this is not an algebra over A, since the right and left mul-tiplications by functions are not the same. The IR-subsheaves Cr definedabove as images of ®T Dl are all submodules for the A-biModule structure.Moreover, it is clear from the definition that these satisfy the condition

Cr.Cs C Cr+s.

This is paraphrased by saying that the algebra C is actually a filtered algebra,of which we recall the definition below. It is easily verified that Cr aresheaves.

7.9. Definition. An algebra A together with subspaces Fr(A), r E 7L+is said to be a filtered algebra if Fr(A) C F,.+1(A) for all r E Z+ andFr(A).Fs(A) C F,.+,,(A) for all r, s E Z+.

Every filtered algebra gives rise to a graded algebra as follows. Considerthe direct sum E)(Fr(A)/Fr_i(A)). It is an algebra under the natural defini-tion which follows. Suppose V E Grr(A) = Fr(A)/Fr_i(A) and w E Grs(A).Then taking representatives for v, w in Fr (A), F3(A), and multiplying themin A, we obtain an element of F,.+s(A). Its image in Gr,.+s(A) is indepen-dent of the choices made and this multiplication defines an algebra operation.This is called the associated graded algebra and is denoted by Gr(A).

Let us now consider the graded R-algebra associated to the filtered al-gebra C. Its 0th graded component, A, is a subalgebra. Indeed, we claim itactually commutes with all elements of Gr(C). Let us prove this assertionby induction. Since A is itself commutative, we can start the induction. As-sume that for every function f, the corresponding element m(f) commutes


with all elements of Cr modulo Cr_1. Any element a of Cr+1 is a linearcombination of images of elements of the form 8 ® (X + m(g)), with ,8 E Cr.Then we have

m(f) (j3 ® (X + m(g))) - (,Q ® (X + m(g)))m(f )m(f)(/3(9 X) - (0 (9 X)m(f)(mod Cr)m(f)(Q®X) -'3®(m(f)X +m(Xf))((m(f ),Q - ,8m(f )) ®X (mod Cr),

but by the induction assumption m(f )Q -,3m(f) E Cr_1. This proves ourclaim. Notice also that the first graded component of Gr(C) is canonicallyDl/A = T. Hence we have a canonical algebra homomorphism of the tensoralgebra over A of T into Cr (C) . We claim that this homomorphism is ac-tually an isomorphism. It is obvious from the definition that C is generatedas an 1I8-algebra by C1 and A so that Gr(C) is generated as an A-algebra byits first graded component, namely T. This implies that the above homo-morphism is surjective.

Since Cr are all sheaves and all these maps are compatible with restric-tions to open sets, it is enough to check the isomorphism statement overopen sets in 1[8'x.

We will define a C-module structure on a specific A-module, namelythe tensor algebra Q of T. We have shown that we need only to define anaction VX on Q subject to equation 7.7. We define Vx(E f,,X.. ) to beX (9 E f«X« + E(X f«)Xa. Here a is a multiindex (i1, ... , ir), Xi denotes

and Xa denotes the element Xi,1 ® ... ® Xiv, in Q. That this satisfiesthe two conditions 7.7 is obvious. It is also clear that if FFQ denotes thefiltration on Q associated to the natural gradation on the tensor algebra,then we have Cr.F.,Q C Fr+3Q Hence we have an induced action of theassociated graded algebra Gr(C) on the tensor algebra Q. Since we have anatural homomorphism of the tensor algebra Q into Gr, we therefore havean action of Q on itself. This action is by definition the juxtaposition action,namely the multiplication on the left in the algebra structure of Q. Sincethis action is faithful, we conclude that the map Q Gr(C(M)) is injective.We have therefore proved the following.

7.10. Theorem. The filtered ia-algebra C has as its associated graded al-gebra an A-module which is in fact the tensor algebra of T.

We shall go on to the definition of the algebra D of differential opera-tors. Consider the two-sided ideal Z in C generated by elements of the formR(X, Y) = VxVy - V yV X - 0(X,y] with X, Y vector fields. The quotientof C by I gives rise to an R-algebra. Again, since this can be done for all U,it follows easily that we have a presheaf. The resulting sheaf of algebras D


comes with a filtration Fk (D) making it a filtered algebra. Now the sheaf Amay be made into a C-module by defining X. f to be X f , for all vector fieldsX. It is clear that this passes down to a D-module structure on A. Un-der this action, elements of Fk (D) act as JR-homomorphisms of A which arelinear combinations of composites of k or less vector fields. In other words,we have given a sheaf homomorphism Fk(D) -+ Diffk. In order to checkthat this is an isomorphism it is enough to verify it on sections over anyopen domain U in R. It is surjective by definition, and in order to proveinjectivity, we note that modulo the relations we have imposed, any elementof .Pk(D)(U) can be written as

E f21 i.-in ( a )" ... (DQn )in, ii +.... + in < k.

The image in Diffk(U)

isO2t1he

differential operator on functions given by theabove expression. Thus we have only to check that it is zero if and only if theaction of the operator on all functions is 0. But this is easily seen by lookingat its action on functions of the form l(xi - ai)'-, where a = (al, . , . , an)is some point in U at which the coefficient functions are nonzero. Thus wehave proved the following theorem.

7.11. Theorem. The algebra D is naturally isomorphic to the quotient ofthe connection algebra C by the two-sided ideal generated by elements of theform R(X, Y) _ V XVy - V yV X - V [X,Y] with X, Y vector fields.

7.12. Exercise. Show that left invariant differential operators on a Liegroup form an algebra and that it is canonically isomorphic to the universalenveloping algebra of its Lie algebra.

7.13. The symbol sequence.Consider the graded algebra associated to the filtered algebra D. As

in the case of Gr(C), we see that the left and right A-module structureson D pass down to identical structures on Gr(D), making it actually anA-algebra. Moreover, in view of the fact that R(X, Y) are all 0 in thisalgebra, one also sees that it is a commutative algebra. We have a naturalinclusion of T in D as the space of homogeneous first order operators andhence we also get a homomorphism of T into Gr(D). By the universalproperty of symmetric algebras, this induces an algebra homomorphism ofthe symmetric algebra S(T) into Gr(D), which is obviously onto.

7.14. Theorem. The algebra D of differential operators on a differentialmanifold M is a filtered algebra whose associated graded algebra is canoni-cally isomorphic to the symmetric algebra of T.


Proof. As in the proof of Theorem 7.10, we have only to check that thenatural map Sk(T) -- Dk/Dk-1 is an isomorphism on a domain U inRn. But then we have proved that Dk(U) has an A(U)-basis consisting of

(TX11z1 .. . (axn)Zn )il... ( a ) Zki 1 + +i, < k and hence the images ofJ n axl 8xn /ii + + in, = k, in Dk/Dk-1 form a basis. But clearly these are images of abasis in Sk (T) under the natural map we have given above. This completesthe proof.

7.15. Definition. In particular, we have shown that there is an exact se-quence of vector bundles:

0-,Dk-1-->Dk,Sk(T)-->0.

It is called the symbol sequence. If D is a differential operator on an openset U of order < k, then its image in Sk(T)(U) is called its symbol or kthorder symbol.

7.16. Remarks.

1) If we consider a differential operator of order < k, and take its imagemodulo lower order operators, it is essentially the same as consideringits kth order terms. The above sequence says in effect that the 'highestorder term' of a differential operator makes invariant sense only as asection of Sk(T).

2) A differential operator D satisfying D(1) = 0 may be said to be anoperator without constant term. In particular an operator of order 1without constant term is a homogeneous operator. For higher orders,of course these two notions are different, and while an operator with-out constant term makes sense in any manifold, there is no invariantway of defining a homogeneous operator in general.

3) An operator of order at most 1 is of the form X + m(f) where X is avector field and f is a function. Its symbol is X.

7.17. Differential operators on vector bundles.We have seen a few examples of differential operators on manifolds,

but they are rarely on functions, or even on systems of functions. Theyare often defined on sheaves of tensors on a differential manifold. We aretherefore obliged to generalise the notion of a differential operator. In fact,we will define a differential operator (of order < k) from one locally freesheaf S into another, say F. This is done by considering an (R-linear)sheaf homomorphism, which has the following property. Whenever s is asection of E and A a section of F*, the map f H (D (f s), A) = A(D(f s))from functions to functions is a differential operator (of order < k). While


this is a practical definition, the theoretically satisfactory definition wouldbe that it is a section of F ® D ® £*, where the first tensor product iswith respect to the left, and the second with respect to the right A-modulestructure on D. A slightly different (but obviously equivalent) formulationis the following.

7.18. Definition. A right A-linear homomorphism of £ into F ® Dk iscalled a differential operator from £ to F of order < k.

Using the left A-linear map Dk -- A which associates to any D itsconstant term D(1) we can associate to any differential operator from Eto T an JR-linear homomorphism E --> T. We will denote the sheaf of suchoperators by D(E, F). In the case when E and F are trivial, we refer to suchan operator as a system. If D is a differential operator from E to F of order< k, its kth order symbol is the A-linear homomorphism E -* F (& Sk(T)obtained by using the symbol map Dk --), Sk(T).

7.19. Remark. One can also define inductively an J-linear map D : E -* Fto be a differential operator of order at most k, by requiring that for everyfunction f, the operator f - [D, f] = D o m(f) - m(f) o D be an operatorof order at most k - 1. If { f;,}, 1 < i < k, are functions, then

[... [[D, f1,], f2],... ,fk]

is a 0th order operator, namely an A-linear homomorphism E -+ F. Thisfunction depends symmetrically on the f2's. For example, this follows fromthe equality [[D, fi], f2] + [[f2, D], f1] = 0 by the Jacobi identity. Moreover,assume that all the fn's vanish at a point rn E M. Then evaluation of thisbracketed function at m goes down to a map Sk(T;,,,) --> Hom(Em, Fm).In other words we get a homomorphism a(D) of Sk(T*) into Hom(E, F).Associated to this, we get also a section of Sk(T) ® Hom(E, F). This canbe easily seen to be the symbol o- as we have defined. We will call either ofthese the symbol, but usually reserve the notation & for one and o for theother.

7.20. Exercise. Show that the above definition coincides with the symbolof a differential operator as defined in 7.15 on functions and hence also onbundles as in 7.18.

Computation of symbols.We will now give some examples of computation of symbols of first order

operators.


7.21. Examples.

1) Consider the exterior derivative d : Ai(T*) --> Ai+l (T*). From its localexpression it is clear that it is a first order operator. In any case, leta be an i-form and X1, ... , Xi+1 be vector fields providing a linearform, namely w w (X 1, ... , Xi+l) on Then in order tosee that d is a differential operator we have only to show that themap f H d(f a)(Xi, ... , Xi+1) is a differential operator, But sinced(fa)=dfAa+fda, we have

d(fa)(X1,...,Xi+1) = (-1)r+ldf(Xr)a(X1...... r..... Xi+1)

+ f(da)(Xi,...,Xi+1)-1)r+1(Xf)a(X1, ... , Xr, , Xi+1)

+ fda(Xi,...,Xi+1)Hence this is a differential operator of order 1 on functions whosesymbol is the vector field E(-1)r+la(Xi...... r, ... , Xi+1)Xr andwhose constant term is da(Xl,... , Xi+1). Now the symbol of d is ahomomorphism a (d) : Ai (T *) 4 Ai+l (T *) 0 T, and so can equallywell be interpreted to be a bilinear homomorphism Ai(T*) x T* -,Ai+l(T*). Then the above computation gives the following:

o(d)(a,8)(X1,...,Xi+1) _((_1)r+1a(Xl,...,Xr,...,Xi+l)Xr)

E(-1)r+la(Xl...... ... , Xi+1),a(Xr)

(0na)(Xl,...,Xi+1)

In other words, a (d) (a, ,(3) = 0 A a.

2) Consider the Lie derivative with respect to a vector field X, say of ani-form a. Then for f E A(M) and vector fields Xr E T (M), r < i, wesee that the operator f L (X) (f a) (XI, ... , Xi) is given by

f F-+ X (f a(X1, ... , Xi)) - E(fa)(Xi, ... , [X, Xr], ... , Xi)r=1

_ (Xf)a(X1,...,Xi)+f(Xa(Xi ...,Xi))

- f a(Xi, ... , [X, Xr], ... , Xi).r=1

This map is obviously a differential operator of order 1 whose symbolis a(Xi, ... , Xi)X. This shows that the symbol of L(X), which isa homomorphism Ai(T*) --y Ai(T*) ®T, is given by a (L(X))(a) _a®X.


7.22. Symbol of a composite of differential operators.Let D1 : E --+ F and D2 : F -+ G be operators of order < k, 1, respec-

tively. By composition we obtain an operator of order < k + l from E to G.We wish to compute the (k+l)th symbol of D2 o D1. Firstly, we remark thatin the case when E = F = G = A, this is easily done. For the symbol is thenatural homomorphism Dk --+ Sk (T), and this is multiplicative in the sensethat °k+l(D2 o Dl) = o-l(D2).Qk(Dl) where the multiplication on the rightside is intended in the sense of symmetric algebras. In general, the symbolsof D1, D2 are homomorphisms o : E -* F ® Sk(T), T : F -* G ® S1(T),respectively. Any two maps o-, r as above give rise to a map (TO ISkiTi) o o' :

E , G ® S'(T) 0 Sk (T), and using the multiplication in the symmetricalgebra, we get a map E --3 G ®Sk+1(T). This is called the composite of thesymbols o and T. Thus our computation above can be restated as

7.23. Proposition. If D1i D2 are differential operators E --> F and F --> Gof orders k, 1, respectively, then the (k + l)th order symbol of the compositedifferential operator D2 o D1 : E --j G is the composite, of the two symbolsak(D1) and UZ(D2).

7.24. Tensorisation with local systems.If D is a differential operator from a vector bundle E to F, and L is

any other vector bundle, there is no natural way of defining a differentialoperator L ® E --> L ® F. The reason is that the tensorisation is over A,while D is not A-linear. For the same reason, it is clear that if L is a localsystem of R- or C-vector spaces, then one can define such an operator. It ismoreover clear that its symbol, considered as a map L ®E -+ L ®F ®Sk (T),is obtained as the tensor product of the symbol of D with 1L.

In particular, if L is a local system, then one can talk of the de Rhamcomplex with coefficients in L, namely

0--,L->L0 A-+ ...--+L0A'(T*)and clearly the de Rham sequence of sheaves remains exact after tensorisa-tion with the local system L. Notice that if L is a local system of 118- (resp.C-) vector spaces and E is a locally free A-sheaf, then L 0 A is a locally freeA-module.

Exercises

1) Determine the space of derivations of A = IR [xl,...,x,].2) Show that the group B of triangular matrices, namely matrices (Azj)

such that Atij = 0 whenever i is greater than j, is a Lie subalgebra ofgl(n, I[8). What is the corresponding Lie subgroup of GL(n, R)?

Exercises 71

3) Let X be a connected topological space, x a point of X and L a localsystem of vector spaces over X. Show that the natural map ,C(X) -Lx is injective.

4) If E is a complex vector bundle of rank n, treated as a real vectorbundle of rank 2n, show that E ® C is isomorphic to E ® E.

5) Show that the Plucker imbedding of the Grassmannian has nonzerodifferential at all points.

6) If a Lie group G acts on a differential manifold M, then there is anatural homomorphism of the Lie algebra g of G into the Lie algebraof differentiable vector fields on M. When is this map injective?Compute this map for the action of GL(n, IR) on 1R'".

7) Show that the universal covering space of any connected Lie group isalso a Lie group. What is its Lie algebra?

8) Show that if Y is a (differentiable) vector field and (pt) is its flow, then

l o(o(Y)t

- Y _ LX(Y))exists uniformly, where Y is a vector field with compact support.

9) Let w be a 2-form on 1R4 with coordinates (x, y, z, t). Write w in termsof coordinates and note that it gives rise to a 2-form a and a 1-form,6 on I[83, depending on t. Compute dw in terms of a and Q, givingrise to a 1-form -y and a function cp on R3 (again depending on t)-

10) Let G be a compact connected Lie group. Consider the differentiablemap g --> g2 of G into itself. Compute its differential at any x E Gand determine at what points it is an isomorphism.

Chapter 3

Integration onDifferential Manifolds

So far we have been discussing differential operators on a manifold. In orderto develop calculus on it, we need also a theory of integration. Fortunately,the abstract theory of integration, particularly that of Borel measures ona locally compact space, is applicable to differential manifolds. But it isconvenient to restrict oneself to what are called densities. These are in thefirst place Borel measures which are in some sense indefinitely differentiable.We will make this notion precise by requiring that Lie derivatives of measuresexist.

1. Integration on a Manifold

We start with the remark that from the point of view of integration, a Borelmeasure on a differential manifold can be defined as follows.

1.1. Definition. A Borel measure on a differential manifold M is a linearforma on the vector space CC ° (M) of indefinitely differentiable functionswith compact support on M, which satisfies the following continuity axiom:If { fk} is any sequence of elements in CC° all of whose supports are containedin a compact set K and sup fkI tends to zero as k tends to infinity, thenp(fk) tends to zero. The scalar µ(f) is also denoted f f dµ.

If cp : M --+ N is a proper differentiable map, then any Borel measureA on M has an image measure cp(µ) in N. Indeed, if f has support in acompact subset K of N, then f o cp has support in cp-1 (K) which is compactby assumption. Hence we may define (cp(µ)) (f) to be i(f o cp) for every

7mm73

74 3. Integration on Differential Manifolds

f E CI (N). That this linear form satisfies the continuity requirement isobvious.

1.2. Remark. For any compact subset K of M, one can provide the spaceCK of indefinitely differentiable functions on M with support in K, with thestructure of a normed topological vector space in such a way that a sequence{ fk} tends to zero if and only if {sup I fkl} tends to zero. A measure is thena continuous linear form on this space. A base for neighbourhoods of 0 inthis topology is provided by {f E CK : sup If I < a}. A linear form 1, iscontinuous at 0 in this topology if and only if there is a positive number Asuch that I A(f) I < A sup I f I for all f. The constant A may depend on K andso it is better to denote it by AK.

1.3. Example. The Lebesgue measure on the Euclidean space R' is a Borelmeasure. The continuity of the linear form follows from the `dominatedconvergence theorem'.

1.4. Exercises.

1) Show that if f is a differentiable function, then the map g H fg of C°°into itself is continuous.

2) Is the linear form f H f (0)' of C,°(W) a Borel measure?

3) Is the linear form f At (0) of CC°(R) a Borel measure?

1.5. Product measure.If µ, v are measures on differential manifolds M and N, then one can

define a natural measure on the product M x N. We note that it is enoughto define in a consistent manner, continuous linear forms on CA for anycompact subset A of M x N. Since any compact set A in M x N is containedin pl (A) X P2 (A), it is clear that we may as well assume that A is itself ofthe form K x L where K, L are compact subsets of M, N, respectively.

To elements f E CK and g E CL I, associate the function on MxN taking(x, y) to f (x)g(y). Clearly this induces a linear map CK 0 CL --* CKXL. Itis obvious that this map is in fact injective.

1.6. Exercise. Prove the above assertion by showing that if E f(x)g(y) _0 with gi linearly independent in CL, then fi = 0 for all i.

The linear form on this subspace, given by the bilinear map (f, g) Hp(f).v(g), is easily seen to be continuous for the induced topology. If wenow show that this subspace of CKxL is dense, then it would follow that thelinear form above extends uniquely to a continuous linear form as required.

1. Integration on a Manifold 75

Let f belong to CKXL. Given e > 0, a E K and b E L, we can find openneighbourhoods Ua, V b such that I f (x, b') - f (a, b') I <e and If (a', y)- f (a', b)< e for all x E Ua, y E Vb, a' E K, and b' E L. Using the compactness ofK and L, we choose finitely many points a;,, b3 such that U, = Ua;, Vj =Vb, cover K, L, respectively. Let (cpj) and (0j) be partitions of unity forthese coverings. We define a function g E CK ® CL by setting g(x, y) =E f (az, bi)ccz (x)0j(y). Then f (x, y) - g(x, y) = f (x, y) (E cpz(x)'f (y)) -E f (az, bj)coj(x)''j (y) = E(f (x, y) - f (az, bj))cpi (x)'tlaj (y)). From the choiceof az and bj and noticing that the support of coj (x),0j(y) is contained inU, x Vj, we conclude that I f (x, y) - g(x, y)I < e for all x, y. This proves ourclaim that CK ® CL is dense in CKxL

Thus we have the following

1.7. Proposition. If u and v are Borel measures on M and N, respec-tively, there exists a unique Borel measure .t ® v on M x N which satisfies

f f (x)g(y)d(t ® v) = ff(x)d(x)J9()dv()for all f , g with compact support.

1.8. Lie derivatives of measures.One can try to define the Lie derivative L(X)(p) of a measure p with

respect to a (differentiable) vector field X as follows. The Lie derivativeL(X)(p) is the functional which takes any f E Cc' to limt,owhere (cpt) is the flow of X. Note that although cot may not be defined for allt, they can be defined for small t in a neighbourhood of the support off . Thisis adequate for the definition above. We see that this limit exists and is equal

to µ(limt-,o f°`°tl-f) = -µ(X f). While it is easy to check that the mapf -+ -µ(X f) is linear, there is no reason why it should satisfy the continuityaxiom. We will assume that it in fact does. Indeed, we will assume that ifX1, ... , Xk are any number of vector fields, then the iterated Lie derivativeof µ is also a measure, that is to say, the linear form f H (-1)kp(Xk ... X1 f )satisfies the continuity axiom. These considerations motivate the followingdefinition.

1.9. Definition. If p is a Borel measure with the property that for anyfamily of vector fields X1i ... , Xk, the linear form f -4 µ(X1 Xkf) onCc' is a Borel measure, then we say that µ is indefinitely differentiable. IfX is a vector field, the measure f -µ(X f) is called the Lie derivative ofy with respect to X and is denoted L(X)(µ).

If U C V are open subsets of M, then a measure on V gives rise toa measure on U since there is a natural continuous inclusion of C°°(U) in


C,' (V), namely, extension of functions by setting them zero outside U. It isalso obvious that differentiable measures restrict to differentiable measures.The assignment to every open set U of M, of the set of all differentiablemeasures on U, is thus a presheaf under these restriction maps. It is infact a sheaf. Let {Ui} be a locally finite family of open sets and pi beBorel measures on them, such that lµil Ui f1 Uj and mi I Ui fl U3 coincide forall i, j. Then we can define a measure p on U = U Ui as follows. Let{cpi} be a partition of unity with respect to the above covering. Then forany f E CC (U) define g(f) = E pi (f Cpi). Note that this summation isonly over the set of i's such that supp(f) fl Ui is nonempty. This set isfinite since f has compact support. If f has compact support containedin Uj, then this summation becomes > i pj(f cpi) because f cpi has supportinside Ui fl Uj and we know that ui I (Ui fl Uj) = aj I (Ui fl Ui). But thenEpj (f cpi) = pj (f E cpi) = pj (f ). This implies that p restricts to µi onUj for all j. It is a straightforward verification that p is a Borel measureon U, that is to say, it satisfies the continuity axiom, and also that it isdifferentiable if all the pi's are. We will denote the sheaf of all differentiablemeasures by B.

In a natural way, X3 is an A-module. If p is a measure and h a differen-tiable function, then hp : f --, p(h f) is a linear form on CC°. This satisfiesthe continuity axiom, for if { f} is a sequence of functions with support ina compact set K and sup I f, l tends to zero, then {h f} is also a sequenceof functions with support in K and sup I h f,, I tends to zero. Hence by defi-nition p(hf,,,) tends to zero as well. Moreover, if p is differentiable, then hpis also differentiable. We have in fact the following formula regarding Liederivatives of measures.

1.10. Proposition. If p is a differentiable measure and h a differentiablefunction, then hp is also differentiable, and for every vector field X, we have

L(X)(hu) = L(hX)(p) = Xh.p + hL(X)(p).

Moreover, if X, Y are vector fields, we have

L([X,Y])(p) = L(X)L(Y)(p) - L(Y)L(X)(p).

Proof. The first assertion is similar to [Ch. 2, Proposition 5.3] and can beproved similarly. In any case, using the formula L(X) (p) = -p(X f ), wecan check it directly:

(L(hX)(A))(f) -,(hX(f))_ -p(X(hf) - Xh.f)

(L(X)p)(hf) + (Xh.p)(f)(h.L(X)(p))(f) + (Xh.a)(f)

1. Integration on a Manifold 77

It is easy to check similarly that L(X)(hp) also gives the same expression.The second assertion is also straightforward. In fact, we have

L([X,Y])(,u)(f) _ -p([X,Y]f)_ -A(XYf)+1a(1'Xf)

(L(X)(u))(Yf) - (L(I')(,u))(Xf)(L(X)(L(I')(p)))(f) - (L(1')(L(X)(A)))(f)((L(X)L(V) - L(I')L(X))(µ))(f)

We will hereafter mean by 13 the A-module of differentiable Borel measures.

1.11. Example. Let M be a domain in a real vector space V. Then any(translation) invariant measure y is infinitely differentiable. In fact, if Xdenotes the vector field 8v, v E V, then its flow is given by translations bytv. Hence the Lie derivative L(X) of the measure µ exists and is actually 0.From this we conclude that the Lie derivative of p, with respect to any vectorfield of the form f 8v with f differentiable, is actually ((9v f )l.c. In particularthe Lie derivative with respect to a differentiable vector field of any measureof the form hµ with h a differentiable function, is again a measure of thesame form.

1.12. Remark. If µ is any measure on a real vector space V such thatL(8v)p = 0 for all v E V, then it is clear that u is invariant under transla-tions.

1.13. Theorem. An invariant measure on a real vector space V exists andis unique up to a scalar factor.

Proof. We will prove the uniqueness by giving an injective map H of thespace of invariant measures into the one-dimensional space An(V*) where nis the dimension of V. We interpret the latter as the space of alternatingn-linear forms on V. In order to give this map, we choose one of the twoconnected components of An(V) \ {0}. Then f o r any vi, ... , vn E V wedefine H(p)(vi, ... , vn) to be ±A ([vi]), where [v2] denotes the cube formedby 0 and the vz's. This set can be defined as the convex closure of all thevectors of the form 7, vj, with the summation running through any subsetof {1, ... , n}. Since the cube is compact, its measure is finite. Note that ifthe vz's are linearly dependent, the set [vi] is contained in a proper subspace.It cannot have nonzero measure, since a compact set contains the disjointunion of infinitely many translates of [vi]. In particular, if two of the vi'sare equal, then H takes the value 0. If they are linearly independent and µis nontrivial, then l.c([vZJ) is nonzero, since any open set can be covered bycountably many translates of [vi]. We assign the sign 1 (resp. -1) in our


definition of H, if vl n A vn belongs to (resp. does not belong to) thechosen component.

It is clear that if vl is replaced by a positive integral multiple rvl, then[rvi, ... , vn] is the union of [(r-l)vi, ... , vn] and the translate of [vi,. .. , vn]by (r - 1)vl, the intersection being of measure zero. Hence one concludesthat µ([rvi, ... , vn]) = ry([vl, ... , vn]) by induction. It is clear that thesame equality persists for rational values of r as well. If r is a positive realnumber, we can approximate it by an increasing sequence {ak} of positiverational numbers and notice that [rvl, ... , vn] is the increasing union of[akvl, ... , vn] and so the same equality is valid for r as well. Finally, thesigns in our mapping are the same for all these cubes and so it follows thatH(µ) (Avl, ... , vn) = \H(1-6) (vl, ... , vn) for all real A, since multiplying vlby -1 changes the sign in our definition, but does not change the measure.By a similar elementary argument, one can also check that it is additive ineach variable. Thus H(µ) is an alternating n-form on V. We have given aninjective linear map of the space of invariant measures into An (V*). As forthe existence, we identify V with Rn and take the Lebesgue measure on 1Rnwhich is of course left invariant.

1.14. Exercise. Prove the above assertion that the map (vi, ... , vn)±µ([v2]) is additive in each variable.

1.15. Corollary. The set of all measures in 13 in a domain U in Rn satisfy-ing LX (µ) = 0 for all vector fields of the form X = 6, is one-dimensional.

Proof. Let µ be a nonzero measure as above. There is no loss in generalityin assuming that U is a unit disc. Then by looking at the flow, we deducethat for any open subset V of U and a E Rn such that the translated setW = Ta(V) is contained in U, we have Ta, (p l V) = ,u I W. Consider the opencovering {TT U} of ll8n obtained by translating U by all points of R. Provideeach of these open sets with the translated measure T,(µ). We will get aglobal measure on R', if we verify that they agree on the intersections. Letz E T.,U n TyU = Ux,y. Then and T_yU.,,y are both contained inU, and T,;_y translates one to the other. From this and the translationinvariance of µ, we check that T_,(µ) and Ty(µ) coincide on the intersection.Thus there is a global measure on R' which restricts to T(U) as T(µ).Clearly this measure is translation invariant. Our assertion therefore followsfrom Theorem 1.13.

From the above, one can surmise that the sheaf B of differentiable mea-sures as a sheaf of A-modules, is very close to the sheaf 1C of n-differentialson M (where n is the dimension of M). For one thing, the Lie derivativesof n-forms behave very much the same way.


1.16. Proposition. Let w be an n-form, X a differentiable vector field andh a differentiable function. Then we have

(L(hX))(w) = L(X)(hw) = Xh.w + hL(X)(w).

Also we have

L([X,Y])(w) = L(X)L(Y)(w) - L(Y)L(X)(w)

for any two vector fields X, Y.

Proof. We use the formula dtx + txd = L(X), proved in [Ch. 2, 6.7, ii)].Applying this on w, and using the fact that the exterior derivatives of n-forms are zero, we get (L(hX))(w) = dthX(w) = d(htx)(w) = dhA (txw) +hdtxw. Using the fact that tx is an odd derivation, we may write this as-tx(dhnw)+tx(dh)w+hL(X)(w). But dhAw is clearly zero and tx(dh) =X h, proving one of the equalities above. The remaining statements havealready been proved in [Ch. 2, 5.8].

1.17. Remark. We will again consider any real vector space V of dimensionn as a differential manifold. Note first that the sheaf A'n(T*) is in this casesimply A®j An (V *) . We may also tensor with A, the one-dimensional spaceof invariant measures. The latter sheaf can be identified with the subsheaf ofB consisting of measures of the form f.µ where f is a differentiable functionand A is an invariant measure. This isomorphism between the space ofinvariant measures and the space of n-differentials proved in Theorem 1.13yields an isomorphism of these two sheaves as well. Since the Lie derivativeswith respect to vector fields of the type 8z, are zero on both invariant formsand invariant measures, the above formulae show that this isomorphism iscompatible with the operation of Lie derivatives. However, one should notethat the isomorphism depends on the choice of a connected component ofA'n(V) \ {0}. Such a choice is called an orientation of the vector space V.

2. Sheaf of Densities

We will take up the relationship between K and B in general now.

2.1. Definition. An A-homomorphism of the sheaf IC = Af(T)* into Bis said to be flat if it commutes with Lie derivatives with respect to all(differentiable) vector fields.

2.2. Proposition. If M is connected, any two nonzero flat homomor-phisms of K into 13 differ by a nonzero scalar factor. If M is a domain inRn, then there does exist a nonzero flat homomorphism.


Proof. It is enough to prove the first statement also when M is a domain inlR'. For any invariant form w and any v E Rn, we have 0. If A isa flat homomorphism of K into B, then clearly we also have L(t9) (Aw) = 0.In other words, a flat homomorphism takes invariant forms to invariantmeasures and is therefore determined up to a scalar factor. On the otherhand, we have observed that the A-homomorphism which maps any nonzeroinvariant form w to the Lebesgue measure is actually flat.

2.3. Remark. We have seen that any domain in a vector space V admits aflat homomorphism IC --r B. This is even canonical once one has chosen anorientation in V. The question arises whether any differential manifold ad-mits a flat homomorphism. It is not so for the following reason. We maycover the manifold with coordinate open sets, and in each of these opensets which are diffeomorphic to domains in R, take a flat homomorphism.These may not patch together since the flat homomorphism that we havegiven above for domains in Rn, depends on the particular coordinate system.Two different choices of coordinate systems may give rise to flat homomor-phisms which are negatives of each other.

The assignment, to any open set U, of the set of all flat homomorphismsKu -> fiu is a sheaf of lid-modules. Actually we may consider the set ofall flat homomorphisms which coincide with an integral multiple of anystandard flat homomorphism on a coordinate subset. Since the two standardisomorphisms are negatives of each other, it is a local system of rank 1 overintegers, which we will denote by OZM.

2.4. Definition. The sheaf OZM is called the local system of twisted inte-gers. If this local system is isomorphic to the trivial local system M x Z,then the manifold M is said to be oriented. In that case, there are twotrivialisations and each is called an orientation on M.

2.5. Remarks.

1) Any connected manifold is either not oriented or has two orientations.

2) Any orientation of a real vector space V (namely, a choice of a compo-nent of An(V) \ {0}), gives rise to an orientation on the differentialmanifold V.

3) If M and N are two differential manifolds, then the sheaf KMXN is thetensor product of the sheaves KM and KN pulled back to M x N.Also the operation of taking the product of two measures yields anisomorphism of the tensor product of 5M and 5N pulled back toM x N, with l3MxN. Hence we also get an isomorphism of local

2. Sheaf of Densities81

systems OZMxN ' pi(OZM) (9 p2(OZN). In particular, if M and Nare oriented, then so is M x N.

4) The local system OR of all flat homomorphisms is obviously OZ ®z A

2.6. Definition. The image of 1C in B by any nonzero flat homomorphismis called the sheaf of densities. We will denote the sheaf of densities by SMor simply S.

By definition then we have the canonical isomorphism IC ®z OZ ^ S.In the case of domains in R', sections of S are simply measures of the

form f dx where f is a differentiable function and dx is the Lebesgue measure.Hence in a manifold these are measures which in any coordinate system canbe expressed as f dx as above.

2.7. Remark. It will turn out as a consequence of a theorem of Sobolevthat S is the same as B (see [Ch. 8, Remark 2.6], but we do not need it here.

Since our definition is intrinsic, it follows that if U and V are domainsin R, and cp a diffeomorphism of U with V, then the pull-back of a densityhdx is again of the form gdx. The formula making this explicit is called `thechange of variable' formula and will be derived presently (Corollary 2.9).

Suppose that U is a manifold with an etale map into an oriented manifoldM (for example Rn). Then the tangent sheaf on M pulls back to the tangentsheaf on U. Hence the sheaf 1C of n-differentials also pulls back to 1C. It isalso easy to check that S on M pulls back to S on U. In particular, thestandard flat isomorphism on M gives rise to a flat isomorphism on U. Inother words, any orientation on M induces an orientation on U. (Is it alsotrue that if U is oriented, then M is oriented?)

Let it : U --> M be a continuous family of etale maps of U into M.Formally this means that there is a differentiable map i : (0, 1) x U -* Mwhose restriction it to each {t} x U is an etale map of U into M. Then it givesan orientation on U for each t. Since t varies continuously, it is natural toexpect that all these orientations on U are the same. To check it rigorously,we proceed as follows. Consider the differentiable map (0, 1) x U --+ (0, 1) x Mtaking (t, x) to (t, it(x)). This map is clearly an etale map into (0, 1) x M.Thus we see that the pull-back of OZ(o,1)xM comes with a trivialisation.Its restriction to each {t} x U is the local system of twisted integers on U.Hence the orientation on (0, 1) x M gives rise to one on each of {t} x U.Since the fibres of the local system are discrete, all these sections coincide,i.e. all these orientations are the same.


2.8. Proposition. If co : U -- V is any diffeomorphism of a domain inRn onto another, then cp is compatible with the orientations on the two opendomains if and only if the sign of det(dco), at one (and hence every) pointof U is positive.

Proof. We may assume, using a translation of U, that 0 E U and then bycomposing by a translation of V, that co(0) = 0. Let L(cp) be the restrictionto U of the linear map of 11 into itself given by the differential dco of co at0. Then we have a family of maps U -3 Rn given by cot = tL(co) + (1 - t)co.Clearly these maps are all etale. At t = 0 this gives co and at t = 1, itis L(cp)IU. From this we deduce that the orientations on U induced by coand L(co) are the same. Since we have identified the orientations of a vectorspace V and the differentiable manifold V, our assertion follows from theobvious fact that a linear transformation of a vector space preserves theorientation if and only if the induced map on top exterior power preserveseach component, that is to say, it has positive determinant.

Since we know how a diffeomorphism of a domain U of Rn with anotheracts on An(T*), we can deduce how it acts on S.

2.9. Corollary (Change of variable formula). If co is a diffeomorphism ofa domain in l1 into another domain D, then for any function f on D withcompact support, we have

f f(cp(x))I det(dco)Idx = ff(y)dy.

Proof. Using the isomorphism K ® OZ -> S and the compatibility of themaps on the three sheaves induced by co with the above isomorphism, wecan compute the induced map on S by computing the induced maps on ICand OZ. Now the isomorphism given rise to on the tangent bundle by co isby definition dco. Hence the induced map on K = An(T*) is multiplicationby det(dco)-1. On the other hand we have computed the induced map onOZ above. Putting these together, we see that the induced map (which isthe pull-back of measures) is given by multiplication by I det(dco)I-1. Theformula claimed gives the image measure.

2.10. Remark. A trivialisation of OZ gives rise to a trivialisation of thelocal system OR. It is easy to see that if OR is trivial then OZ is so,as well. We could therefore have defined orientation as a trivialisation ofOR, considering two trivialisations equivalent if they differ by a positive realmultiple. Moreover, any trivialisation of OR leads to a trivialisation of theA-module A ® OR. Again, the converse is easy to see. Thus an orientationgives a canonical isomorphism K = An(T*) -* S. On the other hand, S


is always trivial (although there is no canonical trivialisation). In fact, wemay cover the manifold with a locally finite family of coordinate open sets(Ui), take the Lebesgue measure eui in each of them, and consider E cpi/ai,where (cpj) is a partition of unity for the covering. This gives a section of Swhich is nonzero everywhere. Thus M is orientable if and only if the bundleA? (T*) is trivial.

2.11. Examples.

1) We have seen that any domain in Il comes with a canonical orientation.2) If the tangent bundle of a differential manifold is trivial, then the canon-

ical bundle is also trivial. Hence the manifold is orientable. In par-ticular, any Lie group is orientable.

3) Consider a connected closed submanifold H of R'ti of dimension n - 1.In view of the exact sequence

0 -> TH -- TRn I H -* Nor (H, RT) -> O

and the fact that the normal bundle is one-dimensional and the middleterm is trivial, we conclude that An-1(TH) is trivial if and only if thenormal bundle is trivial. We recall that this is indeed the case. Thereis a differentiable function f such that all partial derivatives of f donot vanish simultaneously at any point of the hypersurface H givenby f = 0 [Ch. 2, Example 2.6]. The vector field v f = a onIl8' can be restricted to H f to give a section of TRn JH f. Now thetangent space at a point (al, ... , a,,,) E H j is given by the subspace{E yi a

tiff

: E yi a (a) = 0}. The vector v f is therefore nonzero atall points of Hf. The image of this section in the normal bundle isnonzero everywhere and gives a trivialisation. Thus we conclude thatH f gets a canonical orientation. Note that if we replaced f by -f,we would get the opposite orientation on H f. In particular, if f isthe quadratic polynomial E x? - 1, the hypersurface H f is the sphereSn-1. The vector field we have given is the outward radial vector atevery point of Sn-1 and as such it is never tangential to S"Let

M be a differential manifold and (U, x) a local coordinate system onit. Then U has a canonical orientation. If (V, y) is another local coordinatesystem, then we get two orientations on u fl V. They are related by ±1 (ifU f1 V is connected, which we shall assume). It is clear from our analysisabove that these orientations are the same if and only if the determinant ofthe transformation matrix is positive. From this we conclude the following.

2.12. Proposition. The local system OZ on a connected manifold M istrivial, i.e., the manifold is orientable, if and only if there exists a system of


coordinate charts covering M such that on the overlaps the transformationsof the coordinate systems have positive Jacobians. Moreover, any orientationgives rise to a special system of oriented charts as above.

2.13. Example. The real projective space, RPn, is orientable if and onlyif n is odd. In fact, we have a natural map Sn -+ fin. Noting that this isan etale map, we see that the pull-back of OR][$pn is ORsn which is trivialas we have seen above. The antipodal involution also acts on Orsn, If thislocal system is the pull-back of a trivial system on RJPn, then the involutionpreserves the trivialisation on Sn. Hence RPn is orientable if and only if theantipodal involution preserves the orientation on Sn. Using the radial vectorwe see that it preserves the orientation of Sn if and only if it preserves thatof Rn+1 Note that this map is given by (x1,. .. , xn+1) '-' (-xl, , -xn+l)which preserves the orientation in Rn+1 if and only if n is odd.

2.14. Exercises.

1) Show that RP2 cannot be imbedded as a submanifold of R3.2) Let V be a real vector space of dimension n and Grassk(V) the Grass-

mannian of r-dimensional spaces in V. When is it orientable?

2.15. Remark. The vector bundle S associated to S has an interestinginterpretation. For a E M, the fibre Sa is isomorphic to Sa/MaSa whereMa is the ideal of functions f vanishing at a. Any element p of Sa canbe represented by a measure h(x)dx in a coordinate chart (U, X 1 ,-- . , xn)around a. Consider the map which associates to p the invariant measureh(a)dx on the tangent space Ta(M). Note that since a choice of coordinatesystem (x1 x) has been made, there is a canonical basis (a a ),..., n

s

of Ta(M) and hence one can talk of the Lebesgue measure dx on Ta(M) aswell. If a different coordinate system (yl, ... yn) (again with a as origin)is used, then h(x)dx pulls back to h(cp(x)) I det( )I dy. Under our map it

induces the measure h(cp(0))1 det(a )(0)Idy on Ta(M). But this is simplythe pull-back by the linear transformation dcp (the differential Ta(M)Ta(M) of the map cp) of the invariant measure h(O)dx. Thus we have

2.16. Proposition. The vector space Sa, a E M, may be canonically iden-tified with the space of all invariant measures on Ta(M).

Proof. We have already shown that there is a well-defined map of the stalkSa of S at a, into the set of invariant measures on ,,,(M), namely that whichin any coordinate chart around a point a E M is given by h(x)dx -+ h(a)dx.Clearly MaSa maps to 0 under this map and induces an isomorphism of Sawith the space of invariant measures on Ta(M).


2.17. Remark. An important class of manifolds which are orientable is thatof complex manifolds. In fact, we may take the complex coordinate chartsas a special system of coordinates. Then on the overlaps, the coordinatetransformations are given by holomorphic transformations cpz(z) of an openset in C". Considering them as real transformations, we get the Jacobianmatrix

axi ayi7axi ayi

where 1;27 77;, are respectively the real and imaginary parts of cpz(z) and x2, yzare real and imaginary parts of zi. But since cpj are holomorphic, we have theCauchy-Riemann equations = and In order to compute

xa ayi V3 iits determinant, we treat the matrix as a complex matrix and conjugate itby

1 -

to obtaina + ay 0

0 a - 1

whose determinant, namely I det(aza)12, is clearly positive.

3. Adjoints of Differential Operators

In analysis, it is sometimes convenient to work with vector-valued measures,namely measures with values in a vector space. From a global perspec-tive, it is expedient to have a notion of a vector-bundle-valued density on adifferential manifold.

3.1. Definition. Let E be a differentiable vector bundle on a differentialmanifold M. Then a section of £ ®IC OR OR = £ ®,,q S is called an E-valueddensity.

3.2. Twisted forms.We have seen that if D is a differential operator from E to F, where E

and F are vector bundles, then we may also define an operator E®,C -> F®Lwhere L is any local system. All the operators we defined on forms, suchas exterior derivation, Lie derivative, inner product and the correspondingformulae are valid therefore for sections of the bundles obtained by tensoringa tensor bundle with L. In particular, we may tensor the de Rham com-plex by OR and get a `twisted de Rham complex'. Sections of the bundlesAi(T*) ® OR will be called twisted forms. Thus by definition, a density is atwisted n-form. We may also take an exterior product of a differential form


and a twisted form to get a twisted form, or of two twisted forms resultingin an ordinary form.

3.3. Theorem (Stokes). If µ is a density with compact support of the typeda where a is a twisted (n - 1)-form, then fm ,4 = 0.

Proof. Let (Ui) be a locally finite coordinate covering of M and (gi) a dif-ferentiable partition of unity with respect to (U2). If we set pi = d(gia), thenclearly E µi = d(E gia) = da = A. So it is enough to show that f µi = 0.In other words, we may assume that µ = da with support of a containedin a coordinate neighbourhood and show that fm µ = 0. If (xl,... , xn,) is acoordinate system, we have seen that the measure y may be represented byf dx where dx is the Lebesgue measure and f a differentiable function withcompact support. Having trivialised OR over the coordinate neighbour-hood, we may assume that a is given by E fidxl A ... A dxi A ... A dxm,.Hence da corresponds to the measure (E(-1)i-1a )dx, where fi are differ-entiable functions with compact support. Let us compute fu a dx. We mayassume that the support of fi is contained in an open cube with its closureC a subset of U. Then fu a dx = fc a = f (fi(a) - fi(b))dx' where dx' isthe Lebesgue measure in R'-1 and a, b the maximum and minimum of theith coordinates of points in C. But fi(a) = fi(b) = 0, proving our assertion.

If X is any vector field, the operator µ H -L(X )µ of S into itself, iscalled the adjoint adj (X) of X. We will compute adj (X) in local coordinates.Let (xl,... , xn) be a local coordinate system, and let X = J ai(x)&. Itis enough to compute the effect of adj (X) on the Lebesgue measure dx. Infact, using Proposition 1.10, we get

L(X)(dx) L (ai 8xi/-- I (dx) 09xi J dx

/and again,

aL(X)(fdx) _ (ai.t_ + f ax dx = axi(aif) dx.

(9xi

Hence if we trivialise S over U, i.e. identify differentiable functions fwith densities f dx, then adj (X) is the operator - E a om(ai). Incidentally,this shows that adj (X) is also a differential operator of order < 1.

We will next define for any D E DM, its adjoint as a differential opera-tor adj(D) : S --> S of order < 1. If D = m(f), then we define adj(D) to bemultiplication by f. If D = X E ?(M), then we have defined adj(D) above,namely the operator p, -L(X)(µ). We wish to extend this definition todifferential operators of arbitrary orders. In fact, the map D adj (D) is


to be an antihomomorphism of the algebra D(M) into the algebra D(S, S)in the sense that adj(D1 o D2) = adj(D2) o adj(DI). In view of the universalproperty of the algebra D we have only to show that

adj(X om(f)) = m(f) oadj(X),adj(m(f) oX) = adj(X) om(f),

adj (Xi o X2 - X2 o X1) = adj (X2) o adj (XI) - adj (XI) o adj (X2) .

As for the first equality, we have adj (X om(f)) (p) = adj (m(f) X +m (X f )) (µ)= -L (f X) (µ) + (X f) p = -f L (X) (µ) . The second equality is proved simi-larly. The last formula follows from Proposition 1.10. Since all these formu-las are only to be verified locally, one may also verify them from our explicitlocal computation of adj (X), given above.

3.4. Proposition. Let E, F, G be differentiable vector bundles on a dif-ferential manifold M. Then there is a natural isomorphism D --* adj(D)of E)' (E, F) with Dk (adj (F), adj (E)), where adj (E) = E* ® S. If D EDk(E, F), then adj(adj(D)) belongs to Diffk(E, F). With this identification,we have adj(adj(D)) = D. Moreover, if D1 E Dk(E, F) and D2 E D'(F, G),then adj(D2 o D1) = adj(Di) o adj(D2).

Proof. We note first that the map D --* adj(adj(D)) of Dk into itself is actu-ally an algebra homomorphism and so, in order to show that it is the identitymap, we have only to check that adj (adj (X)) = X for any vector field Xand that adj (adj (m(f))) = m (f) for all functions f. The latter is obvious,since adj(m(f)) is simply multiplication by f and so is adj(adj(m(f))). Itis enough to check the first assertion locally, and so we may assume thatX = -, where (xi,... , x,ti) is a local coordinate system. Then with theobvious trivialisation of S we have seen that adj (axa) _ - a

EE

and henceproving our assertions in the case E = F = A.adj (adj ((axi)))

TXT

The general case follows on remarking that the map Dk (E, F)Dk (E, F) given by D adj (adj (D)) is obtained by tensoring the map Dadj (adj (D)) on Dk on the right by F and on the left by E. The proof of thelast assertion is similar.

3.5. Remark. Since the sheaf S is isomorphic to An(T*) ® OR, it followsthat there is an identity between differential operators E ® S - F 0 S andthose from E ® An(T*) to F ® An(T*). Hence the adjoint might just aswell have been defined as above with S replaced by An(T*). However, ourinsistence on using S is explained by the following assertion.


3.6. Proposition (Integration by parts). Let D : E F be a differ-ential operator of order < k and adj(D) its adjoint. If s E E(M) andt E (adj(F))(M), at least one of them having compact support, then

IM(Ds, t) = f(s,adi(D)t).

Here we use the duality to define (Ds, t) and (s, adj (D)t) as elements ofS(M).

Proof. We will first show that this is the case when E = F = A. Theassertion is clear if D is of order 0, namely D = m(g) for some g E A(M).We will assume to start with that D is a vector field X. In this case,we have to show that if f is a differentiable function and p is a density,then f(Xf)fit = f f(-L(X)µ). But fL(X)(p) = L(X)(fjc) - (Xf)p byProposition 1.10, so that we are reduced to showing that f L(X) (f p) = 0.In other words, we need to show that if v is any density with compactsupport, then f L(X)(v) = 0. But L(X) = dtx + txd and since dv = 0,we have fm L(X)v = fm dtxv = 0 by Stokes' Theorem 3.3. Now we proveour assertion for higher order operators by induction on the order. We haveshown above that if D E Dl (M), then the formula holds. It is enough toprove it for operators of the form X. . . . Xk, where Xi E T(M). We havethen

f((Xl...Xk)f)P = f(X2...Xkf)(adi(Xl)P)

ffadi((xl...xk))(P)

as was to be proved. This concludes the proof when E = F = A.The general case is easily reduced to the above. Firstly, note that the

assertion follows easily from the above if both E and F are trivial. Secondly,if s has compact support K, then Ds also has the same support, and so bothsides depend only on tIK. Replacing t by cot where co is a function vanishingoutside a compact neighbourhood of K, we may assume that all the sectionss, t, Ds, D*t have compact support. Hence we may cover this support by afinite open covering (Ui) such that E and F are both trivial on each U.We may then use a partition of unity to write out both s and t as a sum ofsections whose support lie in one of the Ui's. This reduces the problem tothe case when both E and F are trivial.

3.7. Remark. The formula above could have been used to define the adjointof a differential operator. The advantage of our definition is that it is local,and it is a priori clear that adj (D) is also a differential operator. In any


case, adj (D) is uniquely defined by the above formula. For if D1, D2 are twooperators satisfying

f (Ds, t) = f(s,Dit), i = 1, 2,

then for any t E (9 S)(M), we have fM(s, (D1 - D2)t) = 0 for alls E E(M). By replacing s by f s, we see that the measure (s, (Di - D2)t) isitself 0. But this implies that (D1- D2)t = 0 by the duality between E andE*.

3.8. Examples.

1) Consider the exterior derivative d : Ai(T*) -+ Ai+1(T*). We mayidentify adj(A'(T*)) = Ai(T) ® An (T*) ® OR with An-i(T*) 0 OR.Thus the adjoint of d is an operator adj(d) :

An-i-1(T*) ® OR -*An-i(T*) ® OR. We claim that this is simply (-1)i+ld where d isagain the exterior derivative, now on twisted forms. In fact it wouldbe enough to check that if w is a twisted (n - i - 1)-form and a is ani-form, then

fm(da A w) _ (-1)'+' II a A dw.

But then da A w + (-1)ia A dw = d(a A w) so that we have only tocheck that f d(a A w) = 0, which is a consequence of Stokes' formula.

2) If X is a vector field on M, then the Lie derivative L(X) : Ai(T*) -Ai(T*) is a differential operator. As above, adj(L(X)) may be con-sidered an operator An-iT* 0 OR -* An-'T* ® OR. This is indeed-L(X), for we have only to check that

fML(X)a A w = fM a A (-L(X)w),

for all a E AiT* and w E An-iT* ® OR. In other words, we needto check that f L(X)(a A w) = 0. But then L(X) = dtx + txd andd(a A w) = 0 so that our assertion follows again from Stokes' formula.In this case, it is also easy to prove our assertion directly from thedefinition.

3.9. The adjoint of an operator with complex coefficients.The definition of the adjoint of an operator has to be slightly modified

when E and F are complex vector bundles. If D is an operator E --+ F, thenits conjugate D can be defined as an operator E --+ W. In fact, if s E E(M)then Ds is defined to be D. The conjugate of the operator F* ® AnT* ®


OR --* E* ® AnT* ® OR, which can be defined as in the real case, is calledthe adjoint adj(D) of D. It is an operator

F* ® AnT* ® OR = adj(F) --* adj(E) = E* ® ATT* 0 OR.

3.10. Symbol of the adjoint.If D is a differential operator E -* F of order < k, then its (kth order)

symbol is a vector bundle homomorphism E -} F®Sk (T). It maybe thoughtof as associating to any v E T,,*n a homomorphism o(D) : E -- F. Nowadj (D) is a differential operator F*®S --> E*®S of order < k. The symbol ofthe latter assigns therefore to any v E T, Z, a homomorphism F* ® S -+E* 0 S. We claim that it is essentially (-1)k times the conjugate transposeof o(D),,. In fact, S is a line bundle and so the symbol u(adj(D)) may equallywell be considered as assigning to v, a homomorphism F,* --; E. It is inthis sense that we claim that o-(adj(D)) = (-1)ko,(D),. To check this wefirst note that by the definition of the adjoint operator it is enough to checkthe assertion in the case E = F = A. Moreover, by the formula for symbolsof composites [Ch. 2, 7.2], it is enough to prove it for D = m(f), f E A(M),and D = X E T(M). In the former case, the assertion is obvious, whilein the latter case, the claim follows from our computation of or(L(X)) in[Ch. 2, 7.21].

3.11. Remark. Although S is globally trivial, we have not defined theadjoint of an element in V as an element in D. In fact, in order to do so, wewould have to choose a global trivialisation of S. After such a trivialisation,we may define the adjoint of a differential operator E -* F to be a differentialoperator F* -} E*. However, this definition of the adjoint would depend onthe chosen trivialisation of S. Moreover, we can choose a Hermitian metricalong the fibres of E and F to define isomorphisms of F, E with F*, E*,respectively, and actually define the adjoint as an operator F --> E. If we dothis, then we may compute the symbol of the adjoint as follows. If a linearmap o : E -* F ®Sk (T) is given, for every v E Tv , we get an associated mapo-v : E -+ F. Its conjugate transpose (using the Hermitian metrics alongthe fibres) is a linear map F -f E. We define the adjoint symbol adj(v) ofo to be the linear map F -> E®Sk(T*) given by setting adj(o), to be (-1)ktimes the conjugate transpose of a,,. Then our discussion above gives thecomputation of the symbol of the adjoint as saying o(adj(D)) = adj(v(D)).

Exercises

1) Let G be a connected Lie group and X a right invariant vector field.Compute L(X) (A) where u is a left invariant measure.

Exercises 91

2) If µ is a linear form on C,° which takes nonnegative functions withcompact support to nonnegative real numbers. Show that y is then ameasure.

3) Determine the image measure on a Lie group of a left invariant measureby the map x '- x2.

4) Show that the set of unitary matrices which have at least one eigenvaluewith multiplicity > 1 is a set of measure zero with respect to the leftinvariant measure on U(n).

5) Let T be the group of diagonal matrices in U = U(n). Consider the mapT x U/T - U given by (t, x) xtx-1. Write down the differentialform on T x U/T whose image in U is the left invariant measure.

6) Consider the diffeomorphism Ilg+ x S'z-1 - }W' \ {0}. Write downthe pull-back of the Lebesgue measure and write it down as an n-differential form.

7) Prove that the tangent bundle of any manifold, considered as a differ-ential manifold, is orientable.

8) Let G be a connected Lie group and H a closed subgroup. Find anecessary and sufficient condition for the existence of a G-invariantmeasure on G/H.

9) Show that there is no nonzero measure on CIP' which is invariant underthe action of GL(n + 1).

10) Using the Lebesgue measure to trivialise S over 1Rn, find the conditionfor a vector field E fi '9 to be its own adjoint.

Chapter 4

Cohomology of Sheavesand Applications

We have mentioned that sheaves are the gadgets which allow one to passfrom local data to global ones. The obstructions in this passage are kepttrack of, as in a log book, by the cohomology of the sheaves in question. Wewill study the definition and properties of cohomology in this chapter.

1. Injective Sheaves

Suppose F, 7 and F' are sheaves of abelian groups on a topological spaceX and we have homomorphisms .F' ---p and 2 -- .F". Then we say thatthe sequence

is exact if the associated sequence at the stack level, namely

x--+ Fx-+ 'X",

is exact, for all x E X.Note that to say that F --> -> 0 is exact means that at the stalk

level Fx -> 9x is surjective. In long hand, this is equivalent to saying thatif s E 9(U), where U is a neighbourhood of x, then there exist an openneighbourhood V C U of x and an element t of .F(V) such that the imageof tin 9(V) is resuv(s).

If .T is a subsheaf of 9 then one may define the quotient sheaf by takingthe quotient of the etale space of under the equivalence relation: vif they are in the same stalk OJa and v - w belongs to Ta. Note however

93

94 4. Cohomology of Sheaves and Applications

that although we can define a presheaf by associating to any open set U thegroup g(U)/F(U), it is not a sheaf in general.

1.1. Exercise. Consider the constant subsheaf Z over the real line. LetI be the subsheaf which associates sections which vanish at 0 and 1. Showthat the quotient presheaf defined above is not a sheaf.

Of course the sheaf associated to the quotient presheaf is the quotientsheaf as we have defined. This also gives an example of the fact that if

0-}.2''

is an exact sequence of sheaves on X, it is not true in general that theinduced sequence

1.2. 0- 1:'(X)-4F(X)-4 -T"(X)-+0

is exact (see [Ch. 2, 6.16] for another example). The problem is only at theright end of the sequence. More precisely we have the following.

1.3. Proposition. If 0 -+ P --> F -i Jc" is an exact sequence of sheavesof abelian groups over a space X, then the induced sequence

0 -> Y (X) - F(X) -p .F"(X)

is also exact.

Proof. For a section s of F' to vanish as a section of F, we must have thatsx E .fix is 0 for all x E X. But .FFx' -+ Fx is injective, which implies thatsx is also zero as an element of F'x and hence that s = 0. This proves theinjectivity of .:''(X) -+ F(X). If a section s E F(X) induces 0 on F"(X),then sx E Fx maps to 0 in F for all x E X. So sx belongs to F' for allx E X, in view of the exact sequence

0 -* --- FT, .

This means however that s is actually a section of F as claimed.

The cohomology of a sheaf measures the nonexactness of sequence 1.2.We first define some classes of sheaves for which the exactness is neverthelesstrue and will define cohomology in terms of the deviation of the given sheaffrom such sheaves.

1.4. Definition. A sheaf.F of modules over a sheaf 0 of rings on X is saidto be injective if any homomorphism of a subsheaf G' of a sheaf g into Fcan be extended to a homomorphism of the whole of 9 into F.


1.5. Definition. A sheaf F of abelian groups is said to be flabby (resp.soft) if any section of F over an open (resp. closed) subset of X can beextended to a section over the whole of X.

1.6. Proposition. Any injective sheaf is flabby.

Proof. Let U an open subset of X. We will construct in a natural way a,subsheaf .7u of 0 such that Hom(,7u, F) = .T(U), for any sheaf F. Then wemay interpret any section s of a sheaf I over U to be a homomorphism Ju -+1. If T is injective, this homomorphism can be extended to a map 0 --+ T,that is to say, the section s can be extended to the whole of X. Thus theproof will be complete if we show the following.

1.7. Lemma. i) Let 0 be a sheaf of rings on X and.F an 0-module overan open subset U C X. Then there exists a sheaf FU on X which restrictsto F on U and has zero stalks outside U.

ii) If 9 is any sheaf on X, then the obvious restriction mapHom(X, Fu, C) -p Hom(U, F, 9 U) is an isomorphism.

iii) If F is already the restriction to U of a sheaf S on X, then Fu is asubsheaf of S.

iv) In particular, taking S to be 0, we get a subsheaf 7U of 0 such thatHom(Jv, 91U) = G(U) for all 0-modules 9.

Proof. i) Define Fu (V) for any open subset V of X by

.Fu(V) = {s E F(U fl V) : support of s is a closed subset of V}.

V

Support of s

If W C V, then we can restrict s to u fl W. The support of thisrestriction is simply supp(s) fl W, which is a closed subset of W. Thismakes our assignment a presheaf. Since F is a sheaf on U, in order to checkthat FU is a sheaf, we have only to show that if (Vi) is an open covering ofV and s E F(U n V), then the support of s is closed in V if and only if thesupport of the restriction of s to U fl V is closed in V for every i. But this


is obvious. If V C U, then FU(V) _ .F(V) by definition. Hence J7UJU = F.On the other hand if x 0 U and sx c (Fu)x, then sx is represented bys E F(U fl v) with V a neighbourhood of x. The support of s in u n v isclosed in V and its complement N contains x. The restriction of s to U n Nis zero. Hence sx = 0, proving that (.FU)., = 0 for all x ¢ U.

ii) It is obvious from i) that the restriction map in ii) is injective, fora homomorphism f :.Fu -* Q satisfying fx = 0 for all x E U is clearly 0everywhere. To prove surjectivity, suppose f : J -* Q U is a homomorphism.Then f yields homomorphisms ..T(U fl V) -* Q(U fl V) for all open sets V.If a section s of .F over U fl V has support closed in V, then so has f (s).Hence f (s) can be viewed as an element j -(s) of Q(V) obtained by settingf(s)x=0 ifxEV\U and f(s)x= f(s)xif xEUf1V. It is easy to see thats H f (s) gives a homomorphism FU -+ Q which restricts to f.

iii) If.F = S I U, then the identity map of S I U gives rise to a map FU --+ Sby the bijection in ii), in which we take F = S. This is the inclusion whichmakes FU a subsheaf of S in this case.

iv) is a particular case of the above.

On the other hand, it is obvious that over a paracompact space X, flabbysheaves are soft. For, by [Ch. 1, 2.1], any section over a closed subset canbe extended to a neighbourhood and hence to the whole space if the sheaf isflabby.

1.8. Proposition. If F is a flabby sheaf, then an exact sequence

0-> F-+ A--*L3-+0of sheaves leads to an exact sequence

0-4F(X)-A(X)-p 23(X)-0.In particular, this is true for.F injective.

Proof. In view of Proposition 1.3, the crucial thing to check is that anysection s of 13 can be lifted to a section t of A. Consider the set of allpairs (U, r) where U is an open set and r E A(U) lifts s over U. This setcan be partially ordered by defining (U, -r) < (U', -r') if U C U' and therestriction to U of r' is T. It is obvious that it is nonempty (since Ax --> 13,is surjective) and forms an inductive set. Hence by Zorn's lemma, thereexists a maximal element (Uo,To). We have now to show that U0 = X. Ifnot, let x ¢ Uo. There exist a neighbourhood V of x and a section r of Aover V lifting s. We wish to show that there exists a section over Uo U Vlifting s and restricting to To. This would prove our assertion in view of ourmaximality assumption. But To and r may not coincide on U0 fl V. However,the difference ro - r is actually a section of F over U0 fl V, since both of


them map onto s (Proposition 1.3). Since F has been assumed to be flabby,this section can be extended as a section t of F over the whole of X. Wemay alter T by adding to it this extended section t of F, and obtain a newlift -rl of s over V. Now To, Tl coincide on Uo fl V, thereby yielding a lift ofs over U0 U V extending To, which contradicts the maximality of (Uo, To).Thus we must have Uo = X, proving our assertion.

More or less similarly, we have the following analogue for soft sheaves.

1.9. Proposition. If F is a soft sheaf on a paracompact space, then anexact sequence

leads to an exact sequence

0 -.F(X) -3 A(X) -313(X) -> 0.

Proof. As in (1.8), we have only to show that s E 13(X) can be lifted toA(X). Using the regularity of the space X, we see that there exist a family(Ci) of closed sets such that their interiors cover X, and also a family ofsections of A(X) over Ci, lying over s. We may assume that (Ci) is a locallyfinite covering. Consider unions of the sets Ci and sections over such unionsthat lift s. One can check that it is an inductive family and the proof iscompleted as in Proposition 1.8.

1.10. Examples.1) According to [Ch. 1, Proposition 3.13], the sheaf A of differentiable func-

tions is a soft sheaf. In fact, any sheaf of A-modules is itself soft. Moregenerally, any sheaf F of 0-modules (where 0 is a soft sheaf of rings)is a soft sheaf. For, if C is a closed set, U an open neighbourhood ofC and s E F(U), then we have to show that there exists s E F whichcoincides with s in a neighbourhood of C. Consider a section f of 0which is 1 on C and 0 outside a neighbourhood N with N C U. Thenf s is a section of A over U vanishing outside N and hence extendableto the whole of X.

2) Let X be a topological space in which every open set is paracompact.If we consider the presheaf S of singular cochains [Ch. 1, Example1.3, 4)], it is clear that the restriction map S(X) -+ S(U) is surjective.This does not prove that the sheaf S is flabby yet. However, [Ch. 1,Proposition 1.14] shows that S(U) -3 S(U) is surjective. Hence anysection s over U of the associated sheaf S can be lifted to S(U) andthen can be extended to 8(X). Its image in §(X) is an extension ofs, proving that 9 is a flabby sheaf.


1.11. Exercises.

1) The sheaf of holomorphic functions on the complex plane is not soft.

2) The constant sheaf Z over the real line is not soft.

3) The sheaf of differentiable functions over the real line is not flabby.

4) Over a discrete space any sheaf is flabby. Which ones are injective?

1.12. Corollary to Propositions 1.8 and 1.9.

i) If 0 -> F' -+ J- I" - 0 is exact and F, F' are flabby, then so is F".ii) If X is paracompact and .F, F' are soft, then so is F".

Proof. i) It is obvious from the definition that for any open subset U ofX, .):'I U is flabby as well. According to Proposition 1.8, any section s overU of F" can be lifted to a section of T over U. Since F is flabby this liftedsection can be extended to the whole of X. Its image in .T(X) of courseextends s, proving our assertion.

ii) is proved similarly using Proposition 1.9 instead.

2. Sheaf Cohomology

2.1. Proposition. If F is a sheaf of modules over a sheaf R of rings on atopological space X, then there exists an injective sheaf TO of R-modules ofwhich ,F is a subsheaf.

Proof. We will use the fact that the corresponding statement is valid formodules, namely any module over a ring is a submodule of some injectivemodule. For each x E X, let I° be an injective Rx-module containingF as a submodule. Define I°(U) to be the direct product nXEU I..0, forany open subset U of X. If V C U, then there is a natural projectionHZEU I° -' rlxEvl°, which is defined to be the restriction map. It is easilyseen that TO is a sheaf. The inclusions F --j I° give rise to a map rIXEUFx -'

JJXEUI°. Thus we get a homomorphism F I° by composing the naturalhomomorphism F(U) --* rIxEU.Fx with the above inclusion. It is easily seenthat I° is a sheaf of R-modules and that this inclusion is a homomorphism.It remains to prove that I° is R-injective. Let S C T be sheaves of R-modules and f : S --} To an R-homomorphism. Then for every x E X wehave the map Sr -j I° obtained by composing ff with the projection ofI° -3 Ix. These can be extended to Rx linear maps T. - I° since I° isR,,-injective. Hence we get a map T(U) --> IIXEUTx - rHxEUI° = I°(U).It is easy to see that this gives a homomorphism of T into I° extending f.


If Jc' is any sheaf of R-modules, then we include.F in an injective sheaf,say

0->,F -* Z°-+,F1--+0.Then we may include F1 in an injective sheaf 11 to obtain an exact sequence

Proceeding thus, we obtain an exact sequence

where all the sheaves 12 are R-injective.We recalled in [Ch. 2, 6.9-6.12], the concept of complexes and cohomol-

ogy for abelian groups and remarked that the same go through for sheavesas well. We will formally give some of these definitions here.

2.2. Definition. A sequence F° 7G, of sheaves (of abeliangroups)

is called a complex of sheaves or simply a complex if the composite of anytwo successive homomorphisms is 0. The homomorphisms themselves arecalled differentials of the complex. All the differentials are usually denotedby d.

2.3. Definition. An exact sequence of sheaves

0 - - - + J° '71 ._...r y2 _* ... jn ...

is said to be a resolution of F , and the complexJ° : J O _+ j1 __+ '72 __ ...

is called the resolving complex. We denote this by

0-+ .F-+ .7°.

If in addition, all the sheaves Ji are injective, we say that it is an injectiveresolution.

2.4. Remark. The idea is that for many purposes, the sheaf F may, withadvantage, be replaced by the resolving injective complex. A complex ofsheaves is of course a more complicated object than the sheaf itself, but ifits components are all injective, then the resolving complex is in a sensebetter than the sheaf F.

We just showed that any sheaf of R-modules admits an injective resolu-tion. The way we constructed the injective resolution is quite arbitrary, andso for the idea to have any chance of success, we need to prove some kind ofuniqueness for the resolution.


2.5. Definition. A morphism of a complex ..'° into another complex G° isa sequence of homomorphisms J into G', commuting with the differentialsin the sense that the diagrams

commute.

2.6. Proposition. Let 0 --+ T -+ J° (resp. 0 - 9 -* Z°) be any resolutionof F (resp. an injective resolution of 9). Then any R-homomorphism f :Y --} 9 can be lifted to a morphism W : J° -3 Z° of complexes such that thediagram

is commutative.

Proof. Firstly, the construction of a map cp° :.7° --41° making commuta-tive the diagram

If two

is straightforward. For, since Z° is injective, the map F --* 1° obtained bycomposing f with the inclusion 9 -} Z°, can be extended to a homomor-phism WO : ,7° -i Z°. We can successively construct cpZ this way. For, let ussuppose that cps : ,7' -> I has been constructed for all j < i - 1 in such away that the diagram

0

0

'P X70

IfIwo

is commutative. Note that the image of J4-2 in .J 1 is mapped into theimage of Z'-2 in Z2-1, thanks to the commutativity of the diagram above.By the exactness of the horizontal sequences, this is the same as sayingthat the kernel of ,7'-1 ---> J is mapped into the kernel of T'-1 Ii.Again this implies that the map go'-1 induces a map of im(,7'-1 - J') intoim(Z2-1 -* Ii). Now this map of Im J-1 into Z' can be extended to a mapgo' : ,7' --p Z, in view of the assumption that Z' is an injective sheaf. Thisproves our assertion.


There is of course no uniqueness about the extension of f to W. However,by an argument similar to the above, we can also prove the following state-ment of `uniqueness up to homotopy' about the extension cp. Homotopy isa topological notion and we will give its definition soon (4.5). At this point,we simply give the algebraic analogue and work with it.

2.7. Definition. Two morphisms f, g : Z° -. 9° of complexes are said tobe homotopic if there exist (for all i) homomorphisms ki : Zz -> Jz-1 suchthat

dokz+k''+lod= fti - gz.

2.8. Exercises.

1) Let K° and L° be two complexes. Then define K°®L° by setting its ithcomponent to be Ej+k=Z Ki ® Lk and the differential to be the mapinduced by (a, b) H da ® b + (-1)i a ® db for (a, b) E Ki ® Lk. Checkthat this defines a complex and that there are natural homomorphismsEHi(K°) ® Hk(L°) -> Hj+k(K° (9 L°).

2) Consider the complex D° consisting of two members D° = R ® Rand D1 = R. The differential is given by (a, b) b - a. Let £°be the complex consisting of only one member £° = R. Note thatthere are two morphisms D° -> £°, which map (a, b) to a and b,respectively. Consider the tensor product complex 3 ® D and showthat morphisms f, g as in 2.7 are homotopic if and only if there existsa morphism h : Z° --> 3 ® D such that the composites of h with thetwo projections 3° ® D° -3 3° ®£° = 3 are f and g.

2.9. Proposition. Let 0 -* Y -+ 3° and 0 --+ 9 --> 2° be resolutions, withZ° injective. If cp, ' : 3° -+ Z° are morphisms of complexes such that

is commutative, then cp, 0 are homotopic.

Proof. Similar to that of Proposition 2.6.

Since we have defined the notion of morphisms, we also have automat-ically a notion of isomorphism of complexes. Although two injective reso-lutions of a sheaf need not necessarily be isomorphic, we have the followinguniqueness statement.


2.10. Corollary. If 0 -p F - 1, 0 -- J1 --3 J° are two injective resolu-tions of a sheaf F, then there exist morphisms cp : Z° --} J' and V : 3° --> Z°of complexes inducing the identity on .F such that cp o o and '% o cp are bothhomotopic to the identities.

Proof. The identity map F --p F extends to morphisms cp : Z° --+ 3°and 0 j° --+ Z° by Proposition 2.6. On the other hand both cp o V) and0 o cp extend the identity maps and so are homotopic to the identities byProposition 2.9.

We remarked at the beginning of this chapter that exactness of a complexdoes not imply that the corresponding sequence of sections is also exact. Thenotion of cohomology of a sheaf arises when we replace a sheaf by its injectiveresolution and then take the sequence of sections. The latter is no longerexact, but is nevertheless a complex of abelian groups.

2.11. Definition. Let F be any sheaf and 0 -> jc' -* TO an injective reso-lution of.F. The cohomology groups of the complex

z°(x) -> Z1(x) , ... zi-1(x) P(X) , Zi+1(x) - .. .are called the cohomology groups Hi(X,F) of the sheaf Y.

2.12. Remarks.

1) Definition 2.11 is meaningful since any two injective resolutions 1°, J'of F are homotopically equivalent and therefore yield homotopicallyequivalent complexes Z(X)°,,T(X)° by (2.10).

2) Proposition 1.3, applied to the exact sequence 0 -+ F - Z° - Zl, givesthe exact sequence 0 - F(X) -* 10(X) -> Zl (X ). By definition,H° (X, .'F) is the kernel of the map Z° (X) -* 11(X). Consequentlythere is a canonical isomorphism of H° (X, F) with F(X).

3) If F is itself an injective sheaf, then a resolving complex Z° is given byTo = F and Zi = 0 for all i > 1. Hence we conclude that Hi (X,.F) = 0for all i > 1.

2.13. Definition. If f :.F -* 9 is any homomorphism, the induced ho-momorphism Hi(f) : H'(X, F) , Ht (X,9) is defined as follows. Let0 -+ F ---> 3°, 0 --+ G -+ Z° be injective resolutions. If cp : J° --> Z° isan extension of f as in 2.6, then Hi(cp(X)) : Ht(J(X)) -> Hi(Z(X)) isindependent of the resolutions 3, Z according to Proposition 2.9 and this isdefined to be Hi (f) .

Let 0 --> F1 -+ F2 --; F3 --> 0 be an exact sequence of sheaves. Thenwe can actually construct injective resolutions 0 Fi --> Z°, i = 1, 2, 3, in


such a way that we have an exact sequence

0-+Zl -4 12 -*2 --*0of complexes fitting in a commutative diagram

0 0

1 1 1

2.14. 0 - ,F 1 - J 2 -> F3 --> 0

1 1 1

0 -* Zl -+ Z2 -* Z3 -* 0

The procedure that we will adopt for such a construction is the following.Choose inj ective resolutions 11',13' arbitrarily. Then we will construct acomplex 12 to fit in the short exact sequence. For every i, take the sheafT2 = Tj ® T3. Since Zl are all inj ective, any exact sequence

0-+T-->7-,2--*0necessarily splits, and we are led to the above choice of Ij any way. Theinclusion Zl -- Z2 and the surjection I -+ T3 are the obvious ones. We havenow to define the differentials of the complex I. Note that if we make (I)a complex that fits in diagram 2.14, then the exactness of 0 -- F2 -} E20follows from the exactness of the complexes 0 -* F1 -* 11 and 0 -> F3 -- 13'.For, we have only to check this at the stalk-level and there it follows by the`5-Lemma'.

Thus the only thing to be checked is that we can define differentialmaps T 2+1 to fit in diagram 2.14. In the following we will work atthe stalk level and will leave it to the reader to verify that the inducedmaps on the etale spaces are continuous. We will fix a point and denote allthe stalks of sheaves at that point by the corresponding Roman letters. Thedifferential has to be defined by d(x, y) = (dlx+ fi(y), day) for x E If, y E I3,where d1 and d3 are the differentials of the complexes Ii and 13' and fi isa homomorphism II -> Ii+l. The only constraint on fi results from therequirement that d2 = 0. This yields

0 = d2(x, y) = d(dix + fi(y), day) = (dix + di fi(y) + fi+ld3y, (123Y),

or dlfi+ fi+ld3 = 0, since we know that d2 = 0 = d2. To define the (fi),we proceed inductively, noting that in order to define fi+i, we have only toensure that fi+1 I d3 (I3) is given by

fi+l (y) = -dl fi (x) where d3x = y.

We will check that y -+ -d1 fix, where x is any element of 73+1 with d3x = y,is a well-defined map im(d3) -_+ I1+2. Then it would follow that fi+1 can bedefined to be any extension of this map as a homomorphism I3+1 . Pwhich exists since Ii+2 is injective. If d3x = d3x' then by the exactness of I3


we see that x - x' = d3a for some a E I3. Then dl fi(x - x') = dl fi(d3a) _-dl(di fi_la) by our inductive assumption. Thus dl fi(x) = dl fi(x') and ourproof is complete. In particular, we obtain

2.15. Proposition. If 0 -* .27' --+ F -> F" --i 0 is an exact sequence ofsheaves, then we have a long exact sequence of cohomology groups:

0 -> H° (,r'') - } H° (F) -' Ho (Fig) --+ Hl (F') , Hl (.F) -a Hl (.F")

... --> Hi (Y:') ---> Hi (,F) , Hi (F") , Hi+l (Fi) , .. .

Proof. Let Z'°,Z° Z"° be injective resolutions fitting in an exact sequence0 -* Z'° -> Z° -> Z"° -* 0 as above. Then from Proposition 1.8, we concludethat

0 - Z'(X)° -3 Z(X)° --> 1"(X)° -> 0is an exact sequence of complexes of abelian groups. Now our assertion fol-lows from the definition of HZ (X, .17) and the long exact sequence associatedto a short exact sequence of complexes which we recall below.

2.16. Remark. If0--> A°->B°-+ C° -+0

is an exact sequence of complexes, then the two morphisms induce homo-morphisms Hi(A°) --> Hi(B°) and Hi(B°) - Hi(C°). Since the compositeof the two morphisms above is zero, so is the composite at the cohorologylevel. More is true. The sequence

H'(A°) -* HZ(B°) Hz(CO)

is exact. In fact if b E Bi is such that db = 0, and its image in Ci is dx withx E Ci'l, then lifting x to an element yin Bi, we see that b-dy maps to zeroin Ci. Hence it belongs to Ai. Moreover, we also have d(b-dy) = db = 0. Inother words, the cohomology class of b comes from a class in Hi(A°) provingthe exactness as claimed.

The more interesting fact is that there is also a map Hi(C°) - Hi+l(A°)making the sequence

Hi(BO) -a H'(C°) - Hi+l(Ao) -* H'+1(BO)

exact. We will just give this connecting homomorphism and leave out theactual checking of exactness, which is straightforward. Take an element c inCi with dc = 0, representing an element of H. We know that there existsan element b of Bi which maps to c. Moreover, db maps to dc = 0. Fromthe exactness of 0 -.+ Ai -+ Bi - Ci, it follows that x = db belongs to Ai+lMoreover, since dx is zero in Bi+2, it is also zero in Ai+2. Thus x represents

a cohomology class of A°. If we choose a different element b' mapping to c,

3. Cohomology through Other Resolutions 105

the difference b' - b belongs to A' and hence db' = db+d(b' - b) represents thesame cohomology class. This class depends only on the class of c and not onc itself. To see this, we have only to check that if c = dx for some x E C'-1,

then the class we have defined is zero. In fact, if we lift x to some y in Bi-1,

then dy gives a lift of c. Since d(dy) = 0, the cohomology class we associatedto c is zero. Thus we have given a homomorphism H'(C°) , H'+1 (A0).

But conventionally we take the negative of the above map as the con-necting homomorphism. Some rationale for this convention may be foundin Exercise 7) at the end of the chapter.

3. Cohomology through Other Resolutions

Although the cohomology of a sheaf was defined in terms of injective res-olutions, it is seldom practical to make computations of cohomology usinginjectives. The reason why we can make do with other resolutions is thefollowing fact.

3.1. Lemma. Let 0 --> F be an arbitrary resolution of the sheaf F.Suppose H'(X,Ji) = 0 for i > 0 and all j > 0. Then the complex 3(X)°has cohomologies canonically isomorphic to those of F.

Proof. In fact, our argument would actually yield the following more gen-eral statement.

3.2. Lemma. Let 0 -* F --> J° be an arbitrary resolution of the sheaf F.Suppose H' (X, ,7i) = 0 for i +j:5 n, i # 0. Then Hk (X, F) is canonicallyisomorphic to Hk (,7° (X)) for k < n.

Proof. We will prove the lemma by induction on n. We note first thatfor n = 0, the assumptions regarding vanishing of cohomology are vacuous.Thus what we have to show is that the kernel of J° (X) , J (X) is isomor-phic to H°(X,F) _ F(X) (Remark 2.12, 2)). But this follows on applyingProposition 1.3 to the exact sequence

0-->'r -->,7°->,71.Let us then assume the assertion valid for all n < m and prove that it holdsfor n = m as well. Let )C be the cokernel of the inclusion F -> ,7°. Thenby Proposition 2.15, we have an exact sequence

H'-1()7) -> H'-1(J°) -* Hi-1(K)

--> H'(F) -* H'(,7°) -* H'(1C) __> .. .

Take i = 1 to get the following exact sequence (since H'(,7°) = 0):

0-,Ho (F)-4Ho (9°)-->Ho(1C),H'(F)-0.


But then from the exact sequence

0->K j2

we conclude that there is a natural isomorphism of H° (K) with the kernelof ,71(X) --> ,72 (X ). Thus Hl (F) is isomorphic to the quotient of the ker-nel of the map ,71(X) -> ,72(X), namely H°(K), by the image of the mapH° (,7°) -* H° (K), or what is the same, image of ,7° (X) --f J'(X). In otherwords, H1 (X,.77) -- H'(,7(X)°). To prove that H'(X,.F) f-- Hi(,7(X)) for2 < i < m, we use again the long exact sequence. Since by assumption,Hi-1 (,70) = H'(,70) = 0, we deduce that there is an isomorphism Hi (F) ^'Hi-1(K). But we have a resolution 0 -+ K ,7° where we define the com-plex ,7° by setting ,7P = ,7'+' for j > 0 and keeping the same differentials.Moreover, we have Hi(X,,7i) = 0 for i + j < (n - 1), i 0. Hence by theinduction assumption we see that H'(X,K) f-- H'(,7(X)°) = Hi+1(,7(X)°)if 1 <i<m-1. Hence if2<i_<m, as was to beproved.

In order to use the above observation effectively, we need to find criteriafor the vanishing of H' (X, F) for all i > 1. We have already seen (Remark2.12, 3)) that this is true for injective sheaves. There are also other types ofsheaves for which higher cohomologies vanish.

3.3. Proposition. i) If .1' is a flabby sheaf, then Hi(X,.F) = 0 for alli > 1.

ii) If X is paracompact and .F is soft, then H(X, F) = 0 for all i > 1.

Proof. Let F -> I° be an inclusion of F in an injective sheaf and let 9 beits cokernel. By Proposition 2.15, we have a long exact sequence

... -+ H'(-P) ---> H' (21°) - . Hi (g) -a H'+1 (Y) -+ H'+1 (-To) .. .

If i > 1, then H'(10) = Hi+1(I°) = 0 and hence Hi+1(,F) is isomorphic toH'(g). But then both F and I° are flabby and hence so is G (Corollary1.12, i)). Thus we will be through by induction on i, if we can show that fora flabby sheaf F, we have Hl (X, F) = 0. But we have the exact sequence

0-*.F'(X)--3I°(X)---g(X)-*H'(X,F)--->H'(X,I°)=0.

The content of Proposition 1.8, namely that I°(X) --> G(X) is surjective ifF is flabby, is just that Hl (X, F) = 0. This completes the proof

ii) The proof is similar to i), using this time Corollary 1.12, ii).

One can get many refinements and generalisations of Proposition 3.3.We state one of them here without proof since it is quite straightforward.


3.4. Proposition. Let 0 - _T -3 1°, 0 -- )- --> J° be two resolutions ofsheaves satisfying the condition in Lemma 3.2 and f :.F -> !g, f : Z° -, J°be morphisms of complexes fitting in a commutative diagram

.F -+ Z°

Then the map f (X) : 1(X)° -+ J(X)° induces maps H'(X,Y) -* Hi (X,!9)according to the isomorphism in Proposition 3.3, and these coincide withH'(f)

4. Singular and Sheaf Cohomologies

In [Ch. 1, Example 1.3, 4)], we defined the group of A-valued singularcochains (where A is any abelian group.) We will digress here to use thisnotion and define the singular cohomology groups of a space with values inA and to sketch briefly some of its properties with a view to relating themwith sheaf cohomology, via Lemma 3.1.

4.1. Definition of singular cohomology.We recall [Ch. 2, 6.10, 1)] the definition of the singular complex and

singular cohomology of a topological space. If o- is a singular n-simplexin X, namely, a continuous map of the standard n-simplex An into X,then for each i, 0 < i < n, we can define a singular (n - 1)-simplex asfollows. Consider the map Fi : An-1 -* An obtained by restricting thelinear map R - Rn+1 mapping (in the increasing order) the standardbasis (eo, ... , en- 1) of W" into that of ][8n+1 missing only the ith element ofthe latter. More explicitly, we have

Fi (ej) = ej for j < i - 1, and Fi (ej) = ej+1 if j > i.By composing this map with a singular (n -1)-simplex o, o Fi,which may be viewed as the ith face of a. If A is any abelian group, an A-valued n-cochain associates to any singular n-simplex, an element of A. Forany A-valued cochain a, we define its coboundary da to be the (n+l)-cochaindefined by (da)(o-) = Ez ol(-1)ia(o- o Fi). It is then a straightforwardverification to see that d o d = 0. In fact, in this computation, terms of theform a(o o Fi o Fj) occur twice but with opposite signs and so cancel out.Since it only involves routine checking, we will not carry out the computationhere. We thus get a complex

S(X)°:0-Usox -+>SIX -+... Sx_--,SX 1 -r ...

This is called the singular cochain complex of X. Its ith cohomology is calledthe singular cohomology of X with values in A and is denoted Hi (X, A).


4.2. Remarks.

1) We can also define Sz (X) to be the free abelian group over singulari-simplices and define differentials Sz (X) -* S;,_1(X) by mapping anysingular simplex s to E(-1)'s o F. This defines a complex whosehomologies are called integral singular homology groups of X. Theintegral singular cochain complex S°(X) is then the dual of the sin-gular chain complex S° (X) with transposed differentials. If A is acoefficient group, then Hom(S°(X), A) is the singular cochain com-plex with values in A. Similarly we can also define a singular chaincomplex with values in A to be the complex S° (X) ® A. All these canbe understood in the context of complexes in general. If Co is a com-plex of R-modules (with differentials taking CZ into Ci_1), then we candefine complexes (C ® M)° and Hom(C, M)° for any R-module M.However the relation between the homologies, say, of Co and (C*)° isnot just taking the duals again. If A is a field, it is indeed so.

2) The singular complex is unwieldy and not amenable to computation.But it is good for establishing the basic properties. For computationalpurposes, one has to use other complexes which give cohomologiesisomorphic to singular cohomology. For a differential manifold the deRham complex turns out to be one such.

4.3. Exercise. Using the singular complex, compute the singular cohomol-ogy of the space consisting of a single point.

4.4. Effect of a continuous map.Suppose X, Y are topological spaces and f : X --4Y a continuous map.

Then any singular n-simplex a in X gives rise, by composition with f, to asingular n-simplex f o in Y. If a is an n-cochain in Y, then it gives rise toan n-cochain f *a in X by the prescription (f *a) (a) = a(f a). Clearly thisgives rise to a morphism of the singular cochain complex of Y into that ofX and hence a homomorphism H(f) : H' (Y, A) -; H' (X, A).

4.5. Definition. Two continuous maps f, g : X --+ Y of topological spacesare said to be homotopic if there exists a continuous map h : X x I -> Ysuch that h(x, 0) = f (x) and h(x, 1) = g(x) for all x E X. A space is saidto be contractible if the identity map is homotopic to a constant map intoitself.

4.6. Theorem. If f, g : X -* Y are continuous maps which are homotopic,then H' (f) = H' (g) for all i > 0. In particular, a contractible space has thecohomology of a point.


4.7. Remark. The above result that homotopic maps induce the same ho-momorphism on singular cohomology groups may be thought of as a `prin-ciple of continuity'. Since the maps ht : X -> Y obtained by restricting hto X x {t}, depend continuously on t, the induced maps Hi(ht) may alsobe expected to depend continuously on t. The objects Hi being discrete,these maps remain constant under continuous deformation. In particular,HZ(ho) = HZ(hi)

Proof of Theorem 4.6. Let io, it : X -> X x I be continuous inclusionsgiven by x (x, 0), x H+ (x,1). Since h o io = f, and h o it = g, we haveio o h* = f * and ii o h* = g*. Hence it is enough to show that io and itinduce the same maps on cohomology groups.

We will prove this by showing that the induced maps on the singularcomplex of X by the two inclusions are homotopic in the sense of Definition2.7.

In order to give such a homotopy, we note first that for any singularn-simplex in X, we have a canonical singular `prism', namely a continuousmap An x I -* X x I. We first decompose the `prism' An x I into pieceswith an identification of the pieces with the standard (n + 1)-simplex. Letus denote the `lower' vertices (ei, 0) of On x I by eo,... en and the `upper'vertices (ei,1) by eo, ... , en. For each i with 0 < i < n, we consider theconvex closure of e6, ... , ei, ei, . . . , en. This piece can be identified with thestandard An+i by the restriction to On+1 of the linear map which takes thestandard base of I[8n+2 into e0',... , ei, ei, ... , en.

If or is a singular n-simplex in X, then we get a natural continuous mapv x I of An X I into X x I. For each 0 < i < n, we obtain a singular(n + 1) -simplex Pi (or) in X x I by composing v x Id : An x I -- X x I withthe natural (linear) map of An+1 into On x I given above.

For any singular (n + 1)-cochain a in X x I, we associate the n-cochainPa in X by setting (Pa)(v) = E(-1)ia(Pi(a)). An easy verification givesus the formula

4.8. dP(a) + Pd(a) = al - ao

where ao (resp. al) is the pull-back of a by the map x -> (x, 0) (resp. (x, 1))of X into X x I. This gives the required homotopy.

One can easily show that the singular cohomologies with coefficients inan abelian group A of a point, are given by Ho = A, Hi = 0 for i # 0.Hence any contractible space has the same cohomology. Since any vectorspace is contractible, h(t, v) = tv giving a homotopy between the identity


map and the constant map, we conclude that Rn has the same cohomologygroups.

4.9. Barycentric subdivision.If io, il, ... , it are distinct integers between 0 and n, then we will de-

note the centre of gravity or the barycentre of the points e0,. .. , ei, in lRn+1(where eo, ... , en is the standard basis) b y [i0, ... , ir]. If A is a permu-tation of (0, ... , n), then the linear map An -An given by sending eionto [A(0), ... , A(i)] will be denoted ba. The set of all bA where A runsthrough all permutations of (0,... , n), is to be viewed as splitting up Aninto (n + 1)! parts and identifying each piece with An. Now we define a mapb : Sn (X) --4Sn (X) by setting

b(a)(o-) = L(sgnA)a(a o ba).

Geometrically speaking, a singular simplex o- and the collection (a o bA)represent the same object and hence o- and b(a) are not very different. Thefollowing result is therefore only to be expected.

4.10. Proposition. The map a H ba is a morphism S°(X) S°(X) ofcomplexes which is functorial in X in the sense that if f : X -+ Y is anycontinuous map, then we have a commutative diagram

S°(Y) S°(X)lb lb

S° (Y) -* S° (X)

Moreover b induces the identity map on the cohomology groups.

Proof. If A is a permutation of (0,1, ... , n) and 0 < k < n-1, then it is easyto verify that b, o Fk = bAotk o Fk, where tk is the transposition (k, k + 1). Infact, both are maps An_, --+ An induced by linear maps 1[8n -> Rn+1 takingei, i < k - 1, into [A(0), ... , A(i)] and ei, i > k, into [A(0), ... , A(i + 1)]. Ifa is an (n - 1)-cochain, then by definition we have

((bod)(a))(a) _ (sgnA)(da)(o-oba)n

(sgn A) E(-1)ia(U o bA o Fi).i=0

From what we have seen, the surviving terms are (-1)n E,\ sgn(A)a(a ob,\ o Fn). Now it is easy to verify that ba o Fn = F),(n) o b- where A is thecomposite of A and the cyclic permutation (n, n - 1, ... , A(n)), treated as a


permutation of (0, ... , n - 1). Hence

((b o d)a)(Q) _ o o ba)

((d o b)a)(a).

This proves the first assertion. The functoriality of b is obvious from thedefinition. To prove the last assertion, we will give a homotopy operatorbetween the identity and b. This is done in the following lemma. It ismore convenient to work with the singular chain complex S° than with thesingular cochain complex S°. We recall (4.2) that Si is the free abeliangroup on the set of singular simplices and that the differential Si - Sz-1 isgiven by ds = E(-1)is o Fi. Also our definition of b can be interpreted asthe transpose of the map Si -> Si defined by bs = sgn(A) s o ba. It is theneasy to see that it is enough to check that b and Id on S° are homotopic.This is accomplished in the following lemma.

4.11. Lemma. To every singular n-simplex a we associate an (n-1)-chainka such that for any chain a we have a-ba = k(da)+dk(a). Here we denoteby k the linear extension to all chains of the map k defined on simplices. Inother words, b and Id are homotopie and in particular, b induces the identitymap on singular cohomology groups.

Proof. We will not give an explicit homotopy operator but will prove itsexistence by induction.

We will assume that we have constructed such an operator ki : Si (X) ->Si+1(X) for all i < n - 1, satisfying the equality

Id -bi = ki_ld + dki

not only for the given space X, but for all spaces at the same time, andthat the maps ki are functorial in the obvious sense. We will need then toconstruct similar maps kn. This is done as follows. Treat the standard n-simplex On as a topological space and consider Sn(An). There is a canonicalsingular n-simplex un on it, namely the identity map! We have to define inparticular kn(un). Our choice is dictated by the requirement un - bnun) =kn_1(dun) + dknun. Since un - bnun) - kn_1dun is already defined, sucha choice is possible if we show this to be a boundary. But then An iscontractible, and hence by Theorem 4.6, it is enough to check that it is acycle. We therefore compute d(un - bnun - kn-idun). We use now the factthat b commutes with d and the consequence of the induction assumption,namely dun - bn_1dun = dkn_1dun, to conclude that it is indeed a cycle.We therefore define kn(un) to be vn where dvn = un - bnun - kn_1dun.


We have however to define kn on all singular n-simplices of all topologicalspaces. But then un is universal in the sense that given any singular n_simplex s in X, treat the map s : On X as a map of topological spacesand consider the image of un in Sn (X) under the map of singular complexesinduced by s. This image is the composite of the identity map of On and s,and hence is s. We are thus forced to define kns (in view of the functorialityrequirement) to be the image in S,,_1(X) of vn E Sn_1(On) by the morphismAn -+ X induced by s. This proves our assertion.

Proposition 4.10 has lent substance to our claim that splitting up sim-plices to smaller ones does not change the homological picture. We haveonly carried out this systematically, with proper bookkeeping! We will nowshow how it can be used. If U = (U2)zEI is any open covering of X, then sup-pose one takes only singular simplices whose images are contained in someU,, (called U-small simplices) and builds up a singular complex with thosesimplices. The cohomology of the subcomplex should be no different fromthat of the full complex, because given any singular simplex, one can divideit as many times as necessary in order to make the subsimplices U-small.This means that we have

4.12. Proposition. i) Let U = (Ui)zEI be an open covering of X andS"(X) the complex of singular cochains defined only on simplexes with imagein some U. Then the natural surjection S° -+ (SU)° is a morphism ofcomplexes and induces isomorphisms on cohomology groups.

ii) The presheaf U H S(U) gives rise to a complex S° of sheaves on X.The natural morphism S(X)° -* S(X)° induces isomorphisms on cohomol-ogy groups.

Proof. Again we will prove the statement in the set-up of singular chaincomplex. The first remark is that for any singular r-simplex s, there existsN such that bNs is U-small. In fact, considering s as a continuous mapAr -> X we take the open covering s-1(Uz), i E I, of A,. Since the standardsimplex is compact, there exists a number 1 (called the Lebesgue number ofthe covering) such that any subset of diameter less than 1 is contained in oneof the sets s-1(Uz). It is easy to see that the diameter of all the barycentricpieces bN(0r) tends to zero as N tends to infinity. Hence for large enoughN the diameter of all these pieces is less than 1. This shows that bNs is alinear combination of U-small simplices, i.e. bNs E Su(X).

Since all the maps bN leave the subcomplex So invariant, assertion i)follows from the following algebraic statement.

4.13. Lemma. Let K° be a complex and L. a subcomplex, with K2 free andLZ generated by a subset of the basis. Let b : K° -> K° be a morphism which


takes L. into itself. Assume that there is a homotopy operator k between band Id and that k takes L into itself. Finally assume that for every x E Kithere exists N such that kNx E Li. Then the inclusion L C K is a homotopyequivalence.

Proof. It is clear that btm is also homotopic to the identity. Indeed, ifwe take km = (1 + b + b2 + + b''-1) o k where bi denotes the ith it-erated composite of b with itself, then dk, + k,,,,d = Em of bi(dk + kd) _Ei O1 bi(1- b) = 1- b'n. With this in mind, we define a homotopy operatork,,,, : K,. -j K,,_1 by setting (k,,x) = (kN(x)x), on basis elements of K,.,where N(x) is the least integer N such that bNx E L. Then we compute, asabove, the term (dk.+kcd-Id)x = (dkNixl+kN(x)d-Id)x-(kN-kw)dx tobe -bN(x)x - E aj (kN - k,,c,) (yj), where dx = E aj yj. Now (kN - km) yj =

EN(x-1 bikyj. But note that all the terms in the summation as well asz- (yj)

bN(x)x belong to L. Hence we can define a linear map cp : K --- L bysending x to (Id -dkm - kmd)x. It is obvious that it is a morphism of com-plexes. If x were already in L, we have that N(x) is zero by definition, andk... x = kx and similarly k,,.dx = kdx. Hence (Id -dk,, - k,),,d)x is simplybx. In other words, we have given a morphism cp K -* L such that itis homotopic to the identity on L. If we treat cp as a morphism of K intoitself it is clearly homotopic to the identity with k,,,, providing a homotopyoperator between it and the identity.

Proof of Proposition 4.12, ii). For every open covering U we have mor-phisms S° - (SU)° -> (S)°. We have just seen that the first map is ahomotopy equivalence. If V E Hi(S°) and c is a cocycle in S' (X) represent-ing it, we first lift c to an i-cochain c' in S(X) by [Ch. 1, Proposition 1.14].Since dc' maps to dc = 0, every x E X has an open neighbourhood suchthat dc' vanishes on it. Thus, although dc' itself may not be 0, there exists acovering U such that the image of dc' in (Su)i+1 is zero. In other words, theimage of c' in (Su)i is a cocycle. The class in H((Su)°) which it defines,maps to v. Since the map Hi(S°) -+> Hi((Su)°) is an isomorphism, thisshows that the image of Hi(S°) in Hi(S°) contains v. Since v is arbitrary,this map is surjective. If u E Hi(S°) maps to zero in Hi(S°), and x is acocycle in Si representing u, then its image in Si is of the form dv. Now vcan be lifted to a cochain y E Si-1, again by [Ch. 1, Proposition 1.14], andx - dy maps to zero in S1. As above, this implies that there exists a coveringU such that x - dy maps to zero in (Su)i, showing that the class in Hi(S°)defined by x maps to zero in H1((S")°). Hence the cohomology class of xin Hi(S°) is itself zero.


4.14. Theorem. Let X be a topological space which is locally contractible.Then the singular cohomology groups of X with coefficients in an abeliangroup A are naturally isomorphic to the cohomology groups of the constantsheaf A.

Proof. The first thing to notice is that we have a sheaf complex

A-So.'5 'S...-S5 ,.Si+1---+ ....

Secondly by 4.12, ii), the cohomology of the complex §U (X)' is naturallyisomorphic to that of the complex S(X)°. But by assumption, every x E Xhas a contractible neighbourhood U, which has therefore, by Theorem 4.6,the singular cohomology of a point, namely, H°(Ux, A) = A; Hi(UX, A) = 0for i > 1. This means that the sequence

0->A-4So (Ux)-->...-+

is exact and hence also

0 -+ A S°(Ux) - ... 5i(Ux) --+ si+l(Ux) ...

In particular, A --> S° is a resolution. By Example 1.10, 2), the sheavesSi are all flabby. Hence by 3.1 and 3.3, we may use this resolution for thecomputation of the sheaf cohomology of A. This proves our assertion andalso justifies our notation Hi (X, A) for sheaf cohomology.

5. Cech and Sheaf Cohomologies

In algebraic topology there is also another way of associating cohomologyspaces to a topological space, due to Cech. Like singular cohomology, it isalso defined a priori as having coefficients in an abelian group A. But itsdefinition is well adapted to having coefficients in a sheaf of abelian groups.Let us first give the definition, and later show how it is related to sheafcohomology.

Let U = (Ui)iEI be an open covering of a space X. Then we will constructa resolution F -+ C° of any sheaf F on X. For every U C X we will definein a natural way a complex C°(U) and a natural map .F(U) -+ do (U). Wewill do this for U = X and simply observe that for an arbitrary U, we haveonly to replace (U1)iEI by the induced covering (U f1 Uj)iEI of U.

For any finite sequence a : [0, 1, ... , r] -+ I we denote by I a I the integerr and by Ua the set n U ,(i). Then we define Cr = In orderto define the differential d : Cr --> Or+1, we define maps o'k : Cr -> 0r+1

for each 0 < k < r + 1 and set d = E(-1)kQk. To do this we need todefine maps 9ra o Qk : Cr --> .F(U,,) for each a with Ial = r + 1, whereIra : Cr+1 --+.F(Ua) is the projection. For each k, consider the (monotonic)

5. Cech and Sheaf Cohomologies 115

inclusion of [0, 1, . . . , r] in [0, 1, . . . , r + 1] which misses k, and compose itwith a to get ak with I ak I = r. Then Ira o Uk is defined to be the compositeof the projection d'-* .F(Uak) and the restriction F(Uak) --; .77(Ua).

It is easy to see that we have thus defined a complex. In particular, anyelement (' of C° is given by sections (si) of 17(Ui), i E I. If s is a section ofF then of course it gives rise, on restriction, to such a bunch (si) of sections.Moreover, d : C° -* C' can be described as follows. For C E C°, (d()i,j is thesection si - si over Ui n Uj. The sheaf conditions S1, S2 ensure precisely thatthe kernel of this differential is F(X). It is clear that all our constructionsare compatible with restrictions so that we have a complex Co of presheavesand a natural map F giving an exact sequence

0->F-do -, C1.

5.1. Proposition. The complex with respect to a covering U = (Ui)iEIdefined above, namely

is a resolution of F.

Proof. We have already checked the exactness of 0 -> F --+ do -* C'. Itis enough to show that for any x E X and u E Ci (X) with du = 0, thereexist a neighbourhood V of x and v E d'-'(V) such that dv = uIV. Now xbelongs to Ua, for some a E I. Then we will take V = Ua,. In order to definev E Cz-1(Ua,), we have to define va E F(Ua n Ua) for each a : [0, i - 1] --> I.We set va = where t is defined by /3(0) = a(j - 1) for all1 < j < i. Notice that Ua n Ua = Up so that our definition is meaningful.Then we have (dv)a = i(-1)kvak = E(-1)kupk, where /3k : [0, i] --+ I isgiven by Ok(0) = a, (ik(j) = a(j - 1) if j < k,fk(j) = a(j) if j > k. On theother hand, since du = 0 by assumption, we have in particular, du..,k = 0,where yk : [0, i + 1] -; I is given by -yk (0) = k, and yk (j) = a (j - 1) for1 < j < i+1. This gives the equality ua = E(-1)kuok, proving that dv = u.

5.2. Definition. Let F be a sheaf of abelian groups on a topological spaceX and U be an open covering. Then the complex C(X)° is called the Cechcomplex associated to the covering.

Applying 3.1 we get the following consequence.

5.3. Theorem. If F is a sheaf on X and (Ui)iE1 is an open covering suchthat for every i°i ... , i,. E I with Uio n n Ui,. 0 0 and j > 0, we have


H1(Uio,...,j,, F) = 0, then there is a canonical isomorphism of the cohomolo-gies of the Cech complex of F with respect to the covering and the sheafcohomology Hr(X,.F).

Proof. We have seen above that F -f C° provides a resolution. In orderto apply 3.1, we have only to show that Hi(X,Ci) = 0 for all i > 0. Butby definition, Ci is the sheaf U H fJjaj=i .F(UA). The components here are

simply the sheaves F., which are the extensions of Fl Ua to the whole of X,as in Lemma 1.6. From this it easily follows that, for all i > 1, we have

Hz(X,C3) -- II H'(X,.F'a) c H HZ(UA,.F) =0.XI=i IAI=i

One might write down explicit conditions on the covering in order toconclude that the cohomology of the Cech complex and the cohomology ofthe sheaf coincide in some range, using the more precise Lemma 3.2 insteadof Lemma 3.1. In particular, one may conclude

5.4. Theorem. Let (Ui)iEI be an open covering of X. Assume that Ui areall simply connected and that Ui fl Ui are all connected. Then H' (X, Z) canbe computed using the Cech complex. In other words, let c = (cii) be integerscorresponding to i, j E I with Ui fl Ui 0 0, and satisfying cii + cik = cik fori, j, k E I with Ui fl Ui fl Uk 0 0. The set of such c modulo those that satisfycij = di - di, where di are integers indexed by I, gives the first cohomologygroup.

5.5. Remarks.

1) We have defined above the Cech cohomology with respect to a covering.In most cases, we may choose a covering as in Theorem 5.3 and sothis suffices for practical purposes. Indeed we will show later (Ch. 6,Remark 1.16) that if M is a differential manifold, then there existsan open covering which satisfies the condition of Theorem 5.3 for theconstant sheaf. However, from the theoretical angle one has to freethis cohomology from dependence on the covering. That can be doneby defining the cohomology for all coverings, ordering all coverings byrefinement and passing to a limit.

2) When .F is a sheaf of nonabelian groups, we have no notion of cohomol-ogy. One can of course set H° (X, F) = T(X), which is a group. If(Uj)iEI is an open covering, one can also define the Cech cohomologyH' (X, F) for the covering as follows. Consider the set of assignmentssii E F(Ui fl Uj) for all i, j such that Ui fl Uj 0. Assume thatthey satisfy the equality siisik = sik on Uiik, for i, j, k E I withUi fl Uj fl Uk 0. Set (s) - (t) if there exist ai E Fi such that


aisi.7 = tjjo-j on Uij. The quotient set is defined to be H1(X,.F). Itis not a group, but a set with a special point, namely that defined bye = (eij), eij being the unit section over Uij. Again in principle, onehas to pass to a limit, but if the open sets are `good' enough, then onecan make do with a single covering. If M is a compact manifold, thiscan be taken to be a finite covering, so that all the abelian groups oc-curring in the complex for this covering are free of finite rank. Hencethe cohomology groups are finitely generated.

6. Differentiable Simplices; de Rham's Theorem

Consider the de Rham complex associated to a differential manifold [Ch. 2,6.13]. We have shown in [Ch. 2, 6.14] that the complex

0__T**A2T* -3...is a resolution of the constant sheaf R. We note that all the sheaves AiT*are A-modules so that they are soft sheaves by Example 1.10, 1). Hence byLemma 3.1 and Proposition 3.3, ii), we can compute the cohomology of thesheaf I[8 as the cohomology of the de Rham complex

0-->T*(X)-->A2T*(X)- A'T*(X)-->0

where n = dim X. Thus we have

6.1. Theorem (de Rham). The ith cohomology of the de Rham complexof a differential manifold is canonically isomorphic to the ith cohomology ofthe constant sheaf R.

Since we have already shown that the sheaf cohomology is isomorphicto the singular cohomology of the space, it follows that the de Rham andsingular cohomologies are isomorphic. We will give below an explicit iso-morphism between them. In order to do this, the first remark we will need isthat in order to compute the singular cohomology of a differential manifold,one may just use 'differentiable' simplices.

6.2. Definition. Let M be a differential manifold. A singular simplex s :A,, -> M, said to be differentiable if s can be extended to a neighbourhood ofA,, in 118''+1 as a differentiable map.

Following through the same constructions as in Section 4, one concludes:

a) A singular cohomology complex DSM based on differentiable singularsimplices can be defined.

b) The assignment of (DS)U' for every open set U in M and the naturalrestriction maps build a complex DS° of presheaves. This in turngives rise to a complex DS of sheaves.


c) The map A --> DS is a flabby resolution of the coefficient group A.

d) The natural morphism DS(X)° --> DS(X)° induces isomorphism oncohomology.

e) The natural morphism S° -> DS fits in a commutative diagram ofcomplexes of presheaves:

s°S.I 1

DS° -> DSAll these maps are compatible with the inclusions of A in each of thecomplexes. All sheaves in the above diagram are flabby.

In particular, we have

6.3. Proposition. The surjection S(X) -> VS(X) gives an isomorphismof cohomology groups and these are in turn isomorphic to H*(X, A).

Going back to our aim of giving explicitly the de Rham isomorphism,we will give a morphism from the de Rham complex DR° into DS° ratherthan into S°. By the precise isomorphism statement in Proposition 3.4,we have only to give such a morphism which is compatible with the re-spective inclusions of Il8 into DR and into DS in order to ensure that theinduced maps on cohomologies are isomorphisms. Let then w be a dif-ferential form of degree r. In order to associate to it, an R-valued dif-ferentiable singular cochain of dimension r, we proceed as follows. Lets : OT -+ X be a differentiable singular simplex. Then the pull-back ofw is a differential r-form a on (a neighbourhood of) A,.. To produce areal number, we will `integrate' this r-form over Ar and denote this byfs w. Let us now give a precise meaning to this integral. We will iden-tify OT with D, = {(X1,.. . , X,) E ]E8' E=1 Xz < 1, X2 > 0} by theprojection (Xo, X1, ... , X1,) -* (Xi, ... , X,,). Then a can be written asf (Xo,... , X, )dXl A A dX,ti. Then we integrate f with respect to theLebesgue measure, over Dr.

We have thus defined a linear map ATX -> DS' (X ).

6.4. Theorem (Stokes). The map ADT* -- DS' given by integration onsimplices is a morphism of the de Rham complex into the differentiable sin-gular complex.

Proof. What we have to show is that if w is an (r - 1)-form on M ands a singular simplex, then fs dw = E(-1)k Js.Fk w, or what is the same,

f s*(dw) _ E(-1)k hor-1(s o Fk)*(w).


The map w H s*w commutes with the exterior derivative d so that it isenough to show that if a is a form of degree r - 1 on Or, then

6.5. for da = >(-1)k for FF (a).

We identify Or with Dr and Or_1 with Dr_1. Then we have to prove 6.5with Or, Or_1 replaced respectively by Dr and Dr_1. Since both sidesof 6.5 are additive, there is no loss of generality in assuming that a =.f(Xi,...,X, )dX1 A ... AdXj A. .. AdXr.

We will first compute the face operators Fk as maps from Dr_1 to Dr.By definition, we have Fo(X1, ... , Xr_1) = (1 X. Xl, ... , Xr_1),and Fk(Xi, ... , Xr_1) = (X1, ... , Xk_i, 0, Xk) ... , Xr_1) fork > 0. HenceFo (dX1) = - E dXi, and Fo (dXi) = dXi_1 for 2 < i < r. On the otherhand, if k > 0, we have Fk (dXi) = dXi for i < k - 1, Fk (dXk) = 0, andFk dXi = dXi_1 for i > k.

It follows that FF (a) = -(f oFo)(E dXiAdX1A. AdXin AdXr_1) _1)i-1(f oFo)dXlA...AdXr_1. Fork > 0, we have Fk(a) = 0 unless k = j

and Fj*(a) = (f o Fj)dXl A. AdXr_1. Thus we see that the the right side of6.5 is equal to the Lebesgue integral over Dr_1 of (-1)x'1 f oFo+(-1)jf oFj.The left side of 6.5 is the Lebesgue integral of (-1)3-1 over Dr. Thetheorem therefore reduces to proving that if f is a differentiable function of(Xi, ... , Xr), then

fDr= -..,Xr-1) - f(X1,...,0,...,Xr-1))BXj D,_1

(f(i .i- 1

This is just the fundamental theorem of Integral Calculus after the observa-tion that

1

fDr-i f(X1i ... , 1-X, ... , Xr-1)

fD,f (1-X, Xl... , Xr-1)

is proves the precise version ofTh

6.6. Theorem (de Rham). The map which associates to each differentialform of degree r, the differentiable singular cochain obtained by integratingon simplices, induces isomorphisms of the de Rham cohomology groups withsingular cohomology groups.

If f : X -* Y is a differentiable map of one differential manifold intoanother, then f induces a morphism of the de Rham complex of Y into thatof X. We will denote this map by f *. Obviously the de Rham map given


above is compatible with the induced morphism of the singular complexesas well. In particular, we have the following fact.

6.7. Proposition If f and g are homotopic differentiable maps X -+ Y,then the induced maps f*, g* on the de Rham cohomology groups are thesame.

6.8. Exercise. Prove the above proposition directly on de Rham cohomol-ogy, without going through singular cohomology.

We have defined singular cohomology and proved some of its properties.The above considerations say that when the coefficient group is Ilk or C, andthe space is question is a differential manifold M, the singular cohomologygroups can be computed using the de Rham complex. In other words, onecan access the topological information contained in the singular cohomologygroups in terms of analytic data. But the de Rham complex is still quite bigand not very computation-friendly. We will see [Ch. 9, Theorem 2.6] thatwhen we endow M with a Riemannian metric, de Rham cohomology can becomputed more efficiently by analytical means.

6.9. Computation. The de Rham cohomology groups of M = S1 aregiven by

HO=R- H'=R; HZ = 0, for all other i.

Proof. The complex is given by

0 -> A(M) --* T*(M) 0.

Since M is connected, we have already noticed that the equation df = 0for a differentiable function f implies that f is a constant, which proves thecomputation of H°. Writing S' as a quotient of IR by Z, we note that dx isan invariant form on IE8 and so defines a 1-form dx on S' which is nonzero atall points. Hence any 1-form a on S1 can be written as f dx. Since we haveautomatically da = 0, the only question to settle is when it is of the formdg = d dx. In other words, we need to know which functions f are of theform dg/dx. Clearly this is true in lid and the solution for g is unique up toa scalar factor. In particular, if a is the form dx, we get x as the solution inJR for the above equation. But x is not invariant under Z and so does not godown to a function on S'. This shows that Hl 0. In general the solutiong on JR is given explicitly by f0 f (t)dt. This is invariant under translationby 1 if and only if fo f (t)dt = 0. In other words, the map a H fm a inducesan isomorphism of the first de Rham cohomology space with R.


6.10. Remark. The last statement above is indeed a general fact. Considerthe map of the space of densities on an r-dimensional compact, connecteddifferential manifold into R given by µ F-+ f A. Note that all densities arecocycles in the twisted de Rham complex. Moreover, by Stokes theorem,if a density is a coboundary, the integral vanishes [Ch. 3, Theorem 3.3].Hence we get a linear map from the de Rham cohomology space H?(M)into R. It is obviously surjective since if we take an everywhere positivedensity the integral is nonzero. It is a fact that this linear form is actuallyan isomorphism. We do not prove it here, but it is a consequence of each ofmany theorems in Chapters 7 and 8.

6.11. Invariant de Rham complex.Suppose G is a Lie group acting on a differential manifold. Then it is

clear that the exterior derivative of a G-invariant form is again invariant.Hence invariant forms define a subcomplex of the de Rham complex, whichwe call the invariant de Rham complex.

Let G be compact. Then we take the positive invariant measure on Gwhose total measure is 1, and average any form w over the group. Thuswe get a new form by the formula 1(w) = f g*(w). By definition thisform takes on r vector fields X1, ... , XT the value f w(gX1, ... , gX,.). Itis clear that I (w) is itself invariant under G since x* (I (w)) = x*(f g*wdg) =f (gx)*(w)dg = I(w). The map w I(w) gives a morphism of the de Rhamcomplex into the invariant de Rham complex. If w is already invariant, weobviously have I(w) = w. Thus the above morphism is the identity on thesubcomplex of invariant forms.

If G is connected in addition, we would like to show that every de Rhamcohomology class is represented by a G-invariant form. This would provethat the inclusion of the invariant de Rham complex in the total de Rhamcomplex induces isomorphism on cohomologies.

For any g E G, we know by the homotopy theorem that any closed r-formw and g*(w) differ by a coboundary. Since the integral can be approximatedby a finite convex linear combinations of forms of the type g? (w), it followsthat w differs from these combinations by a coboundary. If we knew that insome sense the set of coboundaries is closed (and this will be shown preciselyin [Ch. 9, 1.10], then it would follow that I(w) is closed and defines the samecohomology class as w.

But we can argue more simply as follows. Let w be any closed r-form.Consider the singular cocycle defined by I(w) - w under the de Rham mor-phism. It is enough to show that this is a coboundary. This would followif it were zero on singular cycles. Accordingly, let c = E aici be a cyclewith cc;, singular r-simplices and ai E R. Then I ai J. ci (I(w) - w)) _


Eai fo. fc(g*(w) - w)dg. Now using Fubini's theorem, as well as the factthat g*(w) - w is a de Rham coboundary and hence the corresponding sin-gular cochain is also a coboundary, we see that the above integral is zero.

6.12. Theorem. The invariant de Rham complex can be used to computethe cohomology of a differential manifold on which a compact connected Liegroup G acts.

The immediate application that comes to mind is that of a compact con-nected Lie group acting on itself by left translations. Thus the cohomologycan be computed by using left invariant forms. In other words, consider thecomplex

The differentials are given by

dc(Xl,...,X,) = E(-1)i+ja l([Xi,Xi],X1,...,Xi...... j,...,Xn).

Incidentally, we have managed to compute the cohomology of the Lie groupin terms of this complex, which is defined purely in terms of its Lie algebra.

One can indeed go one step further. One might as well make G x G act onthe Lie group G by (g1i g2).x = g1xg2 1. This implies that one can actuallydo with the biinvariant de Rham complex. Note that the transformationt : x x-1 acts on this complex. This action is easy to compute. In fact,it acts as - Id on the tangent space at 1 and so as (-1)'' on the rth termof the complex. Let w be any biinvariant form. Then t*(dw) = dt*(w).But t acts as (-1)'+1 on the left and as (-1)T on the right side. Hence allthe differentials in the biinvariant complex are zero, and hence we have thefollowing computation.

6.13. Theorem. There is a canonical isomorphism of Hi(G,R) with thespace of adjoint invariant i-forms.

Proof. The only point is to check that the action on the right by G on leftinvariant i-forms is simply the ith exterior power of the adjoint action onAi(g*). But then this is obvious.

The above principle can be used to make many computations, and weshall leave some as exercises.

Exercises

1) Assume given a differentiable map h : M x I -* N, where I is theunit interval. This means that the map h can be extended to a dif-ferentiable map h' from M x I' where I' is an open set containing1. To any differential form w in N, associate A(w) = t a (h'* (w)).

Wt

Exercises 123

Note that A(w) is a form on M depending on t E F. Then definek(w) = fo A(w)dt and check that it gives a homotopy between themorphisms DR(N) ---* DR(M) given by ho and hl.

2) Show that if M and N are differential manifolds, then the tensor prod-uct of the pull-back of the de Rham complexes of M and N to M x N,can be used to compute the cohomology of M x N with coefficientsin R.

3) Let G be a compact connected Lie group. Show that the symmetricbilinear form (X, Y) - tr(ad X ad Y) on its Lie algebra is negative.If G has discrete centre, show that it is nondegenerate. (In this case,one says the group is compact semisimple.) Show that the first andsecond Betti numbers of a compact, semisimple group G are 0 but thethird Betti number is not.

4) Let G be a compact, connected Lie group acting on a differential mani-fold M. Assume that for every m E M, there is an element s E G suchthat m is an isolated fixed point and that its action on the tangentspace is - Id. Show that the odd Betti numbers of G are all zero.

5) Compute the Betti numbers of SU(2) by finding the biinvariant formson it.

6) Let C° be the complex over Z obtained by dualising a complex C°.Show that there is a natural map of the dual of the homology groupHi of the latter complex into the cohomology group of the former.Give an example to show that in general this is not an isomorphism.

7) To any complex C° we associate another complex C[1]° by C[1]i = Ci+1and d[1] = -d. If f : D° -* C° is a morphism of complexes, defineanother complex Cf whose ith term is Ci ® D[1]i and the differentialtakes (x, y), x E Ci, y E D[1]i to (dx + f (y), d[1]y). Check that this isa complex and that there is a natural exact sequence of complexes

0-;C°-+Cf-+D[1]°--+ 0.

If f is an inclusion, show that there is a natural morphism of C Y into(C/D)° and that the induced map Hi(C f) - Hi((C/D)°) followed bythe boundary homomorphism Hi((C/D)°) - Hi+l(D°) = Hi(D[1]°)of the exact sequence

0 --+ D° -->C°-* (C/D)°--;0

° .is the homomorphism induced by the morphism Cf --* D [1]

8) Let M be a complex manifold and N a closed submanifold. Writedown a sufficient condition, in terms of cohomology of sheaves, forany holomorphic function on N to be extendable to M.


9) Let F', F" be two R-modules where R is a sheaf of rings on X. Anextension of F" by F' is an R-module T which sits in an exact se-quence

0,F'- F-*F"-*0.Two extensions are said to be equivalent if there is an isomorphism be-tween the middle sheaves which is the identity on .T' and induces theidentity on):". Show that there is a bijection between the set of equiv-alence classes of extensions which locally split, and H1(Hom(,F'", F')).

10) Assume that a complex E° over C has only finitely many nonzerocohomologies. Define X(E°) to be E(-1)i dimH'(E°). Suppose that

0--+ A°-+B°-+ C°->0is an exact sequence of complexes over C, all of which satisfy theabove condition. Then show that X(B°) = X(A°) + X(C°).

Chapter 5

Connections onPrincipal and VectorBundles; Lifting ofSymbols

There are several angles from which the notion of connections can be lookedat. We will start with the definition in terms of differentiation of tensorfields and later discuss other points of view. Given a vector field we wouldlike to be able to differentiate any tensor field. We have already explainedwhy the Lie derivative does not serve the purpose. A linear connection isa rule by which such a differentiation can be performed. We will deal herewith the more general notion of a connection in any vector bundle E, whichallows one to differentiate sections of E with respect to a vector field. A littlelater, we will deal with the even more general notion of principal bundlesand connections on them. We will turn to linear connections in the nextchapter.

1. Connections in a Vector Bundle

Recall that a differential operator of order < 1 is of the form X + m(f) whereX is a vector field, and f is a function. Its symbol is X. The correspondingsymbol sequence is

1.1. 0 --> A -> Dl -> T -; 0.

125

126 5. Connections on Principal and Vector Bundles

Although D' is an A-bimodule, the maps in this diagram are homomor-phisms for either of the structures. The map D D(1) which associatesto X + M (f) the function f, gives a splitting of the above sequence. Thecorresponding splitting map of T to D' is just the inclusion of a vector fieldin Dl. Note, however, that this splitting is for the left A-module structureon Dz and not for the right structure.

The point of the above observation is the following. Let E be a differ-entiable vector bundle on a differential manifold. Consider its first ordersymbol sequence:

1.2. 0 --* Hom(£, A) -> D'(9, A) -p Hom(£, T) -* 0.

Note that by definition the middle term, namely D'(9, A), is Hom(.E, E)1)where D' is equipped with the right A-module structure. With this un-derstanding, the sequence 1.2 is obtained from 1.1 by applying Hom(£,.).Therefore the left splitting of 1.1 does not lead to a splitting of 1.2.

Being an exact sequence of vector bundles, the sequence 1.2 does split.One can take local splittings and patch them together by a partition of unityfor example. But, unlike the corresponding sequence in the case E = A,this sequence does not split naturally. As a consequence, we cannot give ameaning to the phrase: `a first order homogeneous operator from E to A'.

1.3. Definition. Let E be a differentiable vector bundle on a differentialmanifold M. A splitting of the first order symbol sequence

0 -> Hom(£, A) -> Dl (£, A) , Hom(£, T) -+ 0

is called a connection on E. When E is taken to be the tangent bundle, wecall it a linear connection.

As noted above, a connection always exists. According to the defini-tion, a connection is a section of Hom(Hom(£, T), Dl (£, A)) lying over theidentity automorphism of Hom(£, T). The homomorphism (which we willdenote by V, pronounced `nabla') of Hom(£, T) = £* ® T into D'(.6, A) canbe interpreted in many ways.

For example, it is an A-linear homomorphism T --+ £ ® Dl (£, A) _D' (£, £). The image of a vector field X in this interpretation is denotedVX. The symbol of VX is clearly X ®IdE. Fix a section s of E. Then bothoperators A -> £ defined by f -+ V X (f s) and f -+ X f ® s have the samesymbol. So f VX (f s) - (X f )s is an A-linear homomorphism of A into S.In other words, there is a section cos of E such that VX (f s) - (X f )s = f cpsfor all functions f. Taking f = 1, we conclude that cos = Vx(s).

Thus we have another formulation of our definition, namely,


1.4. Equivalent formulation. A connection in a vector bundle E is anA-linear map X Vx from T to Dl (E, £) satisfying (the Leibniz rule)

Vx(fs) = fVx(s) + (Xf)sfor all f E A(M).

1.5. Remark. A connection gives a rule of differentiation of sections ofE with respect to vector fields. If m E M, then the value of the sectionVxs at m (which is an element of the fibre Em) depends only on the valueof X at m. In order to see this we have only to check that if X vanishesat a point m of M, then Vxs also vanishes at m for all s. Now X can bewritten, at any rate locally, as E f2Xi where ft are functions vanishing at m.But then Vx = E ffVx; and so, (Vxs)(m) = E fz(m)(Vxjs)(m) vanishesat m for all s. Thus for any tangent vector v at m E M, the expression

makes sense as an element of E,,,,, namely (Vxs)(m), where X isany vector field with X72 = v. It is in this respect that the notion of Liederivative of a tensor by a vector field X was found deficient. For while thederivative at a point of a section will naturally depend on the section in theneighbourhood of the point, the value at a point of the Lie derivative of asection, depends also on the vector field in a neighbourhood.

1.6. Definition. Given a connection V on a vector bundle, we call Vx(s)the covariant derivative of s with respect to the vector field X X.

1.7. Remark. Another way to interpret a connection on E is as a sectionof T* ® £ ® Dl (£, A) = DI (£, T* (9£), that is to say a differential operatorof order 1 from E into T* ® E which takes a section s into the 1-formwith values in E given by X -*, Vx(s). Its symbol is a homomorphismT* ® E -* T* ® E. We may compute this as follows. Let X be any vectorfield and s a section of E. Contraction with X gives the operator Vx, whichhas X ® Id as its symbol. This shows that the symbol of the operator wedefined above is the identity homomorphism of T* ® E.

The image of a section s is called its absolute derivative and is denotedby dvs.

Notice that E = A has a natural connection, namely that for which thecovariant derivative of any function f with respect to a vector field X isgiven by X f . The absolute derivative is then the usual differential f H df.

Let V be a connection on a vector bundle E. Let F be any vectorbundle. Then V can be used to `lift' any first order symbol a : E -> T ® Finto a first order differential operator E --> F. In fact, we have only to takethe A-homomorphism & : T* ® E -4 F associated to a and compose it with


the differential operator dv : E --f T* ® E. Since the symbol of the latter isthe identity, the symbol of the composite is a.

Thus we have the following characterisation of a connection.

1.8. Proposition. Let E be a vector bundle. A connection on E givesrise to a lift of every first order symbol v : E -+ T ® F to a differentialoperator V, : E -* F with the property that if f : F - G is any vectorbundle homomorphism, then ((Id)T®flo = f oVQ- Moreover, if F = T*®E,then the natural symbol E --* T 0 T* ® E is lifted to do. Conversely, sucha consistent lifting as above, arises from a connection.

Proof. The last assertion follows by defining a connection V on E by liftingthe natural symbol E -+ T ® T* ® E into an operator E --* T* ® E andchecking that the given assignment of lift for any symbol a : E -* T ® Fcoincides with Vim.

1.9. A generalisation.We will now give a mild generalisation of the procedure of using con-

nections to lift first order symbols to differential operators. Suppose D isa first order differential operator F - + G with symbol o- : F -> T ® G.We have remarked [Ch. 2, 7.24] that D does not give rise to a naturaloperator from F ® E to G ® E since D is not A-linear. However, if weare given a connection V on E, we may use it in order to lift the symbolv ® (Id)E : F ® E -p T ® G ® E to such an operator. Indeed, for s E F(U)and t E £(U) we define D(s, t) = Ds ® t + (as, do (t)). Here ( , ) denotesthe contraction map (T ® G) x (T* 0 E) -} G 0 E. We will now show thatthis 1k-bilinear map, although not A-bilinear, is nevertheless A-balanced. Bythis we mean that for any function f over U, we have D(f s, t) = D(s, ft).In fact, the left side here is by definition D (f s) ® t + (a f s, dot). Notethat o- is A-linear and (, ) is A-bilinear. Hence D(f s, t) - f D(s, t) =(D (f s) - f D(s)) ® t. But then this is the same as (v(D)s, df) 0 t, withthe same contraction notation for the pairing (T (D G) x T* --* G. Nowwe have D(s, ft) - fD(s,t) = (a(D)s,dv(ft) - fdv(t)) = (a(D)s,df (9 t).This shows that the above map is balanced as claimed and hence induces anIR-linear homomorphism D :.F ®A £ ` 9 ®A S. In the course of our com-putation above, we have shown that r f (s 0 t) - f D(s 0 t) = (as, df) ® t =((o (9 (Id)E) (s ® t), df) or, what is the same, D is a differential operator offirst order from F ® E to G ® E with symbol a (9 (Id)E.

1.10. Examples.

1) Let us start with a vector field X. Then we may try to define theanalogue of the Lie derivative with respect to X, of tensors with valuesin E. For instance, one might try to define an operator T* ® E -


T* ® E, using the usual Lie derivative L(X) and a connection V onE. Then the above procedure gives us an operator which takes (a, t)(where a is a 1-form and t is a section of E) to

L(X) (a) ® t + (X 0 a, dot) = L(X) (a) 0 t + a ® Vx (t).

If w is a differential 1-form with values in E given by X * a(X)t,this means that (L(X)(w))(Y) = Xa(Y).t-a([X,Y])t+a(Y)Vxt =-a([X,Y])t+Vx(a(Y)t). We may also write this out as the formula:

(L(X)(w))(Y) _ ([X, Y]) +Vx(w(Y))

for all E-valued differential forms w of order 1.2) Consider the symbol o- of the exterior derivative, say from 1-forms to

2-forms. It is simply the natural homomorphism T* -3 T ® A2(T*).This can be tensored with (Id)E to give a first order symbol of apotential operator from T* 0 E to A2(T*) ® E. If E is providedwith a connection V, we can lift it to an operator which takes anyE-valued 1-form a to the E-valued 2-form which sends (X, Y) toVx(a(Y)) - Vy(a(X)) - a([X' 1'])-

1.11. Remark. We may thus use the same procedure to define the analogueof the exterior derivative at all stages, namely an operator Ai(T*) ® E --*Ai+1(T*) ® E. We will use the notation do for all these. However, we donot in general get an analogue of the de Rham complex this way. That isto say the composite of the operators A'-l(T*) 0 (E) --> Ai(T*) ® E andAi (T*) ® E -- Ai+l (T*) ® E need not be 0. Since the composites of thesymbols gives the second order symbol 0, it follows that the composite isactually a first order operator! It will turn out to be actually an A-linearhomomorphism as one can check directly. For example, when i = 1, thisis a homomorphism E --* A2(T*) ® E or, what is the same, an End(E)-valued 2-form, namely (d2)(X,Y)(s) = V ((dos)(Y)) -VY((do(s)(X)) -dps([X,Y]) = Vx(VY(s)) -VY(Vx(s)) -V1x,Y!(s). In other words, it isthe End(E)-valued 2-form V X o Vy - Vy o Vx - V [x,Y]. This form is calledthe curvature form of the connection. We will return to a detailed study ofthis form later.

1.12. Exercises.

1) Prove that the exterior derivative of exterior r-forms with values in avector bundle E provided with a connection V is given by

da(Xi, ... , Xr+l) = E(-1)i+lVxja(X1,... , Xi) ...) Xr+1)

+ (-1)i+ja([Xi,Xj],X1i...,Xi...... j,...,Xr+1).


2) Compute the Lie derivative of a differential form of degree r with valuesin a vector bundle E.

1.13. Connections in associated bundles.Let VWWW, i = 1, 2, be connections in bundles E. Then one can define in

an obvious way a connection in El ® E2. For any vector field X one simplytakes the operator (s1, 82) H V(2)takes as the new connection.One can also define an induced connection on E = El ® E2 as follows.Corresponding to any vector field X, and sections sz of Ez, consider thesection of El (9 E2 given by

OX)s1 ®s2 + sl

It is easy to check that (f sl, s2) and (S1, f s2) give rise to the same sectionof El 0 E2. From this we conclude that this gives rise to an R-linear ho-momorphism of El ® E2 into itself. We define this to be the operator Vxon El 0 E2. One can also define a connection on the bundle Hom(Ei, E2).Indeed, if X is a vector field and f a homomorphism of El into E2, then weset

(Vx(f))(v) _Vxl (f(v)) - f(VX'(vl))In particular, if E2 is taken to be trivial with the trivial connection, thenone sees that a connection in a bundle gives rise to a connection in its dual.

We may take several copies of the bundles E and E* and use any givenconnection on E to define a connection on (&'(E) ®'(E*) by iterating theprocedure above. Moreover, symmetric (resp. alternating) tensors are leftinvariant under this extension, that is to say, a connection on E gives risein a natural way to connections on the bundle Sk(E) (resp. Ak(E)) as well.We will denote all these satellite connections by the same symbol V.

Thus the structure of a linear connection on a manifold, namely a connec-tion on the tangent bundle T, gives rise to a connection on all the associatedtensor bundles and hence allows us to differentiate tensor fields with respectto tangent vectors.

1.14. Connection as a C-module.A connection on E assigns to every vector field a differential operator

E -+ E. Can we also associate by iteration, natural higher order operators?If f is a function, that is to say, a 0th order operator, then of course wecould define f.s as the obvious product. Notice that this already gives, forevery differential operator D = X + m(f) E Dl, a first order differentialoperator E -> E, namely s -4 Vxs + f s. Moreover the resulting map Dl ->D' (E, E) is a .A-bimodule homomorphism, thanks to the Leibniz condition


we have imposed on the connection. Hence this can be extended as an A-bimodule homomorphism of Dl ® Dl into D2 (E, E). In other words, if X, Yare vector fields, we do get a second order operator E E by composingVX and Vy. Proceeding in this way, we obtain a homomorphism of theconnection algebra C into D(E, E). This makes £ a C-module. Indeed,any C-module, which is locally free as an A-module actually occurs in theabove way. So one can take a purely algebraic point of view and say thata connection is simply the structure of a C-module on a given locally freeA-module E of finite rank.

Notice that we cannot associate a differential operator E -+ E to everydifferential operator in D. The reason for this is that the above C-modulestructure does not in general go down to a D-module structure. Recall thatD is the quotient of C by the two-sided ideal generated by elements of theform R(X, Y) = VX VY - VYVX - [X, Y]. Obviously these elements have toact trivially on a module if it is to become a D-module. (Here VX denotesthe element in C corresponding to a vector field X.)

2. The Space of All Connections on a Bundle

If V and V are two connections on a bundle E, then by definition they arefirst order differential operators E --k T* ® E with the same symbol. Hencetheir difference is an A-linear homomorphism E -f T* ® E, that is to say,a differential form with values in End(E). In other words, there exists a1-form a with values in End(E) such that VX - Vx = cx(X) for all vectorfields X.; One might say that the space of all connections is therefore anaffine space based on the vector space of all 1-forms with values in End(E).We will now make a digression to make precise the notion of an affine space,based on a vector space.

2.1. Definition. An acne space A based on a vector space V, or a V-affinespace, is a set A together with a simply transitive action of the vector spaceV on it. If v E V and a E A, then we denote the action of v on a by v + aand refer to the map a H v + a as a translation.

2.2. Remark. This only formalises the intuitive idea that the affine spaceis `the same as a vector space, except that it has no origin' ! The simplestexample that one can think of is the following. Consider the set of allinstances of time. Then it does not have any `canonical origin'. On theother hand all periods of time do form a one-dimensional vector space, clearlyacting on the former simply transitively.


2.3. Examples.1) Any vector space acts on itself by left translations. It is obvious that

this action is simply transitive and makes it possible to consider avector space as an affine space based on itself.

2) If0-+ V'->V-+ V"-+ 0

is an exact sequence of vector spaces, then the set of all splittings ofthis sequence is an affine space based on the vector space Hom(V", V').

3) Let V be a vector space and f a nonzero linear form on it. Then thespace A = {v E V : f (v) = 1} is an affine space based on the vectorspace N = ker(f). In fact, it is clear that N acts on A by translationsand makes it an affine space.

4) In the above example, the restriction to A of the natural map V\ {0} -*P(V) gives a natural bijection onto P(V) \P(N). Hence P(V) \P(N)is an affine space based on N. The affine space is thus imbedded inthe projective space as an open set. Another way of putting it isthat the projective space is the compactification of the affine space,the points at infinity constituting a projective space of dimension oneless.

Suppose E is a vector bundle. Then a differentiable automorphism A iscalled a gauge transformation. One can transform a connection V on E bya gauge transformation A, which we denote by A*(V). This is by definitiongiven by

2.4. (A*(V))x(s) = A(Vx(A-1s))

There are now two connections on E, namely V and A*(V), and so theirdifference is a 1-form with values in End(E). It is indeed easy to computethis. Use the associated connection on End(E), and write (Vx(A))(s) =Vx(As) - A(Vxs). Substituting for s the section A-1(s), we see thatVx - (A*(V))x takes s to (Vx(A) o A-1)(s). Thus this difference is the1-form do (A) o A-'. We will write it up as a formula.

2.5. Formula. The difference V - A*(V) between a connection V and itstransform by a gauge transformation A, is given by the 1-form

X - Vx (A) o A-'

with values in End(E).

To describe the nature of this action, we again need to digress in orderto define the notion of an afi.ne transformation.


2.6. Definition. An affine map T of a V-affine space S into a V'-affinespace S' consists of a map T of the set S into S', and a linear map l(T) :V -* V' such that T(v + a) = l(T)v + Ta for all a E S and v E V. Thelinear map l(T) is referred to as the linear part of the affine map.

2.7. Remark. The linear part of A is uniquely determined by the mapS --> S'. The composite of two affine transformations is again affine and thelinear part of the composite is the composite of the linear parts. An affinetransformation is invertible if and only if its linear part is invertible.

2.8. Examples.

1) The translations given by an element w E V is itself an affine map S --S with linear part (Id) v, since we have w + (v + a) = Id(v) + (w + a),for all v E V and a E A.

2) If W is a vector space and V is a subspace of codimension 1, thenany automorphism A of W which leaves V invariant, gives rise to atransformation of P(W) \ P(V). It is a routine matter to check thatthis is an affine transformation.

3) Let V, V be vector spaces. Then an affine map of the associated affinespaces is simply a linear map V -> V' followed by a translation.

2.9. Exercise. What can one say about an affine map whose linear part iszero?

2.10. Definition. The group of all invertible affine transformations of anaffine space S, is called the affine group and is denoted GA(S).

In particular if V is a vector space, then GA(V) makes sense. Let S bea V-affine space. Then the set of all affine maps of S into the affine spaceV is a vector space under addition. Moreover, it is a Lie algebra under theoperation [(T, 1), (T', l')] = (l oT' - l' oT, l o l'- P o l). It is denoted by ga(S).If S is a vector space V regarded as an affine space, we will denote it byga(V).

2.11. Remark. We have natural exact sequences

1->V- GA(S)->GL(V)-*1,0--+ V -- ga(S)-->gl(V)->0.

When S is the affine space V, then there is a natural splitting of these twosequences.


The reason why we have interposed all this here is that if A is a gaugetransformation of a vector bundle E, then its action on the space of connec-tions on E is an affine transformation.

2.12. Proposition. Any automorphism of a vector bundle E induces anautomorphism of the space of differential 1-forms with values in End(E). Italso acts on the affine space of connections on E as an affine transformationwith the above action as its linear part.

Proof. The action of a gauge transformation A on T* ® End(E) is clear,namely (Id)T* ®Int(A), while its action on the space of connections has beendescribed in 2.4. Now our assertion is a consequence of the following identity,which follows from the definition of the gauge action on connections:

2.13. A*(a(X) + Vx) = A o a(X) o A-1 + (A*(V))x.

We may make things a little more explicit in the case when E is atrivial bundle. In this case there is a trivial connection on it so that anyconnection can be written as Vx (s) = Xs + a(X) (s), for any vector-valuedfunction s. Here a is an End(V)-valued differential form, so that a(X)is an endomorphism of the trivial bundle. Another way of expressing thesame thing is to say that dos = ds + a(s). Here a(s) is interpreted as thedifferential form X H a(X)(s).

Thus if E is any vector bundle, and V a connection on it, we maytrivialise E in a sufficiently small open set, and express the connection asd + a. If we take another trivialisation, it would differ from the above by agauge transformation A of the trivial bundle. Thus the new expression forthe connection is d + AaA-1 + (dA)A-1. (See formula 2.5.)

2.14. Proposition. If E is a vector bundle which is trivialised on eachset of an open covering (Ui), with transition functions (mij) on the overlapsUi n Uj, then any connection V on the whole of E is given by the operatorsd + ai on U. with the transition formulae

ai = mijajm-' - dmij.mijl = mjilajmji + m. 1 dmji

on Uij.

Proof. Choose the trivialisations ci : it '(Ui) -* Ui x C. The given con-nection goes over to the trivial bundle as d + ai. The transition functionsmij = ci o c3 1 over Uij are gauge transformations of the trivial bundle takingd+aj to d+ai. According to formula 2.5, it transforms the trivial connectionto d - dmij.rn-1. Hence it transforms d + aj to d - dmij.mzjl + mijajra-..


So the connection d + ai coincides with this transform if and only if theequation in the proposition is satisfied.

2.15. Remarks.

1) Gauge theory in Physics treats the space of all connections as a config-uration space with the gauge group as the symmetry group.

2) The action of the gauge group is not faithful. In fact, any nonzeroscalar acts trivially on the space of connections.

3) There is a natural group, which we call outer gauge group, containingthe gauge group as a normal subgroup of index 2 and acting on C byaffine transformations.

2.16. Exercise. If f is an everywhere nonzero function on M and it isconsidered as a gauge transformation of the trivial line bundle, then whatis the linear part of its action?

3. Principal Bundles

There is another point of view, which is useful and more general, to lookat connections. In order to explain it, we need to introduce the notion ofprincipal bundles.

3.1. Definition. Let M be a differential manifold and G any Lie group. Aprincipal bundle with structure group G consists of a differential manifold P,a fixed point free action (conventionally on the right) of G on P and adifferentiable map 7r : P --+ M such that the fibres are simply orbits underthe action of G. It is supposed to satisfy the `local triviality axiom', namely,every point m E M admits an open neighbourhood U such that the map7r : 7r-1(U) -> U can be identified with U x G -+ U together with theG-action on the second factor.

3.2. Remark. The example when P = M x G and it is the projection to Mand the action of G is given by (p, s)g = (p, sg) is called the trivial principalbundle. This justifies the terminology `locally trivial'.

3.3. Examples.

1) Take for P, the space R and for it the map of IR onto M = Sl given byx H exp(2irix). In this case G is the discrete group Z acting on IR bytranslations. This is a particular case of the universal covering spaceof a manifold M, on which the fundamental group acts by deck trans-formations. This gives a principal bundle with it (M) as structuregroup.


2) Let V be a vector space of dimension n and E = V \ {0}. The groupC" acts on it by scalar multiplication and the quotient is the complexprojective space P(V). Take for 7r the natural map of E = V \ {0}onto P = P(V). Let p be a point of E, namely a one-dimensionalsubspace p of V. If f is any linear form on V which does not vanishon p, then the open subspace V \ ker(f) of V \ {0} is invariant underthe action of Cx and defines an open subspace Uf of P containingp. Now 7r-1(Uf) = V \ ker(f) can be identified with the productf-'(1) x GX by the map v i (v/ f (v), f (v)). Since f-1(1) is mappeddiffeomorphically by 7r to Uf, we see that 7r is locally trivial and hencedefines a principal C"-bundle.

3) Consider the space of all r-frames in C'. An r-frame means here anordered set of r linearly independent vectors. This can be made intoa differential manifold, by identifying it with the set of all (n, r)-matrices of rank r obtained by writing out the r vectors as r columnsin any basis (say the standard basis) in (Ctm. This manifold is called theStiefel manifold. To each such matrix, we may associate an elementof the r-Grassmannian, namely the vector subspace generated by thecolumn vectors. The group GL(r, C) acts on the Stiefel manifold bymultiplication on the right, and the quotient is clearly the Grassman-nian. One can check that this gives a principal bundle with GL(r, C)as the structure group.

Local description.By definition, there is an open covering (Ui) of the space M and triv-

ialisations ti, : PI Ui -+ Ui x G. Over the intersection Uij = Ui fl Uj wethen have two trivialisations of P. These give rise to the automorphismAij = (res ti) o (res tj)-1 of the trivial bundle Uij x G over Uij. Thisis given by a diffeomorphism of the form (x, g) H (x, mij(x)g). This fol-lows from the fact that any map A of G into itself which commutes withright multiplication, is left multiplication by A(1). The functions mij satisfymij = mi1 and also, over Uijk = Ui fl Uj fl Uk, the identity

mij.mjk = mik.

The functions mij : Uij -+ G are called transition functions. If we choosesome other trivialisations t2 over Ui, then we get a different set m2 - oftransition functions. The two trivialisations differ by an automorphismcpi = ti o (ti)-1 of the trivial bundle over Ui. Again cp is of the form(x, g) H (x, fi (x)g) for some function fi : Ui -* G. It is clear that thetwo sets mij and mil are related, through fi, by the identity

mij fj = fimi)-

over Uij.


The above data can be interpreted in terms of sheaf cohomology [Ch. 4,Remark 5.5, 2)]. The set of transition functions defines a Cech 1-cocycle withcoefficients in the sheaf G of differentiable functions on M with values in G.As such it defines an element of H'(M, G). Note that G is a sheaf of groupsand not necessarily of abelian groups. Hence the first cohomology groupis only a set with a special point. Conversely, given a 1-cocycle (mij), wemay take the manifolds Ui x G, piece them together by the diffeomorphisms(X19) -* (x, mid (x)g) and define a manifold P. Now it is clear that thisforms a principal bundle. Thus one easily concludes that the set of principalbundles is in bijection with the cohomology set H'(M, G).

When G = C (resp. C"), the sheaf G can be identified with A (resp.Ax). Thus we may say that the set of line bundles is in bijection withHI (M, Ax). In this case, the cohomology H1 (X, A") is actually a group,and the group operation is given by tensor product of line bundles.

3.4. Example. Consider the principal C"-bundle given by the naturalmap 7r : Cn+1 \ {0} , CPn. For every i, consider the open set Ui defined byzi 0 0, and the trivialisation ti : 7r-1(Ui) - Ui x C" given by (z,, ... , zn)((zo, ... , zn), zi). The map ti o ti 1 ((zo, ... , zn), A) H ((zo,... , zn), zi/zj A)takes one trivialisation to the other. Hence the transition functions are givenby the C"-valued functions mi.7 = zi l zj on Uij.

Let 7r : P - M be a principal G-bundle and f : N -* M be a dif-ferentiable map. Then we may define the pull-back of P by f to obtain aprincipal G-bundle on N as follows. Consider the subspace of N x P givenby {(x, l;) : f (x) = 7r(e)J. The group G acts on N x P via its action on thesecond factor and this leaves the subspace invariant. If P were trivial, sayP = M x G, then this subspace would obviously be the same as N x G.In general, if P is trivial, say over U, then the pull-back is therefore trivialover f -'(U). We can put together these local trivialisations to get a dif-ferentiable structure on the pull-back to make it a principal G-bundle. Ifmil is a set of transition functions defining P with respect to a covering(Ui) of M, the pull-back is given by the transition functions mil o f, withrespect to the covering (f -1(Ui)) of N. Our present interest in principalbundles stems from the fact that if P is a principal bundle with structuregroup G, then corresponding to any representation p of G, that is to say, adifferentiable homomorphism of G into GL(V), we obtain a vector bundleE(P, p), called the vector bundle associated to P by the representation p.

In fact, suppose P is a principal G-bundle and p a representation of G in avector space V. Consider the manifold P x V. The group G acts on it by theprescription (p, v)s = (ps, p(s)-1(v)). Take the quotient to be E = E(P, p).There is a natural map of E into M which sends (p, v) to 7r(p). Moreover,


over a locally trivial open neighbourhood U of any point m E M, we mayreplace P by it-1(U) = U x G and hence E by the quotient of U x G x V bythe action of G given by (m, g, v) s = (m, gs, p 1(s)(v)). This quotient maybe identified with U x V via the identification (m, g, v) H (m, p(g)v). ThusE can be provided with the structure of a differential manifold by glueingthese together. One has of course to check that it is Hausdorff. But if (p, v),(q, w) are two points, then it is obvious that they can be separated by openneighbourhoods if 7r(p) ir(q). On the other hand, if 7r(p) = 7r(q), thenthey lie over one local trivial neighbourhood and so can be separated again.Now one can check easily that E is a vector bundle over M.

On the other hand, if E is a complex (resp. real) vector bundle of rank k,then one can associate to it a principal bundle with structure group GL(k, C)(resp. GL(k,1k)) in such a way that the vector bundle is associated to thestandard representation (namely the identity representation) is E.

We will now indicate this construction. Let E be any vector bundle. Aframe is a linear isomorphism of a fibre over any point, say m of M, withCk. Now take the set of all frames at all points of M. If E is trivial, the setof frames can be identified with M x GL(k, C). This is clearly a manifold.One can easily check that by locally trivialising E over an open covering(Uj), one can glue all these manifolds together and get a principal bundle Pwith the required properties.

3.5. Remarks.

1) Let us start with a vector bundle E. Construct the corresponding prin-cipal GL(k, C)-bundle P. Any representation p of GL(k, C), givesrise to the associated vector bundle E(P, p). Thus we have deviseda method of associating to a vector bundle of rank k and a repre-sentation p of GL(k, C), another vector bundle which we may as welldenote p(E). For example, we have already mentioned that if p is theidentity representation, then the associated vector bundle is E itself.If we take the r-fold tensor representation of GL(k), it is easy to seethat the associated vector bundle is actually ®'(E).

2) The association of a vector bundle to a representation can be carried outin greater generality. If the structure group G acts on any manifold F,we can associate a bundle over M which is locally a product with F.

3) Any representation of the fundamental group ir1 gives rise to a vectorbundle, namely the vector bundle associated to the universal coveringspace, viewed as a principal bundle with 7r1 as the structure group.Any covering space of a manifold is in fact a bundle associated to theuniversal covering space considered as a principal 7rl (M)-bundle forits action on the coset space modulo a subgroup H.


4) If F has any additional structure which is preserved under G, then thefibres of the associated fibration over M inherit this structure and thelocal trivialisations respect this structure. If p is a representation ofG, then the fibre is a vector space, and since the vector space structureis preserved under G, the associated bundle acquires the structure ofa vector bundle.

5) Consider the adjoint representation of a Lie group in its Lie algebra.It preserves the Lie algebra structure. Hence the associated vectorbundle, which is called the adjoint bundle (and is denoted ad(P)),carries a Lie algebra structure on all its fibres.

6) If G is a Lie group and H a closed subgroup, then the natural rightaction of H on G gives a principal H-bundle G --> G/H.

Remark 4) above may be applied to the following situation. Let P bea principal G-bundle and G --* H a homomorphism of Lie groups. We canthen associate a bundle with H as fibre, since G acts naturally on H on theleft. But the action does not respect the group structure on H so that thefibres do not come equipped with a group structure. However, if we makeH act on itself on the right by translations, then the left action of G on Hvia the given homomorphism, commutes with the right H-action on itself.Hence the associated bundle with H as fibre comes with an action on theright by H, and it is easy to see that this makes it a principal H-bundle.

3.6. Definition. Let f : G -* H be a homomorphism of Lie groups. If P isa principal G-bundle, then the above procedure gives rise to a principal H-bundle. It is called the bundle obtained by extension of the structure groupthrough f. We will often drop the phrase `through f, particularly when fis an inclusion.

3.7. Remark. Even if the map f is not injective, the phrase of extendingthe structure group is still used.

3.8. Example. Consider any principal bundle P with a discrete group Gas structure group. Then fixing a point m in M and a point p over it in P,we get a homomorphism of 7r, (M) into G as follows. Lift any loop at m toa path in P with p as origin. By the monodromy theorem the end point ofthe lift depends only on the homotopy class of the loop. Thus we get a mapf of 7rl (M, m) into G defined by mapping any homotopy class of loops atm onto the unique element g which takes p into the end point of the lift ofany path in this class.


3.9. Exercise. Assume that G is a discrete group and that P, a connectedtopological space, is a principal G-bundle over M. Show that the associatedG-bundle obtained by extending the structure group via f in the aboveexample is isomorphic to P.

If we start with a principal H-bundle Q and f : G -> H is a homomor-phism, there is no reason to expect that there is a principal G-bundle Psuch that Q is obtained from P by extending the structure group throughf . For example, take G = {1} and f to be the inclusion. Then any principalG-bundle is trivial, and so also is any bundle obtained by extension of thestructure group. But there is no reason why an arbitrary H-bundle shouldbe trivial!

3.10. Definition. Let f : G -+ H be a homomorphism of Lie groups. IfP is a principal H-bundle, then we say that the structure group can bereduced to G through f if there exists a principal G-bundle Q such thatP is isomorphic to the bundle obtained from Q through extension of thestructure group through f. If f is surjective we sometimes say that thestructure group can be lifted to G.

One may think of reduction of the structure group as providing thebundle with some additional structure. We will presently formalise this butwe will give some examples first. The claims made in these examples willbe proved in 3.14.

3.11. Examples.

1) Let E be a vector bundle and P the associated principal GL(k, I!8)-bundle of frames. To provide a metric on all fibres, i.e., a differ-entiable section of the associated bundle S2(E*), which is positivedefinite on each fibre, is equivalent to giving a reduction of the struc-ture group to O(k, II8). Such a structure will be simply referred to asa metric on the vector bundle.

2) In the above example, we may require, instead of positive definitenessof the symmetric form on each fibre, nondegeneracy with a givensignature, say, (k, 1). Here it is a question of reduction of the structuregroup to O(k, l).

3) Suppose that there is a nondegenerate alternating form on each fibre, ora differentiable section of A2(E*) which is nondegenerate on all fibres.It is clear that this is not always possible. For one thing, E has to beof even rank 2k, since there are no nondegenerate alternating formson vector spaces of odd dimension. But there are other topologicalobstructions as well. This is the same as giving a reduction of thestructure group to the symplectic subgroup Sp(k).


4) If E is a real vector bundle of rank 2k, then in order to provide it with acomplex vector bundle structure, what we need is a section of End(E)whose square is -(Id)E. This would give a reduction of the structuregroup from GL(2k, lE8) to GL(k, C).

3.12. Remark. There is a significant difference between Example 1) andthe rest in that, while all vector bundles do admit a positive definite metric,an arbitrary vector bundle may not admit reductions to other subgroupsmentioned above. As for 1), we note that if we trivialise the bundle locallyand provide the trivial bundle with a metric structure, then these can beglued by the use of a differentiable partition of unity, thanks to the fact thata convex combination of metrics is also a metric. This last assertion is notvalid for forms of other signatures. For example in the case of a bundle ofrank 2, the existence of a nondegenerate form of signature (1, 1) would implythat the fibres E, can be written as a direct sum of two subspaces, namelythe two isotropic subspaces for the form. The tangent bundle of S2 does notadmit such a form with signature (1, 1). In Examples 3) and 4), we need atleast that the rank of the bundle is even.

3.13. Examples.

1) Let G be a Lie group and H a closed subgroup. Consider the principalH-bundle G --> G/H. If we now extend the structure group to G,then the new bundle is given by G x G modulo the action of H onit by (x, y)h = (xh, h-1y). But this quotient can be identified withG/H x G by means of the map G x G -p G/H x G given by (x, y) H(xH, xy). In other words, the H-bundle G --> G/H which could wellbe nontrivial, becomes trivial on extension of the structure group fromH to G! We may also say that the trivial G-bundle has a reductionto H which is nontrivial. On the other hand the trivial H-bundle canalso be obtained by reduction from it. Thus two different reductionsof the same G-bundle to a subgroup H, may yield nonisomorphicH-bundles.

2) Suppose

0--+E'-* E-+E"->0is an exact sequence of vector bundles with rk(E') = k and rk(E") _1. Instead of considering all frames, namely isomorphisms of Em,m E M, with a fixed vector space, say IlSk+I, we may restrict ourselvesonly to those frames which take E,,,, into Rk. We get a principal bun-dle all right but the structure group is now the set of all linear auto-morphisms of IIBk+1 which leave the subspace Rk invariant. In other


words, the structure group has been reduced in this case to the so-called `parabolic' group, {g E GL(k+l, IR) : gij = 0, for i > k, j < k}.

3.14. Proposition. There is a one-one correspondence between reductionsof the structure group of P from G to a subgroup H and sections of theassociated bundle with G/H (with the obvious G-action) as typical fibre.

Proof. In fact, in this case, the associated bundle has a very simple de-scription, namely P/H -> M. Indeed, the map of P x G/H to P/H taking(p, gH) to the H-orbit of pg takes (ps, s-1gH) to the same orbit and goesdown to a map of the associated bundle as we have defined, onto P/H.The map which takes the H-orbit of p to (p, H) provides the inverse. It isobvious then that the map P -* P/H is itself a principal H-bundle. Nowany section s : M -> P/H can be used to pull back this H-bundle to anH-bundle on M. It is easy to check that the G-bundle obtained from it byextension of the structure group is simply P. Conversely, if Q is any reduc-tion of P to an H-bundle, then we have an isomorphism (Q x G)/H -* Pof principal G-bundles. Of course, this yields an isomorphism of associatedbundles with typical fibre G/H. But this is given on the left side by passingto a further quotient of (Q x G)/H by the right action of H. This is easilyidentified with the trivial bundle M x (G/H). Now the canonical section ofthis bundle gives a section of P/H.

3.15. Remark. The claim in Examples 3.11 is equivalent, after the aboveproposition, to giving sections of associated bundles with fibres GL(n)/O(n),GL(n)/Sp(n), etc. These reductions in the case of the tangent bundle willbe studied in Chapter 7.

In order to write down things in an analytical fashion, we often trivialisea bundle on a sufficiently small open set. The approach of principal bundlesallows us to do something similar. This is somewhat neater, and the glueingdata gets replaced by `invariance under G'. It is rather like taking the setof all trivialisations, instead of taking one trivialisation, and getting ana-lytic expressions in terms of each of them, with appropriate transformationformulae for a change of trivialisations. This is in some sense closer to theclassical use of `quantities' in physics.

Since the principal bundle consists of `all frames' (assuming it camefrom a vector bundle E), it follows that when E is pulled back to P itself,it admits a frame at every point, that is to say that E pulls back to a trivialbundle. We may rephrase this by saying that when the principal bundle Pis pulled back to the manifold P by 7r, then it is trivial. Although this isobvious, in view of its importance we will enunciate it as a proposition.


3.16. Proposition. The pull-back of a principal G-bundle it : P -+ M byit is canonically trivial.

Proof. This assertion is almost a tautology. Note that the pull-back is givenby the subspace {(PI, P2) : ir(p1) = lr(p2)} of P x P. The trivial principalbundle P x G on the manifold P is mapped isomorphically to the pull-backby the map (p, g) ' (p, pg).

Let p be a representation of G and E = E(P, p) the associated vectorbundle. If a is a section of E, then its pull back to the manifold P gives asection of 7r* (E). But then this is associated to 7r* (P), which is canonicallytrivial. Hence the pull-back of a to P may be considered a section of thetrivial vector bundle or, what is the same, a vector-valued function, on P.Conversely, let f be a V-valued function on P satisfying f (pg) = jo(g)-1f (p)for all p E P and g E G. Then the section of the trivial bundle taking p to(p, f (p)) of P x V induces a section u f of PIG = M to (P x V )/G = E. Wemay sum this up as follows.

3.17. Proposition. Let P be a principal G-bundle, p a representationof G in a vector space V and E = E(P, p) the associated vector bundle.Then there is a canonical isomorphism between the vector space of sectionsof E and the vector space of V-valued functions f on P that satisfy f (pg) _

p(g)-1 f (p) for all p E P and g E G.

In the same way the following statement may also be verified.

3.18. Proposition. Under the same assumptions as above, there is acanonical bijection between the space of differential r -forms on M with val-ues in the vector bundle E(P, p) and the space of differential r-forms w onP with values in V such that

i) w(vl, ... , Vr) = 0 if at least one of the vectors is vertical, i.e. mappedby it to 0;

ii) If w1i ... , wr are tangent vectors at pg obtained as translates by g oftangent vectors vl,... , yr at p, then w(wl,... , wr) = p(g)-1w(vl, , vr).

4. Connections on Principal Bundles

Let S be any manifold on which a Lie group G acts (on the right) simplytransitively. Then the tangent bundle of S is of course trivial, since G can beidentified with S on choosing a point so in S, by means of the map g sog.It is even canonically trivial. Indeed, we have an isomorphism of the trivialvector bundle S x g over S with the tangent bundle given by (s, X) -+ v,where v is the image of X by the differential of the map g -+ sg at 1.


In particular, there is a linear projection of the tangent bundle of S intog. This may be considered as a g-valued 1-form on S.

4.1. Definition. Let G be a Lie group and g its Lie algebra. Let S be anydifferential manifold on which G acts simply transitively on the right. Thenthere is a canonical 1-form on S with values in g, which associates to anytangent vector v at s E S, the unique vector X at 1 which is mapped by thedifferential of g i--> sg to v. This form is called the Maurer-Cartan form onS. In particular, if S = G with G acting on itself by right translations, it iscalled the Maurer-Cartan form on G. We will denote this generally by MG-

4.2. Definition. Let G be a Lie group and P a principal G-bundle. Thena connection form on P is a 1-form on the manifold P with values in theLie algebra g whose restriction to the fibres (which are all submanifolds ofP on which G acts simply transitively) are Maurer-Cartan forms and whichis equivariant for the action of G on 1-forms on P on the one hand and theright adjoint action on g on the other.

4.3. Remark. Any two connection forms on P differ by a 1-form on P withvalues in g which vanishes on vertical vector fields and is equivariant for theaction of G on P on the one hand and the adjoint action on g on the other.In other words, they differ by a 1-form on the base M with values in theassociated bundle ad(P) with g as fibre (considered as a 9-valued form onP). Thus the space of all connections on P is an affine space based on thevector space of ad(P)-valued 1-forms on M.

We will show that a connection form -y on a principal bundle P withstructure group G, gives rise to covariant differentiation on any vector bun-dle associated to a representation p of G in a vector space V. We will denotethe induced representation of the Lie algebra g also by p. As we have seenabove, sections of the associated vector bundle may be identified with func-tions f on P which satisfy f (sg) = p(g)-'(f (s)) for all s E P and g E G.We will first show the following.

4.4. Lemma. If v is any vertical tangent vector at p E P, i. e. 7r : P ->M maps v to 0, and f is a p-equivariant function with values in V, thenv f + p('Y(v))f (p) = 0.

Proof. Note that the map g -+ pg induces an isomorphism of G with thefibre through p, and the Lie algebra g with the space of vertical vectors atp. Therefore the assertion is equivalent to the following. If f : G -> V isgiven by f (g) = p(g)-lx, and X E g, then X f = -p(X)x, which is obvious.


This means that if v is now any tangent vector, and we consider themap f - v f + p(y(v)) (f ), then this value depends only on v modulo thevertical tangent space at p or, what is the same, only on the image by7r of v in the tangent space at m = 7r(p). Next we will investigate thedependence of this value on the choice of the point p over m. Supposethen that q = pg is another point of P over m. If w is the g-translateof v, then w f = v(h), where h is the function defined by h(x) = f (xg).Since f (xg) = p(g)-1 f (x) we have w f = p(g)-'v f . On the other hand,p('y(w))(h) = p(g)-1('y(v)g)(p(g)-1)f = p(g)-1('y(v)f). In other words,for any tangent vector v at a point m E M, we may define Vv(of) bytaking any lift of v at any point p E P over m and setting its value to bev f + p(7 (v)) (f) (p). We will skip the routine check that this does define acovariant differentiation. In sum we have the following theorem.

4.5. Theorem. Any connection form on a principal G-bundle, gives risecanonically to a covariant differentiation in every associated vector bundle.

In view of this theorem, the datum of a connection form on a princi-pal bundle is often said to be a connection on it.

4.6. Remark. Let 7r : P --+ X be a principal G-bundle and f : Y -* Xa differential map. Clearly there is a G-equivariant differentiable map f ofthe pull-back f * (P) into P which induces f on the base spaces. Therefore,if y is a connection form on P, we may take its inverse image f * (y) onf * (P). Since the fibre of f * (P) at y E Y is canonically the fibre at f (y) ofP, the restriction of f * (y) to this fibre is the Maurer-Cartan form. Also theequivariance of f * (y) with reference to the action of G on f * (P) and that ong follows from the G-equivariance of 1. Hence it is also a connection form.It is called the pull-back connection on f *(P).

4.7. Local expression.Consider the trivial principal bundle M x G. The connection form on it

corresponding to the trivial connection is p2(mG), where mG is the Maurer-Cartan form on G. Any connection form is given by Ad(g)-1 opiw+p2 (mG),where w is any 1-form on M with values in g. The form w which determinesthe connection is obtained by restricting the connection form to M x {1}.In other words, if s is the section x (x, 1) then w is the pull-back of theconnection form by s.

Let P be a principal G-bundle and y a connection on it. Let (Ui) be anopen covering and si sections over U2 given by trivialisations ti of the bundleover Ui. If li7 are transition functions given by the above trivialisations, thenwe have sj(x) = si(x).lij(x) over Uij. Note that the action map P x G -> P


pulls y back to Ad(g) o pi(ty) + p2(mG). Here the action by Ad(g) isby the action on g. The pull-backs wi, wj of y by si and sj respectively, aretherefore related by the formula

wj = Ad(1-1) o (wi) + l* (mG).

Compare this with the local expression for connections in vector bundlesgiven in 2.14.

4.8. Remarks.1) In order to define differentiation of a section along a vector at m, we

took an arbitrary lift at a point p over m in the above procedure.However, note that the differential yr, which is a linear map of Tp intog, maps the vertical subspace isomorphically to g, and hence that itskernel (which we will call the horizontal space at p) is supplementaryto the vertical space. So it is also mapped by it isomorphically toT,(M). It makes sense then to lift the given vector v in T,(M) toa vector in the horizontal subspace at p. Thus, given the connectionform, there is a canonical lift of any vector at m to any point in Pover it. Moreover, by the invariance property of y we also concludethat if p and q = pg are two points over m, the horizontal lift at q isobtained by translating the horizontal lift at p by g. From this it isalso clear that any differentiable vector field on M admits a uniquehorizontal lift, that it is differentiable and that it is invariant underthe action of G. Suppose then X is a vector field on M and hXits horizontal lift to a vector field on P. Our definition of covariantdifferentiation amounts to saying that Vx (a f) corresponds to theequivariant function (hX) (f) .

2) If F is a differential manifold on which G acts differentiably, consider theassociated bundle with F as fibre. If (t;, v) is a point of P x F, then wehave a tangent space decomposition T(6,v) = TT ® T. It contains thehorizontal space HH. The differential of the map PxF -+ PXGF mapsHH injectively since it is mapped isomorphically when composed withfurther projection to M. Thus at any point of the associated bundle,we have a subspace of the tangent space mapping isomorphically tothe tangent space in M. In other words, it is a space supplementaryto the vertical space and so may still be called the horizontal space.

3) If V is a connection on a vector bundle, then we can use the horizontalspace in the dual bundle in order to define the covariant derivative. Infact, consider any section s of E. It can be considered as a function fon E* which is linear on fibres. Given any tangent vector v at a pointm E M, we may lift it uniquely to a horizontal tangent vector t atany point on the fibre over m in E*. Then tf is a scalar, and the


association H t f is again a linear map of the fibre and so gives anelement of the dual of namely Em. This is the value of V ,s at m.

4.9. Exercise. Prove the statement in 3) above.

4.10. Extension of the structure group and connection forms.If f : G -> H is a homomorphism of Lie groups, then it induces a

homomorphism f : g -* Cl of Lie algebras. Let 'y be a connection formon a principal G-bundle P. We can then define a connection form on theprincipal H-bundle Q obtained by extending the structure group throughf . In fact, consider on P x H the C -valued 1-form given by pi (w) + p2 (mH),where w is the 1-form obtained by composing p*1 ('y) with f, and mH is theMaurer-Cartan form on H. This is of course a connection form on the trivialH-bundle P x H over P.

Both forms pi (w) and p2(mH) are equivariant for the (diagonal) actionof G and the adjoint action of G on 4 via f. Hence so is the sum. If weevaluate pi (w) on a vector v tangential to an orbit of G, and identify v withan element of g, we get pi(w)(v) = w(pv) = f (7 M) = f (MG M) = f (v).The action of G on H being via f (g)-1, the term p2(mH)(v) gives the value- f (v). Therefore pi(w) + p2(mH) goes down to an Cl-valued 1-form on thequotient Q of P x H by G. It is easily seen now that it is a connection formon the H-bundle Q. We refer to it as the connection form on Q obtained byextension of the structure group through f.

If E is a vector bundle, then we have seen that a connection on it givesrise to a connection form on the principal GL(n)-bundle. Conversely, ifP is the principal GL(n)-bundle associated to a vector bundle V, then aconnection form on P gives rise to a connection on V, by Theorem 4.5.

In particular, if we start with a vector bundle with a connection, theframe bundle gets a connection form, and hence all the vector bundles asso-ciated to it via representations of GL(n) inherit connections. It is a routinematter to check that the satellite connections that we defined on E*, SZ (E),etc. in 1.13 can all be identified with the connections so obtained.

Let P be a principal G-bundle and f : G -* H a homomorphism of Liegroups. Giving a connection form on the H-bundle obtained by extensionof the structure group, is entirely equivalent to giving an 4-valued form onP which is G-equivariant and restricts to f o µG on fibres. We may referto this form as an H-connection form on P, although its structure group isonly G. Now we have the following exact sequence of vector spaces:

0 - 15 --> Ci/ im g -+ 0.


This is actually a sequence of G-modules. If y is an H-connection on the G-bundle P, then we may compose it with the natural map of lj to coker(f) _Cl / im g to get a 1-form which

a) is G-equivariant, andb) vanishes on vertical vectors.

In other words, it gives a 1-form with values in the vector bundle associatedto G for the representation of G on 1)/g.

4.11. Definition. Let P be a principal G-bundle and f : G -; H a homo-morphism of Lie groups. Let y be an H-connection on P. Then the 1-formwith values in the associated bundle with coker(f) as typical fibre, is calledthe second fundamental form.

4.12. Example. Let

0-*E'-->E->E"-*0be an exact sequence of vector bundles, with rk(E') = 1 and rk(E") = k.We have already seen in Example 3.13, 2), that this amounts to givinga reduction of the structure group to the parabolic subgroup G of H =GL(k + 1), consisting of linear maps which leave the subspace spanned bythe first l basis vectors invariant.

Now a connection on E which leaves E' invariant (in the sense that forany vector field X, Vx(s) is a section of E' whenever s is one) may beconsidered as a G-connection on the principal bundle of frames of E. But ifwe are only given a connection on E, that is to say, an H-connection, thenwe may restrict VX to sections of E' and project to E" to get sections ofE". Let f be a function, p the projection E -+ E" and s a section of E.Then we have p(VX(fs)) = p(Xf.s + f VXs) = fp(Vxs). Thus the maps p(VXs) is an A-linear map from E' into E". In other words, we obtainin this situation, a differential 1-form on M with values in the vector bundleHom(E", E').

The quotient Cl/g can be identified with the space of (k, 1) matrices. Theassociated bundle can then be identified with the bundle Hom(E', E"), andthe second fundamental form with the above 1-form.

5. Curvature

5.1. Definition. Let E be a vector bundle and V a connection on it. Theexterior 2-form RV (X, Y) with values in End(E) given by (X, Y) H Vx oV Y - V Y o Vx - V [x y] is called the curvature form of the connection.We will drop the subscript V in RV when we judge that the connection inquestion is obvious from the context.

5. Curvature 149

5.2. Remarks.

1) We have already remarked in 1.14 that a connection endows the .A-module E with the structure of a C-module. The map (X, Y) --*R(X, Y) of T (M) x T (M) into the ring C(M) is .A(M)-bilinear andthe elements R(X, Y) commute with m(f), f E .A(M). Elements ofC (M) which commute with all elements m(f) constitute an .4(M)-algebra. The map (X, Y) -4 R(X, Y) is thus an alternating 2-form onT(M) with values in this algebra and gives rise to an End(E)-valued2-form on every C-module S.

2) We will see in [Ch. 7, Theorem 4.7] that if the manifold is provided witha Riemannian metric (for example, if it is a submanifold of 1Rn), thenone can define a natural connection on its tangent bundle. The asso-ciated curvature form measures the intrinsic curvature of the space.This is the origin of the terminology `curvature'.

We would like now to find a local expression for the curvature form of aconnection in terms of its local expression. This only amounts to computingR(X, Y) in the case of a connection on the trivial bundle given by dv = d+a.Before doing this, we will explain the notation we will adopt in giving thisformula.

Let us suppose that there exists an alternating bilinear product b on avector bundle E with values in another bundle F. If a is a 1-form withvalues in E, then one denotes by b(a, a) the exterior 2-form with values inF which takes (X, Y) into b(a(X), a(Y)). In particular if a is a 1-form withvalues in End(E) (or for that matter, any Lie algebra) then one may use theLie algebra bracket in End(E) to provide a meaning to [a, a]. On the otherhand one may also define w A w, using the composition in End(E). Theselead to the same result. But if we use the Lie algebra bracket and use it todefine the wedge product, then the resulting form would be twice the form[a, a].

5.3. Proposition. Let V, V be two connections on a vector bundle V andR, R' their curvature forms. Then we have

R'=R+dva+[a,a],

where a is the End(V)-valued 1-form defined by X H VX = VX + a(X).


Proof. In fact, let s be a section of V. Then

R'(X,Y)(s) _ (Vx + a(X))(Vys + a(Y)(s))- (Vy + a(Y))(Vxs + a(X)(s))

- (V[x,y] (S) + a([X, Y]) (S))= VxVy(s) + a(X)(Vys) +Vx(a(Y)(s)) +a(X)(a(Y)(s))

- VyVX(S) - a(Y)(Vxs) - Vy(a(X)(s))- a(Y)(a(X)(s)) - V [x,y] (S) - a([X, I']) (S)

= R(X, Y)s + a(X)(Vys) +Vx(a(Y)(s)) - a(Y)(Vxs)-Vy(a(X)(s)) + [a(X),a(Y)](s) - a([X,Y])(s)

= R(X,Y)s+ [Vx,a(Y)](s) - [Vy,a(X)](s)

- a([X,Y])(S) + [a(X), a(Y)](S)= R(X, Y)s + (dva + [a, a]) (X, Y) (s),

proving our assertion.

The above theorem says that

a) If the connection form on the principal bundle of frames of V is givenby 'y, then the curvature form is given by dry + ['y, -y].

b) If the bundle E is trivialised on an open covering (U)i and is given bytransition functions mil on Uij, and the connection is given by theforms ai on the open sets (Ui), then the curvature form is given bydai + [ai, ai] on (Ui).

5.4. Remark. We may now adopt the principal bundle point of view. Lety be a connection form on P. Then the curvature form is defined to be theg-valued 2-form dy + [y, y] on the principal bundle. This can be viewed asa 2-form on M with values in the adjoint bundle. If p is any representationof G in a vector space V and we take the associated connection on E(P, p),then its curvature form is obtained by composing the curvature form of theG-connection with the induced map g -+ End(V) of Lie algebras.

5.5. Curvature of associated bundles.Let V be a connection in a principal bundle P. If f : G -* H is a

homomorphism and Q is the bundle obtained by extension of the structuregroup, then we have seen that Q comes equipped with a connection aswell. What is its curvature form? The homomorphism f gives rise to ahomomorphism f of g to I . If we consider the adjoint action of G on g, andthe composite off and the adjoint action of H on C7, then the homomorphismf respects this action of G. Hence we have an induced homomorphism ofvector bundles from ad(P) into ad(Q). The curvature form of V is a 2-form

5. Curvature 151

with values in ad(P). Composing with the above homomorphism we getan ad(Q)-valued 2-form. It follows from Remark 5.4 that it is actually thecurvature form of the extended H-connection on Q.

Flat connections.

5.6. Definition. A connection on a vector bundle E (or on a principal G-bundle) is said to be flat or integrable if the curvature form is identicallyzero.

5.7. Remarks.1) Sometimes this is defined to be a locally flat connection.2) It follows from the definition that a flat connection form y on a princi-

pal bundle satisfiesdy+['y,y] =0.

If Xi, i = 1, 2, are horizontal vector fields, then we have y(Xi) = 0, sothat the above equation implies that y([X1, X2]) = 0. Hence [X1, X2]is also horizontal. In other words, the subbundle of T(P) given by thehorizontal subspaces at each point, is integrable. Conversely assumethat the bundle of horizontal subspaces is integrable. The curvatureR of the connection is the 2-form on P given by dy + [-y, -y]. Let usevaluate it on two vector fields X1, X2 on P. It is zero whenever oneof the arguments is vertical. Hence we can conclude that y is flat ifwe show that

R(X1, X2) = X1(y(X2)) - X2(7(Xl)) - y([Xl, X2]) - [y(Xl, y(X2)]vanishes whenever both Xl and X2 are horizontal. But then y(Xi)are both zero, and under our assumption, [X1, X2] is also horizontal sothat y([X1i X2]) is also zero. Hence the curvature form is identicallyzero. Thus flatness of a connection is equivalent to the integrabilityof the horizontal bundle.

Moreover, Frobenius theorem [Ch. 2, Theorem 4.2] ensures that throughany point l; in P, there exists a locally closed submanifold of P which isintegral for the horizontal subbundle. The projection map it : P -> Minduces an isomorphism of the tangent space at p to the integral manifold,with the tangent space of M at 7r(e). Hence it restricts (in a possibly smallerneighbourhood) to this submanifold as a diffeomorphism onto an open sub-manifold of M. In other words, there exists a local section at 7r(e) whichtrivialises the bundle, and pulls back the connection form to 0. Thus flatnessof the connection is equivalent to the local trivialisability of the bundle aswell as the connection. Any local section s of P over an open set U trivi-alises the connection if and only if its differential maps the tangent spaces


Tm,(M) of M into horizontal spaces at the image s(m), for all m E U. If sis one such section, then sections of the form t : m '--; s(m)g for some g E Galso have the same property. By the uniqueness of integral manifolds, if twolocal sections which trivialise the connection coincide at a point, then theycoincide in a neighbourhood.

Thus we have

5.8. Proposition. If P is a principal G-bundle provided with a flat connec-tion V, then every point m E M has a neighbourhood U and a trivialisationof both P and V over U. If U is connected, then any two trivialisationsdiffer by translation by an element of G.

If we take for P a trivial bundle, the above conclusion can be restatedas follows.

5.9. Lemma. Let a be a 1-form on a differential manifold M with valuesin End(V) (or more generally with values in the Lie algebra g of a Liegroup G), satisfying

da +[a, a] = 0.Then every point on M has a neighbourhood U and a function cp on U withvalues in GL(V) (or G) such that a is the pull-back of the Maurer-Cartanform MG on G by gyp. Moreover, if cp and V) are two such functions on thesame domain, then there exists g E G such that gy(m) = cp(m)g for all m inthe domain.

5.10. Corollary. If a principal G-bundle has a flat connection V, thenit is given by locally constant transition functions with respect to an opencovering.

Proof. Let (Uz) be a covering with trivialisations of P as well as the connec-tion on these open sets. On the intersections, the two trivialisations differby locally constant transition functions, by the uniqueness statement above.

Representations of the fundamental group.

5.11. Proposition. If a vector bundle is given by a representation of thefundamental group, then the induced connection on it is flat. The same istrue of the principal G-bundle obtained by extension through a homomor-phism f : 7rl -> G.

Proof. Note that the universal covering space, considered as a principalbundle with the fundamental group (with the discrete topology) as structuregroup, admits the connection form 0. Therefore the connection obtained byextension of the structure group is also flat.

5. Curvature 153

We have the following converse.

5.12. Theorem. If E is a vector bundle on M, and V a flat connection onit, then there is an isomorphism of E with the vector bundle Ep associatedto a representation p of the fundamental group of M which takes V to thenatural connection on Ep. A similar statement is also true for a principalbundle with G as structure group.

Proof. In view of Corollary 5.10, our assertion follows from the next propo-sition.

5.13. Proposition. If a principal bundle on a connected differential man-ifold is defined by transition functions which are constant, then it is associ-ated to the universal covering space, via a homomorphism of the fundamentalgroup into the structure group.

Proof. If we provide G with the discrete topology, the transition functionsare still continuous. Hence we may construct a principal bundle Q withdiscrete structure group. Note that each connected component is now acovering space of X. Now using the theory of covering spaces, we get ahomomorphism of the fundamental group into G and a map of the universalcovering into Q inducing an isomorphism of the bundle obtained by extensionof the structure group to G, with Q.

We have seen that associated with any connection on a vector bundle E,there is an `exterior derivative' as a differential operator of order 1 fromAi-1 (T*) ® E into A i (T *) 0 E. If P is a principal G-bundle with a connectionform ry, then for any representation p in a vector space V, we can defineexterior derivatives (with respect to the connection) of forms with values inthe associated vector bundle. If a is an Ep-valued r-form, it correspondsto an equivariant r-form 3 on P (vanishing whenever one of the vectors isvertical). Then by definition, we have

dv,3 (Xl,...,Xr+1)

_ E(-1)i+1QXi ()3 (X13 ... , Xi, ... , Xr+l))

X1,...,Xi,...,Xj,...,Xr+1)_ Y:(-1)i+1Xi/3(X1, ... , Xi, ... , Xr+1)

+ E(_1)i+1 p(ry(Xi))O(Xi, ... , Xi, ... , X,+1)

+E(-1)i+jO(1Xi1Xj], X1i...,Xz,...,Xj)...,Xr+1)

= d (Xi, ... , X,+ 1) + E(-1)i+1p(-Y(Xi))0(Xi, ... , Xi, ... , Xr+1)


The second term in the last line is simply (-y A /3) (X1, ... , X,.+1) where forthe wedge product, we use the bilinear product g ® V -+ V, giving the Liealgebra action on V. Finally we conclude that the form dva corresponds tothe form on P given by

5.14. do +'yA0.

We have also seen that the composite of two successive exterior deriva-tives is not in general zero. However if the curvature is zero, then it is indeedso. In fact, in order to check that the composite of successive exterior deriva-tives is zero, it is enough to verify it locally. But then we may assume thatE is trivial and that the connection is also the trivial one. Thus we get ade Rham complex in this case:

... A'-1(T*) ®E , Ai(T*) ®£ -> Ai+1(T*) ®£ ... .

What can one say about the kernel in the first stage, namely £ -* T® ® £?Indeed, if L is the kernel (which is a sheaf of C-vector spaces), then we cancompute it explicitly on open sets where E and V are trivial. In that case,i.e. when £ = A' and V is the usual connection, the kernel is C'. In otherwords, this C-sheaf is locally free of rank r, or, what is the same, a localsystem of r-dimensional vector spaces. Moreover, the natural map L ®c Ainto £ is an isomorphism. Thus the associated de Rham complex is the sameas the de Rham complex associated to the local system L.

5.15. Remark. The above considerations show that the existence of a flatconnection on a vector bundle imposes severe topological restrictions on thebundle. In fact, suppose the manifold is simply connected. Then any vectorbundle which admits a flat connection would have to be trivial, since thefundamental group is trivial. This suggests that in general, the curvatureform may carry some information about the topological nature of the vectorbundle. Indeed it is true and we will now investigate this question.

6. Chern-Weil Theory

The way we will extract topological information from the curvature is via deRham's theorem. In other words, we will use the curvature form to obtainsome closed forms on M and therefore, thanks to the de Rham theorem,cohomology classes. These may be considered as topological invariants ofthe vector bundle in question.

The first step on the road is

6. Chern-Weil Theory 155

6.1. Bianchi's identity. The curvature form R of a connection V in aprincipal G-bundle P satisfies doR = 0. Here R is a 2 -form with valuesin the adjoint bundle and dv is the exterior derivative of R with respect tothe induced connection on ad(P). This means that we have, for any vectorfields X, Y, Z, the identity

E(Vx(R(Y, Z)) - R([X,Y], Z))where the summation is taken over cyclic permutations of X, Y, Z.

Proof. This is a routine verification following the definition of R. In fact, if-y is the connection form, substituting for R the g-valued 2-form dry + [y,'y]on P, and applying dv to it, we get d(dy+[y, y])+yA(dy+[y, y]) = d[-Y, y]+-y A dy, since d2 = 0 and [y, [-y, 'y]] = 0 thanks to the Jacobi identity. Recallthat by our definition, [y, y] (X, Y) = [y(X ), y(Y)], and (y A dry) (X, Y, Z) =E[y(X), dy(Y, Z)] = E[y(X), YyZ-ZyY-y[Y, Z]], where the sum extendsover cyclic permutations. Substituting d[-y, y] (X, Y, Z) = E X [y, y] (Y, Z) -

[y, y] ([X, Y], Z) = E X [yY, -y Z] - E[y[X, Y], yZ], we get the result.

In particular, if E is a complex line bundle, then the bundle End(E)and the induced connection on it being trivial, do on End(E) is the sameas the usual exterior derivative. Hence the 2-form R with coefficients inC is closed and defines a cohomology class in H2 (M, C). Moreover, if westart with a different connection, then the two connections differ by a 1-form a and their curvatures differ by da by Proposition 5.3. In other words,the class in the de Rham cohomology H2 (M, C) defined by the curvatureform, is independent of the connection. It is obvious that this cohomologicalinvariant of the bundle is functorial in the sense that the invariant of the pull-back of a bundle is the pull-back of the invariant. It is called the geometricChern class of the line bundle.

6.2. Computation.We will make a computation of the Chern class of a specific line bundle,

namely the Hopf bundle. Consider the principal C'-bundle E = Cn+1 \ {0}over CIPn. Consider the C-valued 1-form y on E given by z z; . It isclear that multiplication by any nonzero scalar leaves this form invariant.Also if (ap, ... , an) is any point in E, the identification of CX with a fibreby the map A H (aoA, ... , an.X) pulls y to dz/z, which is the Maurer-Cartanform on C'. Hence -y is a connection form on E. Its curvature form is dysince the Lie algebra is abelian, and is given by Ej d( z,z.) A dzj. It is a2-form on CIEDn pulled up to E.

Now we will assume that n = 1 and integrate this form on CIP1. Itis enough to perform the integration on CX since its complement in P1consists only of two points. Take the section of the bundle over C'< which


takes z to (z, 1). The pull-back of the curvature form is then A dz =dz A dz. We will use the orientation given byax )d-z A dz = (l(l+

I -Z17dx A dy in order to compute its integral. Recall that integrating top degreeforms gives an isomorphism of the top de Rham cohomology with R or C.

We will use polar coordinates z = reie on C \ {0} and convert the aboveCdr A d6. Since dr A dO and dx A dy give the same orientation,form to 2(1+ r)

2 21r(i(1+ )2 drdO = 27ri. In particular, the Chern class isits integral is fo L

nonzero.

6.3. Exercise. Compute the first geometric Chern class of the tangentbundle of 1P1.

6.4. Remark. Since the Chern class of the trivial bundle is zero, it followsthat the Hopf bundle is not trivial. Even this very simple case illustratesthe power of studying the topological invariant of a bundle.

6.5. Integral Chern class.Consider any line bundle on M. Suppose it is given by transition func-

tions (mid) with respect to a locally trivial covering (Ui). We would like toidentify the first Chern class defined above in terms of (mid). The curvatureform is a closed 2-form and gives the Chern class as a de Rham cohomologyclass in H2 (M, C). In order to identify this in the sheaf cohomology withcoefficients in the constant sheaf C, we will recall the explicit description ofthe de Rham isomorphism. How does a closed 2-form give rise to a class insheaf cohomology? We will denote the sheaf of closed i-forms by Zi here.Consider the following two exact sequences of sheaves (here A is the sheafof differentiable complex-valued functions):

0->Z1T*-+Z2-_ 0,

0-->C,A.->Z1--+ 0.Since T* is a soft sheaf, all its positive cohomology spaces vanish [Ch. 4,Proposition 3.3, ii)]. Hence we get an exact sequence

T*(M) --> Z2(M) - H1(M,Z') , 0.This identifies the space of closed forms modulo the exact forms, namelythe second de Rham cohomology, with H1(M, Z1). The second sequenceidentifies it in turn with H2(M,C) since A is soft. We wish to understandhow to associate to a closed 2-form R, a Cech 1-cocycle with values in Z1.In fact, if we can lift R locally to sections of T* (Ui), then the differenceson UZj of the lifted sections over Uj and Ui give a family of sections of Z'over Uij, namely the required Cech cocycle. In other words, we write R asdai on open sets U. and take the differences ai - ai on Uii. Actually, by


our convention for boundary homomorphism in [Ch. 4, Remark 2.16], therequired class is its negative.

Now if (mid) are transition functions for the line bundle in question,the local forms dmij/mi7 are closed. A choice of forms ai on Ui such thataj - ai = (dmij)/mid is just a connection on the line bundle! Its curvatureform is given by the 2-forms dai, which coincide on the intersections andtherefore give a global 2-form. This is the 2-form of which we set out to findthe cohomology class, under the de Rham isomorphism. By construction,we have already written it locally as dai. The difference of aj and ai isactually dmij/mi7, which is the negative of the Cech 1-cocycle with valuesin Z1 that we needed to identify at the first stage.

Therefore we conclude that the first Chern class of the line bundle issimply the image of this class in H2 (C) by the connecting homomorphism ofthe second sequence above. This can again be restated as follows. Take theimage of the element of H' (AX) in H1(Z1) by the homomorphism f --+ df If.Then take the image under the connecting homomorphism of the secondsequence above. But now consider the diagram

0 --; 27riZ --* A -> A" -* 0

0 -+ C -> A --* Z' --* 0

All the maps are the obvious ones; the top right horizontal arrow is theexponential map and the right arrow downwards is the map taking f todf If. The Cech 1-cocyle in 21 given by dmij /mij is the image of the Cech1-cocycle in A" given by mid by the right vertical map. In other words,we take the class in H' (A") given by the line bundle. Instead of takingits image in H' (21) and then go by the connecting homomorphism of thelower sequence to H2(C), we may as well take its image by the connectinghomomorphism of the top sequence to H2 (2iriZ) and then go down by theleft vertical map. Thus we have proved the following statement.

6.6. Theorem. The (geometric) Chern class of any connection in a linebundle represents the negative of the image in H2(M,C) of the element ofH2(M, 2iriZ) obtained as the image under the connecting homomorphism ofthe exponential sequence, of the element in H' (M, A") corresponding to theline bundle.

6.7. Remarks.1) The Chern class of a line bundle is essentially an integral class. One

might have actually defined it as an element of H2 (M, Z) using theexponential sequence as in the theorem above and dividing it by -27ri.One may think of it as the topological definition. This class is clearly


functorial for pull-backs. The above theorem then states that thecomplex cohomology class so defined is the same (up to multiplicationby 27ri) as the curvature class. If E is any vector bundle, we can definethe first Chern class as the integral Chern class of det(E). Again wehave just shown that it is the same (up to a factor of 27ri) as the classdefined by the first Chern form of a connection.

2) The topological Chern class of a line bundle is therefore obtained bytaking the image of the boundary homomorphism of the sequence

where the surjection is given by f H e27rif.

Can we also get a similar class for arbitrary principal bundles? Let E bea principal G-bundle with a connection. Since R has values in the adjointbundle, in order to obtain an ordinary form, we need a homomorphism ofad(P) into the trivial bundle. Suppose we have a linear map t : g -> C,which is equivariant for the adjoint action of G on g on the one hand, andthe trivial action of G on C on the other. Then the induced homomorphismof the associated bundle ad(E) (namely, the associated vector bundle withg as fibres) into the trivial bundle respects the induced connection on ad(E)and the connection on A induced by the trivial representation, namely thetrivial connection. The image of the ad(P)-valued do-closed curvature formR of P is therefore a closed C-valued 2-form and defines a class in H2 (M, C).

The map t is also equivariant for the induced g actions, namely thebracket action on g and the trivial action on C. So the map is zero on [g, g].If we take any other connection on E, then its curvature form differs from Rby an additive term of the type da + [a, a]. Applying t to it, we get d(t o a).Thus the class we defined is independent of the connection and depends onlyon E and t.

6.8. Remarks.

1) The supposed equivariant map oft : g -+ C is actually a homomorphismof Lie algebras. If it integrates to a group homomorphism G -). C',then we can induce a connection on the line bundle (obtained byextension of the structure group). Its curvature form is obviouslyt o R. So in this case, the class defined above is simply the Chern classof the associated line bundle.

2) The following is a special case of the above. Let E be a complex vectorbundle with a connection. For any vector space V, the trace mappingfrom End(V) into C is a GL(V)-homomorphism (taking the adjointrepresentation on g = End(V) and the trivial representation on C).(This only means that tr(TAT-1) = tr(A) for all T E GL(n).) Hence

6. Chern-Weil Theory159

it gives rise to an induced homomorphism End(E) -> Ac. Thus theimage of the curvature form is a 2-form with values in C. This iscalled the first Chern form of the connection. It is closed, as wehave observed above, and gives a class in H2(M, C). This class isindependent of the connection and is called the first Chern class of thevector bundle E and is denoted by cl (E). This is clearly compatiblewith the definition we gave for line bundles. In this case, the tracehomomorphism we started with actually integrates into the grouphomomorphism det : GL(r, C) -- C'. So we conclude that cl(E) =cl(det(E)) where det(E) is the line bundle associated to the bundleE by det : GL(n) --> C*, or what is the same AT(E).

3) Why have we defined the first Chern class for GL(k, C)-bundles and notfor GL(k, R)-bundles? In fact, formally one can take the curvatureform of a connection and apply the trace and obtain a cohomologyclass in H2 (M,118) independent of the connection. But then we havealready remarked that any real vector bundle E admits a positivedefinite metric along the fibres. If we take an orthogonal connection inE, its curvature form takes values in the Lie algebra of the orthogonalgroup, namely the space of skew-symmetric matrices. In particular,the traces of all R(X, Y) are zero. So we conclude that the cohomologyclass thus defined is 0 for all bundles E! This shows the slippery natureof building formal theories without examples.

Notice again that in the case of a real vector bundle, if we define theclass using curvature, we will have obtained a class in H2 (M, R). The corre-sponding element of H2 (M, C) is then the Chern class of its complexificationand is therefore in the image of H2 (M, 21riZ). It is no wonder that it is zeroalways!

6.9. Remark. While the above discussion gave rise to an interesting in-variant, namely, the first Chern class of a vector bundle, we cannot obtainany more classes by that simple procedure, for there are not many GL(V)-homomorphisms from End(V) into the trivial representation space C. Infact, up to a scalar factor, the trace map is the only such homomorphism!However we can modify the construction by starting out with R' insteadof R itself. Notice that R belongs to A(+) ® Sym(End E), where A(+) de-notes the (commutative) subalgebra of elements of even degree in the ex-terior algebra A(T*). Hence it makes sense to talk of R' as an element ofA2r'(T*) ® Sr (End(E)). It is the 2r-form with values in S'(End(E)) givenby

Rr(X i, ... , Xr.) = E EQ fjR(Xo(2i), XQ(2i+1)),


where we take the summation over permutations which satisfy a(2i) < a(2i+1) for all i < r. Then we have the same statement as the Bianchi identity forif, that is to say it is a 2r-form with values in S'(End(E)) which is closed forthe exterior derivative associated to the induced connection on S''(End(E)).This follows from the derivation property of the exterior derivative.

Thus we start with R' as an element of A2'"(T*) ® S'(EndE), and takeits image in any A2, (T*) by means of a homomorphism S'(End E) -+ Agiven rise to by a GL (V )-homomorphism of S(End V) into the trivial rep-resentation space (C. This then gives a closed, complex-valued 2r-form asbefore.

Are there such homomorphisms? Yes, indeed! Actually there are manyand we can in fact write them all out. We will first define such a homo-morphism for each r < dim(V) = k. The map which associates to eachendomorphism A of V, the trace of its lift, Ar (A) : A'' (V) -+ A( V),, isthe homomorphism we have in mind. Up to sign, this map is the same asthat of associating to an element of End(V), the coefficient of tk-T in itscharacteristic polynomial. Another way of saying it is that it associates toany matrix the rth elementary symmetric function of its eigenvalues. Whenr = 1, this is the same as the Chern form we defined earlier.

6.10. Theorem. For every r < dim(V) = k, the coefficient of tk-r ofthe characteristic polynomial det(t + A) of elements of End(V) gives riseto an adjoint invariant polynomial of degree k. It may be considered asgiving a GL(V) -invariant linear map ST (End(V)) -+ C. Let E be any vectorbundle of rank k and V a connection on it. By taking the image of ifunder such a linear map, we obtain a C-valued closed form of degree 2r andthus a cohomology class H2r'(M, C). The cohomology class so obtained isindependent of the connection.

Proof. In fact, consider the manifold R x M and the bundle p2E on it. Theprojection to the first factor may be considered as a real-valued function onthe product, and we denote it by t. Let V1, V2 be two connections on E.Consider the connection V = tV1 + (1 - t)V2 on p2(E), where we havedenoted by Vi the pulled back connections as well. Consider the Chernforms of this connection, on R x M. By the functoriality of the constructionand the obvious fact that the pull-backs of V to {0} x M and {1} x M areV1 and V2 respectively, we conclude that the Chern form on the productalso pulls back to the Chern forms of the two connections on these twosubmanifolds. Now our assertion follows from [Ch. 4, Proposition 6.7].

6.11. Definition. The rth geometric Chern form of a complex vector bun-dle of rank k is the 2r-form obtained by substituting the curvature form in


the (k - r)th coefficient of the characteristic polynomial t H det(tI + A).The cohomology class in H2r(M, C) defined by it, is called its rth geometricChern class and is denoted c,.(E).

6.12. Proposition. If E is a direct sum of two vector bundles El and E2,then the rth Chern class of E is given by

cr(E) = E ci(Ei).cj(E2)i+j=r

where the multiplication is taken in the cohomology algebra.

Proof. We take connections in each of the Ei's and take the direct sumconnection in E and will compute the Chern form of this connection. Thecurvature form in E associates to any two vectors X, Y the endomorphismR(X, Y) which is the direct sum of R1 (X, Y) and R2 (X, Y). The 2r-formRr is therefore given by the sum of the terms Ri.R2 with i +j = r, where wehave used the inclusion of S2i (End El) ®S2j (End E2) in S2r (End E) and theexterior multiplication A2i (T*) ® A2j (T*) - A2r (T*). Applying the lineartrace map, we get our assertion.

This also shows that the rth Chern class is in general nontrivial. Thesimplest thing to do is to take M to be the r-fold product (?1)r and E tobe the direct sum of the pull-backs of the Hopf bundle from all the factorsand use the above theorem.

6.13. Remarks.1) We have seen that the first Chern class is essentially an integral class.

It can also be proved that the rth Chern class comes from a classin H2r(M, (27ri)rZ). From the topological point of view therefore itmakes sense to define the topological Chern class as the class obtainedby dividing our class by (-2iri)r. These are therefore in the imageof H2r(M,Z) in H2r(C). Such classes can indeed be defined as in-tegral classes from a purely topological standpoint and proved to befunctorial and having property 6.12. The general case can actually bereduced to the case of line bundles.

2) We cannot define any more invariants in the case of vector bundlesby this procedure. It is easy to see that any adjoint invariant mapEnd(V) -+ C is generated by the above, namely the coefficients of thecharacteristic polynomial. Hence any cohomological invariant we candefine by the above procedure, is a polynomial in the Chern classesCr.

We can apply much the same ideas for any principal G-bundle P. Thecurvature form R of any connection on it, is a 2-form with values in the


vector bundle ad(P). The 2r-form Rr has values in Sr(adP). If we have anadjoint invariant homogeneous polynomial from g into C of degree r, thenwe can define a cohomology invariant in Her (M, C) exactly as above.

We will now illustrate this for the case of orthogonal bundles. Let E be aprincipal SO (n, C)-bundle. The curvature of a connection on E, consideredas a 2-form on the total space E, takes values in the Lie algebra of SO(n, C).We recall that if the quadratic form is taken in the standard form, namelyq(zl, ... , z,,,) = E z2 , the Lie algebra of SO(n) consists of skew-symmetricmatrices. We then look for equivariant maps S"(so(n,C)) --> C, where theaction of G is the adjoint action on the first and trivial on the second factor.The obvious maps to take are the coefficients of the characteristic polynomialagain, since they are after all invariant under the adjoint action even underGL(n). However, the coefficients of the characteristic polynomial of a skew-symmetric matrix are zero whenever the parity of the degree is different fromthat of n. This is because the lift of a skew-symmetric endomorphism of avector space to Az(V) is again skew-symmetric for odd i and consequentlyis traceless. Thus we get, for every even number 2k < n, a cohomology classin H4k(M,C). These are of course not very different from what we definedabove, treating the orthogonal bundle just as a vector bundle. In fact, ifwe ignore the fact that E is an orthogonal bundle, but treat it as a vectorbundle (that is to say, consider the GL(n, C)-bundle obtained by extensionof the structure group), then these are just 2kth Chern classes.

6.14. Definition. The class (-1)kc2k(E) is called the kth Pontrjagin classof the special orthogonal bundle.

In the case when n = 2r, we have therefore defined r Pontrjagin classes.But we can also define one more class by our procedure. In other words,we will give one more invariant polynomial called Pfaffian from the spaceof skew-symmetric matrices. The space of skew-symmetric matrices can beidentified with A2 (V), compatibly with the adjoint action of the orthogonalgroup on the former and the second exterior of the natural representationon the latter. By taking the rth power of an element of so(2r) = A2(V), weget an element of A2r (V) = C. Thus we get a polynomial map of the Liealgebra into det(V) = A2r (V) of degree r, which is equivariant for the adjointaction of SO(n) on the first factor and the trivial action on the second. Thispolynomial is called the Pfaffcan. Thus we get an invariant in H2r(M,C),using this invariant polynomial. It is called the Euler-Poincare class. In fact,one can show that all SO(2n)-invariant polynomials are generated by theeven coefficients of the characteristic polynomial and the Pfaffian. Thesehowever are not independent. In fact, the square of the Pfaffian is thedeterminant, namely the constant term of the characteristic polynomial.


Again this class can be shown to be (27r)'r times an integral class. From thetopological viewpoint the integral class is defined to be the Euler class.

6.15. Remarks.

1) If an oriented vector bundle of rank 2n admits a section nonzero every-where, then one can introduce a metric along the fibres such that thesection has unit length at all points. In other words, the structuregroup can be reduced to SO (2n - 1). Its Lie algebra consists of ele-ments of the Lie algebra of SO (2n) which are zero on the first vectorof the standard basis. But the Pfaffian of a skew-symmetric matrixwhich is not invertible is of course zero. Hence the substitution in thePfaffian of the curvature form of an SO(2n)-connection obtained byextending one on the reduced bundle, is zero. We thus conclude thatthe Euler class of a bundle admitting an everywhere nonzero section,is zero. If one could compute the Euler class of the tangent bundleof a differential manifold and show that it is nonzero, it would followthat the manifold does not admit a vector field nonzero everywhere.This question was at the root of the development of the characteristicclasses from the topological point of view.

2) If we start with a complex vector bundle of rank n, we can reduceits structure group to the unitary group by introducing a Hermitianstructure along its fibres. Disregarding its complex structure, we maytreat it as an SO(2n)-bundle and compute its Euler class. The resultis that this class is just (-1)' times cr,,(E). This is only a matter ofcomputing the Pfaffian of a skew-symmetric matrix which is associ-ated to a skew-Hermitian matrix A, namely

Re (A) - Im(A)Im(A) Re(A)

The Pfaffian of the latter is computed (for example by diagonalis-ing A) to be det(iA) = in det(A). Now note that to get the topologicalEuler class we have to multiply the geometric Euler class by (1/2ir)nwhile the Chern class is obtained by multiplying by (-1/27ri)'.

3) We can define characteristic classes in somewhat greater generality. Sofar we have been taking maps from S'(g) into C. Suppose G is notconnected. Then we can take a (possibly nontrivial) representation pof G which is trivial on the connected component G° of G containing1. Let V be such a representation, and we take a map S'(0) intoV, equivariant for the adjoint action of G on the first factor and theaction p on V. Then for any principal G-bundle and a connectionon it, we can take the image of the rth power R' of the curvatureR as above and obtain a 2r-form with values in the vector bundle


associated to p. But this bundle has G/G° as structure group. Thelatter being discrete, the associated connection is flat. The 2r-formthat we obtained is then closed for the exterior derivative associated tothe flat connection. By de Rham's theorem this defines an invariant inH2r(M, L(p)) where L(p) is the local system defined by the associatedbundle p(P) and the flat connection on it.

6.16. Example. Consider an orthogonal bundle P, i.e. a bundle withstructure group G = O(n). This group has two connected components, andthe connected component of 1, namely SO(n), is of index 2. Take n = 2r asabove, but notice that the Pfaffian is actually a map Srg -+ A2r (C2r) =

But it is equivariant for 0(2r) only if we take the determinant characteron the second factor and the adjoint on the first. Hence the Euler classtakes values in the cohomology of the local system given by the determinantcharacter of 0(2r).

6.17. Remarks.

1) If E is a real orthogonal bundle, the same construction as above leadsto Pontrjagin classes in H2i(M, IR). In fact, the odd Chern classesof its complexification vanish and the even ones give the Pontrjaginclasses (up to sign).

But the Euler class cannot be accessed directly for a vector bun-dle by our construction. This is because the Pfaffian which is anorthogonal invariant does not extend to an invariant polynomial onall matrices. However, we can introduce a metric along the fibres,that is to say, reduce the structure group to the orthogonal groupO (n, R). When n = 2k is even, the Euler class is clearly defined fora real S0(2k)-bundle as a cohomology class in Hk(M, IR). Since allmetrics form a convex set, one can easily check that the class so de-fined does not depend on the metric. Thus the Euler class is definedfor any (oriented) real vector bundle.

2) If we take the tangent bundle of a compact oriented manifold M of evendimension 2r, the topological Euler class is an element of H2r(M,R).Since H2r(M, Z) = Z, it is an integer. It can be proved to be theEuler-Poincare characteristic, namely, E(-1)ibi where bz =dim Hi (M, III) This is known as the Gauss-Bonnet theorem. See[10].

7. Holonomy Group; Ambrose-Singer Theorem

We will indicate a geometric understanding of connections and curvature inprincipal bundles. Let us begin by recalling that flatness of a connection on a

7. Holonomy Group; Ambrose-Singer Theorem165

simply connected manifold implies that the bundle is trivial. All connectionson bundles over the interval are flat, as there are no nonzero 2-forms and sothe curvature form is always 0. Hence any bundle with a connection admitsa trivialisation. Suppose P is a principal bundle over any manifold M, andV is a connection on it. If q : [0, 1] - M is a differentiable or piecewisedifferentiable path in M, then the pull-back of P to the curve is trivial.Indeed, given a point p on the fibre of P over q(0), there is a canonicalsection of the pull-back q*(P) C I x P, which takes the value (O, p) at thepoint 0. This is the same as saying that there is a lifted path I Pwith origin p.

7.1. Definition. A path in P is said to be horizontal if its differential atany point of I = [0, 1] is a horizontal vector.

Since the pull-back of the connection to I is also trivial, it follows thatthe section q is horizontal. This amounts to saying that any path in Madmits a horizontal lift. The lift is unique if the origin is prescribed. Thismay be referred to as a moving frame, since any point of the principal bundlecan be thought of as a frame. Moreover, if we have a differentiable familyof paths, in the sense that we have a differentiable map of f : T x I into M(where T is a differential manifold), then the map T x I -* P obtained byhorizontally lifting the paths f{t} x I -* M is differentiable.

Even if the path in M is a loop, there is no guarantee that its horizontallift would be a loop. (One knows that this is not the case even for a coveringspace.) There is a unique element of the structure group which takes theinitial point to the end point of the lifted path. This is called the holonomyof the loop at a given initial point p of P. Unlike the case of a covering space,we do not claim that the end point of the lift depends only on the homotopyclass of the loop. The end point of the horizontal lift is the translate of p byan element of G. By taking all possible loops at a point m E M, and liftingthem with initial point at p, we get a family of elements of G. The constantloop gives 1 E G, and the inverse of a loop has the inverse as its holonomy.Let ql, q2 be two loops and q their composite. If both of them are piecewisedifferentiable, so is q. Let qi be the horizontal lifts of qj with origin p. If s isthe holonomy of ql, i.e. 41 (1) = p.s, then the horizontal lift of q2 with originps is given by 42s. So the composite of gl and 42.s is piecewise differentiable,horizontal, lifts q and has origin p. Hence if the holonomy of the loop q isu, then pu = q2 (1) . s = pt. s where t is the holonomy of q2. In other words,u = ts. Thus all the holonomy elements at p form a subgroup of G.

7.2. Definition. The group formed by the holonomy of all loops at a pointm E M obtained by lifting them with initial point p E P is called the


holonomy group at p. The subgroup formed by taking only loops homotopicto the identity is called the restricted holonomy group.

7.3. Proposition. The restricted holonomy group at p E P is an arcwiseconnected subgroup of G.

Proof. In fact, let q be a loop at m = 7r(p) which is homotopically trivial,i.e. there is a family of loops qt at m parametrised by I, with qo = q, and qlbeing the constant loop at m. Their lifts with p as origin have end pointspgt, say. For t = 1 it is the constant loop at p. Then t F-> gt is a pathconnecting 1 and the holonomy of q at p.

7.4. Remark. It can be shown that any arcwise connected subgroup of aLie group is a Lie subgroup.

The dependence of the holonomy group, on the point p is explained inthe following proposition.

7.5. Proposition. If M is connected, then the holonomy groups at anytwo points in P are conjugate.

Proof. Take any point p' E P, at first, on the fibre through p. In otherwords, let p' = pg for some g E G. If q is any loop at m = ir(p) = 7r(p'), thenits horizontal lift q' with origin at p' is simply the right g-translate of thehorizontal lift 4 at p. From this we conclude that if the holonomies of ry atP I P ' are h, h' respectively, then q' (1) = q(1).g = ph.g = p'g-'hg. This showsthat h' = g-'hg and hence the holonomy groups at p and p' are conjugate.

If p' is not necessarily on the same fibre, let m' = ir(p'). Connect mand m' by a path 6 and take its horizontal lift 6 with origin at p. Its endpoint lies over m' and so is of the form p'g. We have just shown that theholonomies at p' and p'g are conjugate. So we may as well replace p' by p'g.In other words we will assume that the horizontal lift of 8 has p' as its endpoint. Then to any loop q at m, we associate the loop q' at m' by composing8-1 first and then q and 6 again. Its horizontal lift is obtained by composing8-' first, the horizontal lift of q and then S. This shows that the holonomysubgroups at p and p' are identical.

7.6. Theorem. Let M be a connected differential manifold and P a princi-pal G-bundle on it with a connection form a. Then the structure group canbe reduced to the holonomy group G p at any point p E P. The connectioncan also be reduced to the reduced bundle in the sense that there is a Gp-connection in the reduced principal bundle such that the given connection isextended from it.

7. Holonomy Group; Ambrose-Singer Theorem 167

Proof. Let p be any point of P. Consider the union of all piecewise smooth,horizontal curves with initial point p. Let us call it Hp. Since any point of Mcan be connected by a path to 7rp and the path can be lifted to a horizontalone starting at p, the map 7r : Hp --a M is surjective.

The holonomy group at p leaves Hp invariant. For if p' E Hp, then thereis by definition a horizontal path y joining p and p'. If g c Gp, then p andpg can be connected by a horizontal lift of a loop q at 7r(p). By translatingthe curve ry by g we see that pg can be connected by a horizontal curve top'g. In particular it follows that p and p'g can also be so connected. In otherwords, p'g belongs to Hp, proving our assertion.

Conversely, if p1 and p2 = p1g are two points in Hp on the same fibre,then p can be connected by horizontal paths C1, C2 to them. By translatingthe curve Cl by g we get a horizontal path connecting pg to p2. Composingthe path C2 with the inverse of this path, we get a horizontal path joining pand pg. This implies that it is a horizontal lift of a loop at 7r(p) and hencethat gEGp.

Now we wish to show that Hp is a submanifold of P. Let p' E Hp and m'be its image in M. Connecting p and p' by a horizontal path, we can identifyGp with Gp'. Consider a coordinate neighbourhood U of m'. For every pointx of U let S., be the line segment (in the coordinate system) joining x andm (which is 0 in the coordinate system). Thus we get a differentiable familyof paths in M, parametrised by U. Hence there is also a differentiable mapof U x I to P, its restrictions to {v} x I for all v E U being horizontal pathswith initial point p'. The restriction to U x {1} gives rise to a differentiablesection s of P over U which has its image in Hp. This gives an isomorphismof 7r-1(U) with the trivial bundle U x G, namely H whereg(e) is determined by the equation 6g(6) = s(irl;). The restriction of g(6) toit-1(U) n Hr, goes into G.

From our remarks above it gives an isomorphism of Hp onto U x Gp.From this it follows that Hp is a differential manifold, that it is a principalGP bundle and that the bundle obtained by extension of the structure groupto G can be identified with P.

The relationship between holonomy and curvature is given by the fol-lowing theorem, due to Ambrose and Singer.

7.7. Theorem. The Lie algebra of the restricted holonomy group at apoint p in a principal G-bundle is the subalgebra of g generated by the valuesR(v, w) where v, w are tangent vectors at points of the reduced bundle asgiven above.


Proof. As we have seen above, the structure group of the bundle and theconnection may be reduced to the holonomy group. Therefore we may as-sume, without loss of generality, that the holonomy group is itself G andshow that the Lie subalgebra g' generated by R(v, w) is actually g. Fromthe G-equivariance of R we see that g' is invariant under the adjoint actionof G. Hence it is also invariant under the adjoint action of g.

The vertical subbundle of the tangent bundle of P admits a subbundle E,namely that corresponding to the Lie algebra g'. Consider the subbundle ofT (P) which is the direct sum of E and the subbundle of horizontal subspaces.We claim that it is an integrable subbundle. If we take two horizontal vectorfields X1, X2, the vertical projection of their bracket is given by R(X1, X2)and so it is a section of E. On the other hand, if we take two sections ofE and bracket them, it is again a section of E since g' is a Lie subalgebra.Finally, the bracket of an invariant vertical vector field and an invarianthorizontal vector field is always zero. These assertions prove our claim. Thissubbundle is also invariant under the action of G. Therefore any integralmanifold for this integrable subbundle has the property that it is invariantunder G and the tangent space at any point contains the horizontal subspaceat that point. This shows that its dimension is the same as that of P. Inparticular, we have g' = g.

Exercises

1) Let E be a vector bundle and s a nonzero section. Show that thereexists a connection V such that Vxs = 0 for all vector fields, if s isnonzero at all points. Is it also true if s has zeros? What does thismean for a trivial bundle?

2) Let M be a connected manifold. Show that the action of the gaugegroup factors to an, effective action of its quotient by the group ofnonzero scalars.

3) Give a faithful representation of the group GA(V) in a vector space ofdimension n + 1 where dim (V) = n.

4) Let !y be a connection form on the trivial U(1)-bundle on any manifoldM. Show that there exists a U(1)-invariant differentiable map M xU(1) --+ SN, for some N, such that 'y is the pull-back of > zidzi.

5) Show that given an open covering (Ui) of a differential manifold, thereexists a family of differentiable functions fi which satisfy E I f 12 = 1and E f i fi = 0. Hence conclude that if M is a compact manifold,and E a U(1)-bundle on it, then any connection on E is induced by adifferentiable map into the complex projective space from the aboveconnection on the Hopf bundle on it.

Exercises 169

6) Let C be the connection algebra and B the subspace given by X E Csuch that X f = f X for all elements f in A. Show that B is asubalgebra which is invariant under brackets by elements of C.

7) Let G be any connected Lie group. Show that a unique connection canbe defined on its tangent bundle with the property that VxY = 0 forall left invariant vector fields X and Y. Compute its curvature form.

8) Let a connected Lie group G act transitively on a manifold M, and letI be the isotropy subgroup at a point m. Find the condition in termsof I, for the G-action to lift to a principal H-bundle.

Chapter 6

Linear Connections

We take up here the special features when we deal with a connection in thetangent bundle of a differential manifold. These are called linear connections.

1. Linear Connections

Let V be a linear connection on a differential manifold M. Let r : T ---+ M bethe tangent fibration. Then there is a canonical vector field on the manifoldT defined as follows. Take any point t of T. It is by definition a tangentvector v of M at the point m = ir(t) E M. This vector can be liftedhorizontally to a tangent vector of the manifold T at the point t. Thevector field we have in mind associates to the point t, this horizontal lift.

1.1. Definition. The vector field defined above on the manifold T is calledthe geodesic vector field associated to the linear connection.

If (U, (xi)) is a local coordinate system in M, there is a natural localcoordinate system (7r-1(U), (xi, yj)) in T, consisting of the functions xi =xi o 7r, yj = dxj . Here dxj are 1-forms on U, considered as functions onit-1(U). A point of 7r-1(U) with coordinates (xi,yj) is then the tangentvector at the point (xi) of U given by E yj. We will now compute thelocal expression for the geodesic vector field in it-1(U). Let V, the givenlinear connection, be given by the law:

v a (axj ak/

.

171

172 6. Linear Connections

The companion connection acting on sections of T*, namely differential1-forms, is therefore given by

Car ' Paz(dxj)) = C a

), dxj> = -rzkk i Caxk

In other words,

V as (dxj) r'ikdxkti

Thus we have Ei k yirZkdxk. Interpreting dxk as thefunctions yk on the tangent bundle, the right side may also be written as

>i,k yiykl ak. By [Ch. 5, Remark 4.8, 3)] the covariant derivative of dxjwith respect to the vector field E yi a1; (considered as a function on T) isobtained by lifting the vector field horizontally and letting it act on the func-tion yj. So if the lift is expressed in local coordinates upstairs in it-'(U) asi yi + E cpk (x, y) as , then it acts on the function yj and yields co j (x, y).

Thus we have coj (x, y) Ei k yiykrik. This then gives the local expressionfor the geodesic vector field to be

1.2. M axti - Y Ykrik a'Yji i, j,k

In particular, the geodesic vector field is differentiable.

1.3. Definition. Given a linear connection, we may consider the flow of thegeodesic vector field. The image in M of any curve in T which is integralfor the geodesic flow is called a geodesic of the linear connection.

1.4. Remarks.

1) If we take any path c : I -* M, then for every t E I, the image of dt bythe differential of c, gives a tangent vector c(t) at c(t), which is a pointof the manifold T. In other words, we get a lift of c to the manifold T.This is called the canonical lift of a curve. If c is a geodesic, and y isthe integral curve of the geodesic vector field with c'(0) as its origin,then c = 7r o y, by definition. Now y' (t) is horizontal, and the map7r maps it on the vector c'(t). But since y is integral to the geodesicvector field G, it follows that the vector -y(t) is the horizontal lift ofthe vector y(t) at the point 2r(y(t)) = c(t). Thus c(t) is the vectorcorresponding to -y(t). In other words, the canonical lift of c(t) isthe same as the horizontal lift. This clearly characterises geodesics.This observation allows us to write down a differential equation thata geodesic must satisfy. See 1.5 below.


2) Let c be a geodesic of a linear connection. In order to translate a vectorv at c(O) parallelly along c, we have to take the horizontal lift of cwith v as origin. In particular, take v to be c'(0). Then the vector atc(t) obtained by parallelly translating c'(0) along c itself, is given byy(t), where y is the horizontal lift of c starting from c'(o). But thisis the canonical lift as we have seen above, and so we conclude thatthe translated vector at c(t) is simply c'(t). We may express this bysaying that the tangent to c(0) remains tangent to c when translatedalong c. For this reason we say that geodesics are autoparallel. In thisrespect it is like a straight line in the Euclidean space.

3) To any Riemannian manifold, which will be defined in Chapter 7, onecan canonically associate a linear connection as we will see in [Ch. 7,Proposition 4.8]. The geodesics for this connection turn out [Ch. 7,5.7] to be precisely curves which minimise distances between closeenough points. This is the origin of the terminology.

1.5. Proposition. A differentiable path c(t) where t varies in an openinterval containing 0, is a geodesic if and only if it satisfies the followingdifferential equation:

d2ci ; dck dck =t2 + >2 rjk(c(t)) dt dt

0.

Here c(t) = (cl(t), ... , cn(t)) with respect to a system (U, (xi)) of local coor-dinates of M.

Proof. We will use the associated local coordinates (7r-1 (U), (xi, yj)) on T.Then the canonical lift of c is given locally by the functions (ci(t), -d-"). Forit to be integral for the geodesic vector field, its tangent at any point shouldbe given by the value of the geodesic vector field at that point. This meansthat > dt a

z2+ (d) is the same as the geodesic vector at the point

(ci (t), L). Equating the coefficients of -y

in this and in 1.2, we get thesystem of differential equations

d2ci i dck dck _dt2 + rjk(c(t)) dt dt -

o

for the functions ci(t).

1.6. Remark. Notice that the above equation only depends on the func-tions Fik + F. Its significance will be explained presently.

By the definition of geodesics, we see that, given any point m E M and avector v at m, there exists a unique geodesic c starting at m with c'(0) = v,namely the image of the integral curve of the geodesic vector field starting


at c'(0). We denote the unique geodesic with m as origin and v as the initialtangent vector by c,. Clearly we have

c and d are two geodesics whose images coincide, thenthere exist constants a 0 and b such that d(at + b) = c(t) for all small t.

1.8. Exponential mapping.We wish to construct a map T(M) -- M x M as follows. To any

tangent vector v at a point m E M, associate the pair (m, This maynot make sense, since the geodesic vector field may not give a global flow.Consequently c,,(t) may only be defined for small values of ItI. If we restrictourselves to a neighbourhood U of M imbedded in T (M) as the zero section,then it is well defined and differentiable. This is called the exponential mapassociated to the linear connection.

The exponential map fits into the diagram:

M Id MIzero I diagonal

UCT(M) O-x-+p

MxM

If we restrict the exponential map to a neighbourhood of 0 in the one-dimensional subspace Cv of the tangent space at a point m E M, thenwe get the map tv Hence the differential at thezero vector at m of the exponential map takes v to (0, v) E T,,,, ® T,,,,. Onthe other hand, from the above diagram we conclude that the image ofthis differential contains all vectors of the form (v, v) in Tm(M) ® Tm(M).Hence the differential of the exponential map is actually surjective, and fordimensional reasons, an isomorphism.

1.9. Proposition. The exponential map takes a neighbourhood of M inT(M) diffeomorphically onto an open set in M x M. Also by restricting toone fibre, the exponential map v -+ c,,(1) at a point m E M is a diffeomor-phism of an open neighbourhood of 0 onto an open neighbourhood of m inM.

Proof. This is now simply the inverse function theorem.

1.10. Corollary. For every neighbourhood W of m, there exists a neigh-bourhood V C W with the property that any two points in V can be joinedby a geodesic contained in W.


Proof. Given W as above, we choose an open subset U of T (M) containingthe zero vector at m such that exp I U is a diffeomorphism onto an open setcontained in W x W. The image of U is an open subset containing (m, m)and so contains an open set of the form V x V, where V is an open set inM containing m. For any x E V, the exponential image of it-1(x) fl u iscontained in {x} x W and contains {x} x V. Hence if y E V, then x and ycan be connected by a geodesic in W.

We will use the exponential local diffeomorphism to identify an openneighbourhood U of m with a neighbourhood V of 0 in Tm,(M). The tangentbundle is in particular trivialised and all the tangent spaces can be identifiedwith Tm(M). Then the equation of geodesics in V has a nice expression.Take the local expression for the connection as d + a where a is a 1-formwith values in End(T,(M)). It fact we will consider it as a function withvalues in the set of bilinear maps T,(M) x Tm(M) -* Tm(M). For everym E M, let /3m denote the form obtained by symmetrising the correspondingbilinear map. Then the equation is

1.11. c" (t) + '3*l (c' (t), c' (t)) = 0.

(Here c', c" denote the first and second derivatives of c.)

1.12. Definition. Take any linear isomorphism of Tm,(M) with R. Thencomposition of the inverse of the exponential map at m with the isomorphismof Tm(M) with R' gives a coordinate system in a neighbourhood of m, calleda normal coordinate system.

1.13. Remark. Consider a normal coordinate system. On the one hand,the curves t H tv are geodesics in this system, and on the other hand, thispath satisfies the differential equation 1.11. Note that since the map is lin-ear the term a2 ipc(t) is zero and the equation reduces to Eikrjk(tv)xjxk =0, where (xi) are the coordinates of v. Letting t tend to zero, we get>rik(0)xixk = 0. This vanishes for all values of xi,xk and so the coef-ficients rk; (0) of this quadratic polynomial are all zero. Thus wehave the following

1.14. Theorem. The functions r23

+ r;i, where r are defined byV a ( ) _ >k r 9 in normal coordinates around p, are all zero atp.

We have the following improvement of Corollary 1.10.


1.15. Proposition. For any m E M, there exists an open nneighbour-hood U such that any two points in U can be connected by a unique geodesiccontained in U.

Proof. We will again use normal coordinates (U, x) at m E M, and usethe Euclidean distance from m in these coordinates. For any x, y in thisneighbourhood we will denote IIxII2 by N(x) and the inner product by b(x, y).

We have seen that there is an open ball V of radius r such that any twopoints in V can be connected by a geodesic in U. If c is any geodesic insideU that has c(O) and c(1) in V, but is not contained in V, then there exists towith 0 < to < 1 such that N(c(t)) attains the maximum at to. This impliesthat dN(c(t)) Ito = 0 and also that d2 Nt ° t) I to < 0.

We now compute d2Nd(c(t) to be

2 tb(c(t), c'(t)) = 2N(c'(t)) + 2b(c(t), c"(t)).

If c is a geodesic, then by 1.11, this becomes

2N(c'(t)) - 2b(c(t), /3 (t)(c'(t), c (t))).

Now consider the function which takes any v to the quadratic form Q : w HN(w) - b(v, 0 (w, w)). But Qo = 0 by Theorem 1.14 and so Qo is positivedefinite. Hence so is Q for all v in a neighbourhood of 0. We may cut downour neighbourhood U to an open ball such that Q, is positive definite for

2all v in it. Taking w = c(t) in the above computation, we see that ddis positive at to. This contradicts our assumption.

An open set such as guaranteed by Proposition 1.15 above is called aconvex neighbourhood.

1.16. Remark. We make two comments. Firstly, there is a fundamentalsystem of convex neighbourhoods. Secondly, an open set which satisfies theabove condition is contractible. If we are interested only in this, we couldhave taken any suitable exponential neighbourhood. But if U and V aretwo convex open sets, then clearly their intersection is also convex. Hencewe conclude that there is an open covering of M such that the intersectionof any finite set of the members of the covering is contractible. In order tocompute the cohomology of a constant sheaf, we may use the above type ofcovering, thanks to [Ch. 4, Remark 5.5, 1)]. From this one can conclude

1.17. Theorem The integral singular cohomology groups of a compactdifferential manifold are finitely generated.


2. Lifting of Symbols and Torsion

2.1. Torsion of a linear connection.Let V be a linear connection on M. Then we may try to lift some

natural first order symbols, using the procedure we have explained in [Ch. 5,Proposition 1.8]. Thus we may try and lift the first order symbol T* ®T* -*A2(T*) given by the wedge product. (According to our theory this can belifted to a differential operator if we are given a connection in T*, but aconnection in T gives a canonical dual connection in T*.) Thus we get afirst order operator from T* into A2 (T*) with this as its symbol. Notice thatthe exterior derivative which is defined without reference to any connectionalso has the same symbol.

Let us compute the lift which uses the linear connection. According toour definition, the lift is the composite of the operator V : T* T®®

T* and the wedging map T* ® T* -+ A2(T*) given by the symbol. Inother words, it associates, to a 1-form w, the 2-form given by (X, Y) H(Vx(w))(Y) - (VY(w))(X) = Xw(Y) - w(VxY) - Yw(X) + w(VYX) _dw(X, Y) + w([X,Y]) - w(VxY) + w(VyX). Thus this coincides with theexterior derivative if and only if VXY - VyX - [X, Y] vanishes for all vectorfields X, Y. Since the symbols of d and the above operator coincide, it followsthat the form (X, Y) - VxY = VyX - [X, Y] is an A-bilinear form withvalues in T.

2.2. Definition. Let V be a linear connection on a differential manifold M.The exterior 2-form with values in T given by (X, Y) H V xY-V yX - [X, Y]is called the torsion of the connection and is denoted by r(V).

It is often convenient to use a linear connection whose torsion tensorvanishes identically. Suppose V is a linear connection with torsion r. Anyother linear connection differs from V by a 1-form with values in End(T).If VX = Vx + a(X), then the torsion form r' of V is given by T'(X, Y) _VX(Y) - VY(X) - [X,Y] = Vx(Y) + a(X)(Y) - Vy(X) - a(Y)(X) -[X, Y] = T(X, Y) + l3(X, Y) - 3(Y, X) where 3 is the bilinear form definedby 0(X,Y) = a(X)(Y).

If 0 is a bilinear map into T such that ,4(X, Y) - i3(Y, X) = -T(X, Y),then r' is identically zero. For example, we can take 0(X,Y) _ -1r(X,Y).Also V and V' have the same torsion form if and only if their difference iis symmetric. Thus we have proved the following assertion.

2.3. Proposition. There exists a torsion free connection. Any two torsionfree connections differ by a symmetric 2-form with values in T.

Often we require a torsion free connection which leaves some structureon the manifold invariant, but this is a more delicate question and will be


dealt with in Chapter 7. We will only illustrate this by a simple examplehere.

A linear connection gives a connection on all the tensor bundles as well.In particular, it gives a connection on K = A''(T*). Since the sheaf S ofdensities is the tensor product of )C with the local system OR, there is alsoa natural connection on S. We have seen in [Ch. 3, Remark 2.10] thatthe sheaf of densities on a differential manifold is actually trivial, althoughnoncanonically. Let m be an everywhere positive density, thus trivialisingthe sheaf S. We will now see if there is a torsion free linear connection whichleaves m invariant.

Let V be any torsion free linear connection. Define a differential 1-formw by setting V x (m) = w(X)m. By Proposition 2.3, any other torsion freelinear connection V' is given by VX = V x + a(X) , where the bilinear map6 : T x T -p T given by (X, Y) H a(X)Y is symmetric. Now OX - OX,acting on S, associates to X, the trace of Y H 3(X, Y). Given any linearform w on T we can find a symmetric bilinear map a : T x T -f T suchthat w(X) = tr(Y -f /3(X,Y)), for all X. For example, the map (X, Y) H1/(n+1)(w(X)Y+w(Y)X) has this property. This proves the following.

2.4. Proposition. There exists a torsion free linear connection whichleaves a given everywhere positive density invariant.

2.5. Exercises.

1) Determine all linear connections with the above property.

2) Find the lift using a linear connection of the symbol of the Lie derivativewith respect to a vector field.

The computation in 2.1 above can be used to get the following generalresult. Let V be a connection on a vector bundle E. Then we have defined in[Ch. 5, 1.10, 1.12] an exterior derivative map Ai-l (T*) ®E -} Ai (T *) ®E. Onthe other hand, using a torsion free connection and the satellite connectionsV on Ai-' (T*) and the connection on E, we may lift the symbol of theabove, into a differential operator A'-'(T*) ® E -> Ai(T*) ® E. These twocoincide if the torsion of the linear connection is zero. If a is a k-form withvalues in E, then the exterior derivative da which uses the connection on Ecoincides with (XI.... , Xk+l) H E(-1)i+l(QXxa)(XA, ... , Xi, ... , Xk+l).

In particular we may take for E the bundle End(T) and for a the cur-vature form R. Then the (second) Bianchi identity [Ch. 5, 6.1] says thatdoR 0. Thus for linear connections, the identity can also be stated asfollows.


2.6. Identity (Bianchi). If V is a torsion free linear connection and R itscurvature form, then

VXR(Y, Z) = 0

where the sum is taken over cyclic permutations of the vector fields X, Y, Z.

2.7. Remark. Let C be the canonical 1-form with values in the tangentbundle T given by ((X) = X for all vector fields X. Then do( is the torsiontensor. In fact, by definition, dvC(X,Y) = VX((Y) -Vy((X) - (([X, Y]),which is just the definition of the torsion tensor.

2.8. Bianchi identity. The curvature form R of a torsion free linear con-nection satisfies the following identity:

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0

for any three vector fields X, Y, Z.

Proof. This is just a question of plugging in the value R(X, Y)Z = VxVyZ-VyVXZ - 0[X y]Z and summing it cyclically. Using the vanishing of thetorsion tensor, the left side of the above identity reduces to the cyclic sum

EVX([Y, Z]) +Vy([Z, X]) - V[XY]Z

which in turn simplifies to

[X, [Y, z]] + [1', [Z, X]] + [Z, [X, Y]]

This is zero, thanks to the Jacobi identity [Ch. 2, 2.1]. One can also givea slick proof by observing that the expression ER(X,Y)Z is the exteriorproduct of R and the 1-form ( with values in T given by ((X) = X forall vector fields X. To give a meaning to the exterior product, we use theobvious pairing End T ® T -> T. On the other hand, dvc is simply thetorsion tensor T which has been assumed to be zero. Hence our assertionfollows from the Bianchi identity in [Ch. 5, 6.1].

2.9. Remark. This is known as the first Bianchi identity. But since thesecond one involving derivatives, stated in [Ch. 5, 6.1], is valid for any con-nection on a principal bundle, it was stated earlier.


2.10. Lifting of a symbol and its adjoint.Let s : E --+ T ® F be a first order symbol. Its adjoint has been defined

to be the negative of the natural transpose map adj(s) : F* - T ®E*. Ifwe choose connections VE and VF on the vector bundles E and F, then wecan lift these into differential operators DS : E -* F and Dadj(s) : F* -' E*respectively. It is natural to ask how Dadi(8) is related to the adjoint operatoradj(D,). Firstly, the adjoint of D,s is actually an operator from F* 0 S toE* 0 S. But then S can be trivialised. Let us in fact fix an everywherepositive density (trivialising S) and a torsion free linear connection whichleaves it invariant. We would like in this case to compare Dam(s) and adj (D,).By definition DS = s o dyE. Hence its adjoint is given by adj(dyE) o adj(9).So we need only determine adj (dv,) : T ® E* ® S -* E* ® S.

We may apply evaluation e(X) on a vector field X and then computethe adjoint, namely adj(Vx). Then the adjoint of V itself is obtained bythe equation adj(Vx)(s, µ) = (adj do)(X, s, Ec). As for adj(Vx) we have thefollowing easy computation.

2.11. Proposition. The adjoint of Vx, where V is a connection on avector bundle E and X a vector field, is -L(X) acting on E* 0 S. HereL(X) is the Lie derivative of densities with values in E* with respect to thedual connection on E*.

Proof. Since the statement is local, there is no harm in assuming that Eis trivial. We will then compare the adjoint of the given connection andthe trivial connection V tr on E. We will identify S with IC and make thecomputation. Let V = Vtr + a, as usual. Then adj Vx = adj(Vtr)x +a(X)t. The assertion for the trivial connection is actually the definitionof the adjoint of a vector field. On the other hand, the Lie derivative withrespect to the dual connection and that with respect to the trivial connectiondiffer by -at, so that we have -Ly(X) = -Ltr(X) + a(X)t. This provesthe proposition.

We have seen that if we take a torsion free linear connection and usethe induced connection on S, then the Lie derivative using the connectionis the same as the canonical Lie derivative. Hence in this case, we concludethat the adjoint of V is obtained by lifting the adjoint symbol, using forthe lifting, the dual connection on E* and a torsion free linear connection.From this we get in general the following result.

2.12. Theorem. Let s : E --+ T ® F be a first order symbol and D,s theE --> F which lifts s using a connection V on E. Lift the adjoint symboladj (s) : F* 0 S -+ T ® E* 0 S using a connection on F and the connectionon S induced by a torsion free linear connection on the manifold. If the


symbol s is invariant under the connections chosen, then the lift of adj (s) isthe same as the adjoint of Ds.

Proof. We can write DS ass odvE. Hence adj(D8) = adj(dvE) o ((s)t (9 Is).We just observed that adj (dvE) is the lift of the adjoint of the symbol of VE,namely -IT®E*®s. The composite symbol is -(s)t ® Is = adj(s). Underour assumption then, the lift of the composite symbol adj(s) is adj(D3).

2.13. Linear connection on a submanifold.Let M be a differential manifold and N a submanifold. Then we have

the following exact sequence of bundles on N:

O-TN ->TMIN --Nor(N,M)-->0.We know that this sequence always splits. Assume given a specific splittingt : TM I N -> TN. If V is a linear connection on M, then it induces aconnection i*V on the restriction TMIN. Using the splitting we can definea connection on TN. In fact, if X, Y are vector fields on N, we defineV Y = t((i*V)XY), where i is the inclusion of N in M.

2.14. Exercise. If the linear connection on M is torsion free, then findthe condition on the splitting which will ensure that the induced linearconnection on N is also torsion free.

2.15. Cartan connections.We conclude this chapter by giving the definition of a related notion

called Cartan connection. Suppose G is a (connected) Lie group and H aclosed subgroup. Consider the principal H-bundle G -+ G/H. If P is anyprincipal H-bundle on a differential manifold M, then we may consider thevector bundle associated to P by the isotropy representation of H on g/C .

The set-up for a Cartan connection is an isomorphism of this associatedbundle with the tangent bundle of M.

2.16. Examples.

1) The structure group of the tangent bundle can be extended to the affinegroup G = GA(n, R), using the splitting H = GL(n, R) - GA(n, R).The isotropy representation of GL(n, ]l8) in g/Cl is simply the naturalrepresentation of GL(n, R) in R n. Take for P the bundle of frameson the tangent bundle. Clearly the bundle associated to the isotropyrepresentation is the tangent bundle.

2) Let V be an (n + 2)-dimensional real vector space with a quadraticform q of signature (n + 1, 1). The group SO(q) is not connected buthas two connected components. Let G be the connected componentcontaining 1. Then there is a natural G-action on the quadric Q =


{v E V : q(v) = 0} in P(V). It is easy to check that G acts transitivelyon Q. Let H be the isotropy subgroup at a point vo of Q. Then theisotropy action of H is an n-dimensional representation. There is anatural homomorphism X of H into Rx, defined by hvo = X(h)vo. Thekernel acts on vo /lEBvo and maps isomorphically onto SO(q') where q'is the induced positive definite metric on v1/Rvo. Now it is easy tosee that the isotropy representation in g/Cj of H can be identified withthis space and the action satisfies q(hv) = (X(h))2q(v). Given a metricon T (M) we can get an H-bundle to which it is associated via theisotropy representation. A G-connection is then a Cartan connection.

2.17. Definition. A Cartan connection is a connection on the G-bundlesuch that there is an isomorphism a of the tangent bundle with the isotropybundle. This isomorphism considered as an 1-form is called the solderingform.

The curvature of the Cartan connection is a 2-form with values in theadjoint bundle with g as fibres, and so gives also a 2-form with values in theisotropy bundle. This is called the torsion tensor of the Cartan connection.

Let us consider Example 1) above. In this case we take the soldering formto be the identity. The Cartan connection in this case is called an affineconnection. Since GL(n, Ilk) is a quotient of GA(n), the affine connectionhas as its image a linear connection. Therefore the curvature of the affineconnection has also two components, namely the curvature of the linearconnection, and a 2-form with values in the tangent bundle. The latter iseasily computed to be the absolute derivative of the 1-form a. Thus we haveda(X, Y) = Vxa(Y) - Vy (a) (X) - a([X, Y]). Noting that a is the identityform, we conclude that the torsion form of the affine connection, consideredas a Cartan connection, is the torsion of the linear connection.

Exercises

1) Let M be a manifold provided with a linear connection. Write downthe condition that the horizontal lift of a vector field X on M tothe principal tangent bundle leaves the connection form invariant, interms of the Lie and covariant derivatives with respect to X.

2) Show that the set of all vector, fields on M which leave a given linearconnection invariant is closed under brackets and forms a Lie algebraof dimension < n2 + n, where n is the dimension of M.

3) Consider the principal bundle of frames of the tangent bundle of adifferential manifold. Construct a canonical differential 1-form onit with values in IE8'. Show that it is equivariant under the action

Exercises 183

of GL(n) and is zero on vertical vectors. Given a connection form,compute its exterior derivative with respect to the connection.

4) Suppose given a diffeomorphism A of M of finite order. Show that thereexists a linear connection on M invariant under it. If m E M is anisolated fixed point, then show that it is also infinitesimally isolated,that is to say, the differential of its action at T,,,,(M) does not have 1as an eigenvalue.

5) Suppose G is a connected Lie group and H a closed subgroup. Assumegiven a vector subspace m of g which is invariant under the isotropyaction of H and is supplementary to lj. Then show that there is acanonical connection in the principal H-bundle G -+ G/H. Computeits curvature form in terms of the Lie algebra structure.

6) Show in the above example that the bundle associated to the principalH-bundle G --> G/H for the isotropic action of H is the tangentbundle of G/H. Under the same hypothesis as above, compute thetorsion tensor field of the linear connection given by the canonicalconnection.

7) A linear connection on a differential manifold M is said to be locallysymmetric if for every m E M there is an involution with m as anisolated point that leaves the connection invariant. Show that a linearconnection is locally symmetric if and only if it is torsion free and thecovariant derivative of the curvature form is zero.

8) Let M be a differential manifold and let the tangent bundle be thedirect sum of two subbundles E and F. Assume that there existsa J-structure on E. Then find the condition for the existence of alinear connection which preserves the decomposition as well as theJ-structure on E.

Chapter 7

Manifolds withAdditional Structures

We are interested here in a general study of differential manifolds which areendowed with some additional structures. Many of these structures involve,in the first place at least, a reduction of the structure group of the tangentbundle from GL(n, II8) to some standard subgroup.

1. Reduction of the Structure Group

1.1. Orientation.We have already encountered a few examples of such structures. We

start with the case when the subgroup is GL(n, R)+, consisting of lineartransformations with positive determinant. A reduction to this subgroup ofthe structure group is equivalent to providing the differential manifold withan orientation. We know that such a reduction is not in general possi-ble. In fact, we have seen in [Ch. 5, Proposition 3.14] that a reduction isequivalent to giving a section of the bundle P/GL(n, R)+, where P is theprincipal bundle of frames of T. This has fibre type GL(n, R)/GL(n, R)+.The determinant homomorphism into R" induces an isomorphism of thisquotient with RI /R+. Thus P/GL(n, R)+ - M is a 2-sheeted coveringspace of M. If the total space of this covering is connected, it admits nosection and so there is no such reduction. If it is not, it consists of two con-nected components both of which map isomorphically onto M. Thus in thelatter case, there are two possible sections, that is to say, two orientations,said to be opposites of each other.

185

186 7. Manifolds with Additional Structures

1.2. Riemannian structure.To give a symmetric 2-form g, which is positive definite at every point,

is equivalent to a reduction of the structure group of the tangent bundle toO(n). This datum is called a Riemannian structure on M. We have alreadyremarked [Ch. 5, Example 3.1, 1) and Remark 3.12] in the general contextof arbitrary vector bundles that such a structure always exists. Indeed,locally we may choose a coordinate system (xl, ... , xn) and take the form

dx?. This gives a Riemannian structure locally. If gi are Riemannianstructures on open sets Uj of a covering, and cpi a partition of unity withrespect to this covering, then E pigi is a global Riemannian metric, since aconvex combination of positive definite forms is again one. If the manifold isoriented, then a Riemannian structure actually gives a reduction of thestructure group to SO(n).

If we replace the condition of positive definiteness of the symmetric ten-sor by nondegeneracy, then, over a connected set its signature does notchange, and so we may as well say that the form has a fixed signature (p, q)at all points of M. This is equivalent to a reduction of the structure groupto O(p, q) for a fixed pair (p, q) of positive integers with p + q = n. We haveremarked that such a structure (if p 0, n) need not exist in general. If itdoes, the structure is called a pseudo-Riemannian structure with signatureof type (p, q).

1.3. Definition. A manifold provided with a Riemannian (resp. pseudo-Riemannian) structure is called a Riemannian (resp. a pseudo-Riemannian)manifold.

1.4. Examples.

1) The tangent space at any point of any real vector space V of finitedimension is identified with the vector space V itself. Hence any posi-tive definite symmetric bilinear form on V gives rise to a Riemannianstructure on it. Usually on Rn, we take the Euclidean inner productg(x, y) = E xiyi. On the other hand, if we take the bilinear form onR4 given by g(x, y) = -xi+x2+x3+x4, we get a pseudo-Riemannianmanifold with signature (3, 1). This is the Lorentz metric on R4, basicto the study of special relativity.

2) There is a natural metric on the unit sphere S. The tangent space ata point x E I[8n+1 with E x? = 1 consists of the vectors v = (vi) withE xivi = 0. If v is such a vector we define g(v) = E q?. It is clearthat this gives a Riemannian structure on S. Whenever we refer toSn as a Riemannian manifold, we have in mind this structure.

1. Reduction of the Structure Group 187

1.5. Exercise. Carry out the construction of the second example in thecontext of a vector space V provided with a symmetric nondegenerate bi-linear form 0 of signature (p, q), p 54 0. Show that the subset W = {v EV : /3(v) = 1} is a closed submanifold of V. What is the signature of theinduced pseudo-Riemannian structure on the submanifold W?

1.6. Remarks.

1) Obviously the second example above is derived from the first. Thisis a general procedure. If M is a Riemannian manifold, and N isa submanifold, then there is a natural Riemannian structure on thesubmanifold. In fact, for any x E N, we may restrict the metric onthe tangent space TA(M) to the subspace T__(N) and obtain in thisway a Riemannian structure on N. The Riemannian manifold thusobtained will be called a Riemannian submanifold. In fact, for thisconstruction, it is enough if N is taken to be a manifold togetherwith a differentiable map into M such that the differentiable map isinjective at all points of N. The structure obtained in this way is saidto be induced from that of M.

2) Notice, however, that a similar construction does not work in the gener-ality of pseudo-Riemannian manifolds. For, while the restriction of apositive definite form to any vector subspace remains positive definite,the restriction of a nondegenerate form need not remain nondegener-ate. However, if M is provided with a pseudo-Riemannian structure ofsignature (p, q), and N is a submanifold of dimension q such that therestriction of the bilinear form on T (M) to Tx (N) is positive definitefor any point x of N, then N gets a natural Riemannian structure.Although this is quite an artificial situation, it does occur in the fol-lowing case. Consider the closed subspace H of 1[8n+1 given by theequation En

i=1 xi - xn+1 = -1. It is easily verified (using Ch. 1, Ex-ample 3.2, 4)) that it is actually a differential submanifold of Rn+1It is not connected since xn+1 is never zero. Indeed it has two con-nected components given by xn+1 > 0 and xn+1 < 0. Consider therestriction of the projection (x1, ... , xn+i) (x1, ... , x,) of Rn+1 to]Rn to each of these. Clearly it is bijective and the inverse is given by(x1, ... , xn) H (x17 .... xn, Z 1 x? + 1). It is also differentiable

and has the map (vi) --> v v aivi as its differential at ap (a) is ... nat+1

point (a2) E 11n. We now provide Rn+1 with the pseudo-Riemannianstructure given by the quadratic form EZ 1 xi - xn+1 This gives

the quadratic form (vi) H (> v2)2 - aivi 2on the tangent space atE ai+1

(ati) E 118n. It is always positive whenever at least one v2 is nonzero. In


other words, the induced quadratic form on the tangent space at anypoint of H is positive definite. Hence we get a Riemannian structureon H. This space is called the hyperbolic space of dimension n. Thisexample will come up again later in 6.13.

3) Clearly the Riemannian structure on Rn is invariant under translations.Since the translatory action is transitive, it is determined by the met-ric on the tangent space at 0. If T is any linear transformation, thenT takes the Riemannian metric to another metric, also invariant un-der translations. The new metric has, on the tangent space at 0, thetransform by T of the standard metric, namely T'T. In particular, theEuclidean metric is invariant under the natural action of the orthog-onal group. Putting these together we conclude that the Riemannianmetric is invariant under the action of the Euclidean motion group,namely the group generated by orthogonal transformations and trans-lations.

Also the sphere is invariant under the above action of O(n). Theinduced Riemannian structure on Sn is clearly invariant under thisaction.

4) If M and N are Riemannian manifolds, then one can provide M x Nwith a natural Riemannian structure, called the product of M and N.In fact, the tangent space at any point (m, n) of M x N is canonicallythe direct sum of Tm,(M) and Tn(N). Since both of these spaces comewith natural metrics, we can provide the direct sum with a metricstructure with respect to which the two subspaces are orthogonal.

5) Consider a compact Lie group G acting on a manifold M. Let g be aRiemannian structure on M. Then we may average the metric overthe group G to obtain a G-invariant Riemannian structure. In otherwords, if v, w are tangent vectors at a point m E M, then we defineg(v, w) to be f g(xv, xw)dx where the integration is taken over G withrespect to the invariant measure dx on G, and xv, xw are the vectorsat xm obtained as translates of v, w by the action of x E G. It is easyto verify that the Riemannian structure is G-invariant in the obvioussense.

6) We may construct on any Lie group G a metric invariant under lefttranslations. As in the case of the Euclidean space, this metric de-pends only on the metric on the tangent space at one point, say 1.If we need this to be right invariant as well, this metric ought to beinvariant under the action of G on the tangent space at 1, given byleft translating by g and right translating by g-1, in other words, theadjoint representation of G in its Lie algebra g. If G is compact, such


a metric exists in view of 5) above, and thus we can find a biinvariantRiemannian metric in this case.

1.7. Almost symplectic structure.Let w be an alternating 2-form on T. We assume again that this form

is nondegenerate. This is called an almost symplectic structure on M.

1.8. Definition. An almost symplectic manifold is a differential manifoldtogether with a nondegenerate (exterior) 2-form.

The existence of a nondegenerate alternating bilinear form on a vectorspace implies that the dimension of the vector space is even. In particular,it follows that an almost symplectic manifold is even-dimensional. Howeverthis alone will not ensure that the manifold admits an almost symplecticstructure. For example the symplectic form on a 2-dimensional manifoldgives an everywhere nonzero section of A2 (T*). This implies that the differ-ential manifold is oriented.

1.9. Exercise. Show that an almost symplectic manifold M of dimension2m has an everywhere nonzero 2m-form. Deduce that M is oriented.

1.10. Remark. This datum is equivalent to a reduction of the structuregroup of the tangent bundle to the symplectic group Sp(n, Ia). For, givenany nondegenerate, alternating bilinear form w on a real vector space V,one can find a basis (ei, fi) such that w(E aiei + E bjfj, E aiei + E bzfi) _

aibi - E biai. In fact, choose any nonzero vector (if V (0)), say el.Since w is nondegenerate, there is at least one vector v such that w(el, v) 0.Dividing v by a nonzero scalar, we may as well assume that w(el, v) = 1.Define fl to be such a vector. Clearly el and fl span a two-dimensionalsubspace V'. Take the subspace {w E V : w(el, w) = 0 and w(fl, w) = 0}.If dim V is greater than 2, this space is nonzero and supplementary to W.The restriction of w to it remains nondegenerate. Thus we find a basis asabove by an induction on the dimension.

The group GL(2n, If8) acts on the space of all nondegenerate alternatingbilinear forms by the prescription g(w) (v, w) = w(g-lv, g-1w). Our remarkabove shows that this action is transitive, and the isotropy group is bydefinition Sp(n, Il8). Thus an almost symplectic structure is simply a sectionof P/Sp(n, R) where P is the principal tangent bundle of frames.

1.11. Examples.

1) Consider I[82n. The tangent space at any point can be canonically iden-tified with R2n itself. Hence the choice of any nondegenerate alter-nating bilinear form on the vector space IE82n, for example, w(x, y) _


iz 1(xiyn.+i - xn+iyi), gives rise to a symplectic structure on themanifold ][82n. This differential form has the coordinate expressionEi=1 dxi A dxn+i.

2) Let N be any manifold. Consider its cotangent bundle as a manifold M.Let 7r be the bundle projection M -+ N. Any point m E M consists,by definition, of a point a E N and a 1-differential a at a. Then wehave rr(m) = a. If v E Tm(M) is a tangent vector at m, then thedifferential of 7r at m maps v to a tangent vector at the point a. Thescalar (drr(v), a) depends linearly on v and so defines a tautological1-form 0 on M.

It is easy to compute it in local coordinates. Let (xi) be a coordi-nate system on an open set U on N. Then -

-may be considered aaxi

function on 7r (U) which is linear on fibres. A coordinate system forM in it-1(U) = U x W" is given by (xi o -7r, as ). Let us denote this

7

system of coordinates by (qi, pi). We will now compute (d7r)(agti). Its

value on the coordinate function xj is given by ag- (xj o7r) _ $ (qj) _Si... Hence dir(-) = xi. Also d7r(j)(xj) _ -(xj oir) =app (qj) = 0.

If v is the vector E ai aez + E b3 app at the point (qi, pj), then d,7r(v) _

>ai . On the other hand the point (gi,pj) represents the differen-tial E pj 0. Hence i(v) = (> ai&, > pj dxj) = E aipi. In otherwords, we have Q = > pjdqj. Incidentally this also shows that theform 0 on M that we defined is differentiable. The exterior derivativeof this form is the 2-form E dpi Adqj, which is clearly nondegenerate.Thus there is a canonical almost symplectic structure on M = T* (N).

1.12. Remarks.

1) Notice that in the examples we have given above, the symplectic formis actually a closed 2-form. In that case, the manifold is said to be asymplectic manifold. We will return to this question later (4.11, 4.12).

2) In the Hamiltonian formulation of classical mechanics, the cotangentbundle with the symplectic form is the basic manifold on which thedynamical equations are written. In this context, if M is the config-uration space, the manifold T* (M) is called the phase space.

1.13. Almost complex structure.

1.14. Definition. Let M be a differential manifold. An automorphism Jof the tangent bundle (that is to say, a gauge transformation of the tangentbundle) satisfying j2 = -I is called an almost complex structure on M. A


manifold provided with such a structure will be called an almost complexmanifold.

As in the case of a symplectic manifold, an almost complex manifold isnecessarily even-dimensional. This follows from the fact that any transfor-mation J of a (finite-dimensional, real) vector space V with j2 = - Id issemisimple and has eigenvalues ±i, the multiplicities of these two eigenval-ues being equal. The vector space V OR C is a direct sum of the eigenspacescorresponding to i and -i (which we denote respectively by V1"0 and V°'1)The complex conjugation taking v+iw E V ®(C, with v, w E V, to v-iw, in-terchanges the two eigenspaces. Hence dim V ®C = 2 dim V 1,O = 2 dim V°,1

Taking V to be the tangent space T,,,, at a point m, we deduce that the di-mension of M, which is the same as the dimension of Tm for any m E M, iseven. The complex dimension of an almost complex manifold is defined tobe half the dimension of the differential manifold.

An almost complex structure is simply a reduction of the structure groupof the tangent bundle from GL(2n,18) to the subgroup GL(n, (C). HereGL(n, C) is imbedded in GL(2n, R) by the map

(azj) C Re(aaz))

It is clear that GL(2n,R) acts by inner automorphisms on the set of alltransformations J of 1182n satisfying j2 = - Id. We have shown above thatwith respect to a suitable basis of V, the matrix of J takes the form (I nn ).Hence the above action is transitive. Identifying R2n with Cn we get onesuch transformation, namely multiplication by i. The isotropy group atthis point is the set of real linear transformations of JR2n, commuting withmultiplication by i, that is to say, the set of complex linear transformations.Thus we may identify GL(2n, lib)/GL(n, (C) with the set {J E GL(2n,1[8) :J2 = -I}. From this we conclude that an almost complex structure on adifferential manifold is simply a section over M of the principal bundle offrames of the tangent bundle, modulo the action of GL(n, C). This amountsto a reduction to GL(n, C) of its structure group.

We note that if V is as above, then V* also has a transformation J withJ2 = - Id. In particular, it is a direct sum of two subspaces Vl;o and Vo,,as above. These may also be characterised as the spaces of (complex) linearforms on V ® C that vanish on V°'1 and V1"0 respectively. If we take theexterior powers A' (V* ® C) = Ar (V *) ® C, then they break up into a directsum of subspaces AP(Vl*o) ® Aq(V0,l) with p + q = r. These subspaces maybe denoted AP,q(V*). If we regard elements of A' as alternating r-forms onV 0 C, then AM (V*) consists of those forms which vanish whenever morethan p of the vectors belong to V 1,O or more than q of the vectors belong to


V°'1. It is easy to check that the complex conjugation in V ® C induces anisomorphism of AP, (V*) with A",P(V*) (as real vector spaces). Thus in analmost complex manifold it makes sense to talk of a differential form of type(p, q). It is a section of the bundle A' (T*) ®C which belongs to AM (T;,) forall m E M. The real vector bundles AM (T*) and Aq°P(T*) are isomorphic,thanks to complex conjugation.

1.15. Remark. Any complex manifold has a natural almost complex struc-ture. For if (U, zi, ... , zn) is a complex coordinate system, then the realtangent space at any point in U can be identified with Cn. Hence there isan operator J on the tangent space, obtained by transporting the operationof multiplication by the complex scalar i. The key point is that if we takeany other coordinate system, the two differ by a holomorphic map (ff(z))of a domain in Cn with another. The induced map on the tangent spaceconsidered as R2n is given by a matrix of the type

8 Re (f2) 8 Re(fi)x 8y8I8m(?fz)

8xj 9yj

Here xj, yj are respectively the real and imaginary parts of zj, and the matrixhas been written with respect to the basis ax; , eye of the tangent space. Inthis basis the transformation J we have defined is given by the matrix

(1-1

1 0

The condition that these two matrices commute is equivalent to the condi-tions:

8 Re(fz) _ 8Im(fz)8xj 8y?

8Im(fz) _ 8Re(f2)8xj 8yj

These are the Cauchy-Riemann equations satisfied by holomorphic func-tions. Since our transition functions are holomorphic, it follows that the al-most complex structure we have defined using a coordinate system, is indeedcanonical.


1.16. Almost Hermitian structures.

1.17. Definition. A differential manifold which has a Riemannian struc-ture g as well as an almost complex structure J such that g(Jv, w) +g(v, Jw) = 0 for all pairs of tangent vectors v, w at any point, is calledan almost Hermitian manifold.

Notice first that if we consider the real tangent space as a complex vectorspace, using J, then g is the real part of a unique Hermitian form on it.This is again a statement on vector spaces. Let V be a finite-dimensionalcomplex vector space and g : V x V -+ R a symmetric I[2-bilinear formsatisfying g(iv, w) + g(v, iw) = 0 for all v, w E V. Then the III-bilinear formw given by (v, w) H g(v, iw) is actually alternating. In fact, setting v = win the above equality, we get g(v, iv) = -g(iv, v), while symmetry impliesthat g(v, iv) = g(iv, v). Moreover, w also satisfies w(v, iw) + w(iv, w) = 0.Now the IR-bilinear map V x V -* C given by

B : (v, w) H g(v, w) + iw(v, w)

is directly verified to be a Hermitian form, namely C-linear in v, antilinearin w and Hermitian symmetric, i.e. B(v, w) = B(w, v).

1.18. Exercise. Prove that any alternating form w which satisfies w(iv, w)+ w(v, iw) = 0 is the imaginary part of a unique Hermitian form on it.

Now the data given by the pair (J, g) as above is equivalent to givinga reduction of the structure group from GL(2n, IR) to U(n). In fact, thealmost complex structure gives a reduction to the subgroup GL(n, C). Toevery Hermitian form H one can associate the matrix H(e2, ej) where (ei) isthe standard basis of C'. This gives a bijection between Hermitian forms andmatrices A that satisfy A = A . The group GL(n, C) acts on all Hermitianmatrices by the prescription (g, H) -- gH(gt)-1. This action is transitiveon the set of positive definite Hermitian matrices, and the isotropy group atthe identity is the unitary group U(n). Thus a Hermitian structure onthe reduced complex vector bundle gives a further reduction to U(n). Onthe other hand, we have remarked that the form w(v, w) = g(v, Jw) atevery point is actually an alternating form. Moreover, this form is alsonondegenerate. For, if w(v, w) = 0 for all w, i.e., g(v, Jw) = 0 for all w, weconclude, since g is nondegenerate, that w = 0. Thus an almost Hermitianstructure induces an almost symplectic structure as well.

1.19. Exercises.

1) Show that any almost symplectic manifold admits an almost complexstructure.


2) Show that any almost complex manifold admits an almost Hermitianstructure and hence an almost symplectic structure also.

2. Torsion Free G-Connections

Since all the structures we have discussed above are simply reductions ofthe structure group of the tangent bundle to a subgroup G of GL(n, R), it isclear that given such a reduction, there exists a connection on the reducedprincipal G-bundle. A connection on the reduced bundle gives rise to aGL(n, I[8)-connection on the tangent bundle, namely, a linear connection. Ineach of these cases, they may be characterised as linear connections whichleave a suitable tensor invariant. For example, in the case of a Riemannianstructure, an O(n)-connection is simply a linear connection which satisfies17x(g) = 0 for all vector fields X. This can be expanded to the equation

X9(Y, Z) - 9(VxY, Z) - g(Y,VxZ) = 0

for all vector fields X, Y, Z.

2.1. Exercise. Write down explicitly the condition that a linear U(n)-connection on an almost Hermitian manifold satisfies.

However, in general, G-connections which, as linear connections, aretorsion free may not exist. We will first investigate the question whether,given a finite group H of gauge transformations of the tangent bundle, thereexists a torsion free connection which is H-invariant. We have already seen[Ch. 5, 2.12] that H acts by affine transformations on the space C of alllinear connections. The linear part of this action on (T* ® T* ® T) (M) isgiven by the trivial action on the first factor and the natural action on theother two.

In addition there is also an involution r acting on C. For any linearconnection V, we define T(V) by the formula

(r(V))xY =VYX + [X, Y]

for any two vector fields X, Y. In order to convince ourselves that r(V) isalso a linear connection, the additivity in X and Y being obvious, we needonly check that (r(V))x(fY) = (X f )Y + f (r(V))xY, and (r(V)) fxY =f (r(V))xY for any f E .A(M). But these follow directly from the definition.Moreover this action too is affine, the linear part being the transposition ofthe first two factors in (T* ® T* (9 T) (M). We will collect these facts in thefollowing.

2. Torsion Free G-Connections 195

2.2. Proposition. There is an affine involution acting on the affine spaceof all linear connections, whose linear part is the transposition of the firsttwo factors of (T* ® T* 0 T) (M). The fixed point space of this involutionis the space of torsion free connections.

Proof. The only thing that remains to be checked is the last statement.But obviously, saying that a linear connection is fixed under r is only arestatement that the torsion tensor is zero.

The question whether a torsion free linear connection invariant underH exists is therefore the same as finding a connection fixed under the affineaction of H as well as T. Now the action of H and T can be put together.Let K be the free product H * Z/2 of H and the group {1, r} of order2. Then the action of H and of r together means that K acts by affinetransformations of C. The linear part of this action of K on T* (9 T* 0 T ischaracterised by a) H acting as the tensor product of the trivial action onthe first factor and the natural (gauge) action on the other two, and b) Tacting as the transposition of the first two factors. If some element of K haslinear part zero, then it acts as translation. If this translation is nontrivial,then there can be no K-fixed connection. So we will try and analyse thelinear action.

We first show that the linear part of the action of K factors throughto a smaller group. Indeed, consider the semidirect product P of Z/2 byH x H x H where the nontrivial element of Z/2 acts by transposition of thefirst two factors. Thus P consists of elements of the form ((hl, h2, h3), 0)and ((hl, h2, h3),1). It contains H x H x H x {0} as a normal subgroupwith quotient Z/2.

The homomorphism of H into P given by h ((1, h, h), 1) and thehomomorphism Z/2 -> P given by r H ((1, 1, 1), r) together define (by theuniversal property of free products) a homomorphism of K into P. Now Pacts on (T* ® T* (9 T) (M) by H x H x H acting by the tensor product ofthe natural actions, and T acting by transposition of the first two factors.Clearly the action of K factors through to this action of P. In fact one caneven compute the image of K in P.

2.3. Proposition. Consider the semidirect product P of Z2 = {1, T} byH x H x H for the action of T on H x H x H given by the transpositionof the first two factors. The subgroup Q of H x H x H consisting of ele-ments of the form (a, b, c) with c = ab mod([H, H]) is invariant under thetransposition. Let R be the corresponding semidirect product of Z/2 by Q.The homomorphism of K = H * Z2 into P which maps h to (1, h, h) for allh E H and the generator T of Z2 to the transposition, has R as its image.


Proof. If c = ab mod([H, H]) and c' = a'b' mod([H, H]), then cc' =aba'b' = aa'bb' mod ([H, H]) and thus one checks that Q is a subgroup. Thetransposition of the first two factors leaves it invariant, for (a, b, c) belongsto Q if and only if (b, a, c) does, since ab = ba in H/[H, H].

Consider the subgroup k of K consisting of elements in H * 7L2 where Toccurs an even number of times.

In order to show that the image of K is R, it is enough to verify that Qis the image of K. Clearly, k contains H, and is in fact generated by H andelements of the type rxT with x E H. The image of H is clearly containedin Q and the image of TxT is (x, 1, x) which again belongs to Q. Hence theimage of k is contained in Q.

Any element of Q is a product of elements of the form (a, b, ab) and(1,1, aba-lb-1) by definition. But (a, b, ab)(a-1, b-1, a-1b-1) = (1,1,aba-'b-1) and (a, b, ab) (a, 1, a)(1, b, b), so that Q is actually the image ofK, proving the proposition.

Let N be the kernel of the surjective homomorphism of K into R. Bydefinition, in the affine action of K on the space of linear connections, allelements of N have linear parts 0, that is to say, we have a homomorphismof N into the group of translations, namely the space of sections of T* T®

* ® T. Since this group is abelian, this homomorphism factors through tothe derived group Nl [N, N]. In fact it actually gives an 118-linear map for ofthe vector space N/[N, N] ®1R into the space of tensor fields of the abovetype. We have already made the following simple observation.

2.4. Proposition. A necessary condition for the existence of an H-invari-ant torsion free linear connection is that the linear map for : N/[N, N] OR ->(T* ®T® (9 T)(M) be 0.

2.5. Remark. The action of the gauge group on C is not effective. Thesubgroup consisting of nonzero scalars acts trivially on C. So if the group Hcontains nontrivial scalars, it is better to pass to the quotient H' of H, bythe intersection of H with R'. Correspondingly K needs to be replaced byH' * Z/2 and R by the quotient of R by the subgroup consisting of (a, b, ab)with a, b scalars in H.

The question arises whether the necessary condition in Proposition 2.4is also sufficient. In fact, if for is 0, then the affine action of the group Kon C also factors to R. If H is finite, R is also finite. Now a finite group ofaffine transformations always has a fixed point. For example, we may takeany connection V and take the barycentre of the polygon (g(o)), g E R.Thus we have

3. Complex Manifolds 197

2.6. Theorem. If H is a finite subgroup of the gauge group, a necessaryand sufficient condition for the existence of an H-invariant torsion free con-nection is that the linear map for vanishes.

2.7. Remark. Notice that if H is finite, the subgroup N of H * Z2 is offinite index since R is finite. Hence it is finitely generated, implying thatN1 [N, N] OR is finite-dimensional. Thus the vanishing of for is equivalentto the vanishing of finitely many tensor fields.

2.8. Exercise. Show that if the subgroup H of the gauge group containsany element of the form m(f), namely multiplication by a nonzero (non-constant) function, then there does not exist any torsion free connectioninvariant under it.

3. Complex Manifolds

Let M be an almost complex manifold. We start by asking for conditionsfor the existence of a torsion free linear connection which leaves the almostcomplex structure invariant. This is the particular case of the discussion inthe previous section, in which the subgroup H of the gauge group is Z/4 andthe action is given by the almost complex structure J. Then H' = Z/2 andthe nontrivial element acts via J. In this case, we will compute tor. Firstlywe have to compute N = ker : K' = Z/2 * Z/2 --+ Q'. Let j, T denote thegenerators of the two cyclic groups of order 2. Then T j -r maps to (j, 1, j),while j maps to (1, j, j). From this it is obvious that (r j )4 is in the kernel.Indeed, this element generates the kernel, since the quotient of Z/2 * Z/2by the normal subgroup generated by (r j )4 is of order 8, which is also theorder of the group R'.

If V is any connection, then 'i-j (V) is the connection v(1) defined byV ")(Y) = [X,Y] + (jV)yX = [X,Y] - J(Vy(JX)). Hence (-rj)2(V) _V 2) is given by VX)Y = [X, y] - J(DYl)(JX)) = [X, y] - J([Y, JX] -JVjx(JY)) = [X,Y] - J[Y,JX] - Djx(JY). From this we deduce that((rj)4(V))XY = [X, y] - J[Y, JX] - V j2X(JY) _ [X, y] - J[Y, JX] -[JX, JY] - J[JY, X] + V XY. This is just the translation of V by the tensorfield [X, Y] - J[Y, JX] - [JX, JY] - J[JY, X]. In other words, the element(rj)4 of N is mapped by for to this tensor field.

3.1. Definition. If J is an almost complex structure, then the tensor fieldtaking vector fields X, Y to

[X, Y] - J[Y, JX] - [JX, JY] - J[JY, X]

is called the torsion tensor field of the almost complex structure.


From Theorem 2.5 and the above computation, we conclude

3.2. Theorem. The vanishing of the torsion tensor field of an almostcomplex structure J is a necessary and sufficient condition for the existenceof a linear torsion free connection invariant under J.

If M is an almost complex manifold, then the complexified tangent bun-dle breaks up into a direct sum of two bundles T1"° and T°'1 according tocomments following Definition 1.14. In the following we will extend thebracket operation to complex vector fields C-bilinearly. If the torsion is zeroand if X, Y are vector fields of type (1, 0), then [X, Y] is also of type (1, 0).For JX = iX and JY = iY and the torsion tensor field takes the value[X, Y] - iJ[Y, X] + [X, Y] - iJ[Y, X] on (X, Y). Its vanishing implies there-fore that J[X, Y] = i[X, Y]. Conversely, if the bracket of any two vectorfields of type (1, 0) is again of type (1, 0), then take two real vector fields Xand Y and note that the vector fields X - iJX and Y - iJY are of type(1, 0). By assumption, the vector field [X - iJX, Y - iJY] is also of type(1, 0) and so we have J[X -iJX, Y-iJY] = i[X -iJX,Y-iJY]. Equatingthe real parts, we get J[X, Y] - J[JX, JY] - [X, JY] - [JX, Y]. This is thesame as the value of the torsion tensor on (X, -JY). Thus we have shownthe following.

3.3. Proposition. The vanishing of the torsion of an almost complex struc-ture J is equivalent to the requirement that the bracket of any two vectorfields of type (1, 0) is also of type (1, 0). Similarly the bracket of any twovector fields of type (0,1) is again of type (0,1).

In the case of a complex manifold, J is given in local coordinates by

a__; F-4 Viand '-' -ax;. Hence N;+i =i(a 3-iay) by . They form a basis over A forWe denote the vector fields 2 (a -i

TyT NY+i ),vector fields of type (1, 0). Similarly the complex vector fields .1 (aj

which we will denote by ea- , form a basis for the space of vector fields oftype (0, 1). Hence it is clear that their brackets are zero, and consequently,the bracket of any two vector fields of type (1, 0) is again of type (1, 0).We conclude therefore that a necessary condition for an almost complexstructure to come from a complex structure is that the torsion should vanish.

3.4. Remark. It is true that the above condition is also sufficient for agiven almost complex structure to come from a complex structure, thanksto a theorem of Newlander and Nirenberg [13].

3. Complex Manifolds 199

3.5. The Dolbeault complex.The (complexified) de Rham complex A(T*) of a complex manifold has

a natural decomposition. We have seen that Ai(T*) is the direct sum ofcomponents A(p,q)(T*) consisting of forms of type (p, q). For simplicity ofnotation, we will simply write AP-q for these spaces.

A form of type (p, q) has the expression E fa,pdz« A dzo. Here, a isa multiindex of length p and Q one of length q. If a = (i1, ... , ip), thenas usual we denote by dza the type (p, 0) form dzil A ... A dzip. Fromthis, we may conclude that actually the bundle AM is nothing other thanAP(T(l,o))* ®Aq(T(o,l))* Now we wish to see the relationship of the exteriorderivative to the type decomposition. From our definition, namely

dw(Xl,...,X,) = i+1

+ .... ,Xr+1),

we conclude that if w is of type (p, q), the evaluation of dw on r + 1 vectorfields of which more than p+1 (resp. more than q+1) are of type (1, 0) (resp.(0, 1)) gives zero. This means that when w is of type (p, q), we get only twoterms in the type decomposition of dw, namely one of type (p+ 1, q) and oneof type (p, q + 1). We will denote these two components respectively by d'wand d"w. On iteration of d which gives 0, we get the following identities:

d'2 = 0, di2 = 0, and d'd" + d"d = 0.

Consider the complex Dol(M)°, given by

3.6. 0--aA->A°,1'...--+ Ao,n--+ 0

and the inclusion 0 --> 0 -i Dol(M)°. Here the differentials are supposedto be d", the sheaf 0 is the sheaf of holomorphic functions and the mapis the inclusion of holomorphic functions in complex-valued differentiablefunctions.

3.7. Definition. The complex defined in 3.6 is called the Dolbeault complexof the complex manifold M.

Then we have the following analogue of the statement which we provedfor the de Rham complex in [Ch. 2, Proposition 6.14]-

3.8. Proposition. The Dolbeault complex is a soft resolution of the sheaf0.


Proof. All the sheaves A',P are A-modules, and consequently soft [Ch. 4,Example 1.10, 1)]. Let f be a local section of the sheaf A such that d" f = 0,i.e. f = 0 for all j in local coordinates. Then it is holomorphic, provingthat the sequence at the left end is exact. We have then to show in a polydiscU in Cn that if w E A°'P satisfies d"w = 0, then it is of the form d"a wherea E A°'P-1(V) and V is a neighbourhood of 0 contained in U. Following thesame inductive argument as in [Ch. 2, 6.14] we can complete the proof if weshow the following (in one variable).

3.9. Lemma. If f is any differentiable function of one variable z in theunit disc, depending differentiably on real parameters t and holomorphicallyon complex parameters s, there exists a function g on some disc D around0, depending differentiably on z and t and holomorphically on s such thataNg- -f on D.

Proof. This is proved by using what is called the Cauchy kernel. We willignore the parameters in what follows since from our construction of g,its differentiable dependence on t and holomorphic dependence on s willbe obvious. By multiplication by a differentiable function which is 1 ina neighbourhood of 0 and zero outside a bigger disc with closure con-tained in the unit disc, we may assume that f has support inside theunit disc U. Then with D(c) denoting the disc of radius e around a, wehave f (a) = lim,,O

2 i J8D(e) f xa dz. This integral can also be written as1 Z-

f (a) _ -2 ti fu\D(E) (f a)dzdz. This is because the latter integral is the

difference of the integrals off() dz over aU and over OD (e) and the formeris zero since f vanishes on 8U. But .1 a is holomorphic in the domain ofintegration and so we may write the integrand also as df z1adzdz. Makingthe substitution z - a = w, we get

f (a) _ f df (w a) (1/w)dwdw.

Now if a is allowed to vary, then df (- a) can also be written as of (ww+a)

So we deduce that the function g(a) defined to be f f w+a dwdw has theproperty that d = f (a).

cta-

Note that d" satisfies d" (f w) = d" f A w + f d"w and in particular, all thedifferentials in the complex are 0-linear.

As in the case of differentiable vector bundles and locally free A-sheaves,we have an identity (proved in exactly the same way) between locally free0-sheaves and holomorphic vector bundles. If E is a holomorphic vectorbundle over a complex manifold, then we denote the associated 0-module

4. The Outer Gauge Group 201

by Eh, retaining the notation £ for the corresponding A-module. Thus wehave E - Eh ®o A. We can tensor the Dolbeault resolution over 0 by Ehand obtain a resolution of the following kind:

O--+ A0,1®E--+ ...--+ A°,n®£-* 0.

This will be called the Dolbeault resolution for Eh. Since it is a soft resolutionas well, we may apply global sections to this complex of sheaves and computethe cohomology of the resulting complex of vector spaces in order to computeH'(M, Eh) [Ch. 4, 3.1, 3.3]. Thus we have

3.10. Theorem. Let E be a holomorphic vector bundle over a complexmanifold M of dimension n. Then the cohomology vector spaces of E arecanonically isomorphic to the cohomology spaces of the Dolbeault complex, inwhich the components are (0, q) -forms with values in E and the differentialsare d". In particular, H'(M, Eh) = 0 for i > n.

3.11. Definition. The bundle T1"0 has a natural holomorphic structure,and it is called the holomorphic tangent bundle and the corresponding 0-sheaf is denoted by Th.

We can tensor the Dolbeault complex by the holomorphic tangent bun-dles AP(T1,°) and get the complex

0-+AP

3.12. Remark. The direct sum of all these complexes over all the p's con-sists of the same components as the de Rham complex but with the differ-ential d", instead of d.

4. The Outer Gauge Group

There is a group which contains the gauge group as a subgroup of index 2. Inorder to define this, we will first associate to any (finite-dimensional) vectorspace V a group which contains GL(V) as a subgroup of index 2. In fact,consider the (set) union of GL(V) and the set I of all isomorphisms V -p V*.On this set we will introduce a group structure. The group structure on thesubset GL(V) will be the same as the standard structure on GL(V). On theother hand, GL(V) acts on the right on I by composition. It acts on theleft by composition with the action A --+ (A')-1 of GL(V) on V*. Finallywe define composition of B1, B2 E I by setting B1.B2 = (Bi)-1 o B2. It iseasy to check that this makes the union a group, which we call the outerlinear group and denote OL(V). This group realizes the outer automorphismA H (A')-1 (defined after choosing a basis) as the restriction of an innerautomorphism.


Now the group GL(V) acts on V* ® V by g ' (gi-1 (9 g). As for theother coset, any element b in it takes V* ® V to V 0 V* by the same map,namely b H (b'-1 (9 b). We can compose it with the transposition t of thetwo factors to get an automorphism of V* 0 V. We will denote this byg L(g). The element L(bg) is then to ((bg)'-1 (9 bg) = to (b'-lg'-1(9bg) _to(b'-1(9b)o(g'-1(9g) = L(b)L(g). Also L(gb) = to((g'-lob)'-1®g'-lob) byour definition of the group structure on OL(V ). Hence L(gb) = to (gob'-1®g1-1 (9 b) = t o (g ®g'-1) o (br-1® b) = (g'-1(9 g) o t o (b'-1® b) = L(g)L(b).Similarly one also checks that L(b)L(b') = L(bb') whenever both b, b' do notbelong to GL(V). In other words we have a representation L of OL(V) onV® ® V. We will modify it a little. Define p(x) = L(x) if x E GL(V) andp(x) = -L(x) if not. This is the representation of interest to us.

4.1. Exercise. Imbed the group GL(V) as a subgroup of GL(V (D V*) bymapping any A E GL(V) to the automorphism (A® (At)-1). Show that itsnormaliser in GL(V ® V*) is generated by OL(V) and the scalars.

Now it is clear how to define the corresponding outer gauge group OL(E)of a (real) vector bundle E. It is simply the assignment to each point m of M,of an element of OL(E,) (depending differentiably on the point m). If Mis connected, it is the union of the gauge group with the set of differentiableisomorphisms of E with E*. From the above linear algebraic considerationwe also see that there is a natural action p of OL(E) on E* 0 E.

4.2. Proposition. The outer gauge group OL(E) of a vector bundle E actsby affine transformations on the space C of connections on E. The linearpart of the action on T* ® E* 0 E is the tensor product of the trivial actionon T* and the action p on E* ® E described above.

Proof. The action of GL(E) on C has already been discussed earlier. If cpis an isomorphism of E with E*, then it carries a connection V on E to oneon E*, which in turn induces a connection on its dual, namely E. This is theaction envisaged. Explicitly, let b be an element of OL(E) \ GL(E) and Ba nondegenerate bilinear form on E representing b. In other words, b is theisomorphism E - E* which maps v to the linear form b(v) : w H B(v, w).Then the action of b on C is determined by the following. For any connectionV, a vector field X and sections s, t of E, we have

B(s, (b(V))xt) = XB(s, t) - B(Vxs, t).

From this it follows that the map V --+ b(V) is an affine transformationwhose linear part, which is an endomorphism a -+ b(a) of T* ® E* 0 E,satisfies the characterizing condition

B(s, (b(a))(X)t) +B(a(X)s,t) = 0.


Finally one can check that this is a group action of the outer gauge groupwith the linear part as claimed.

4.3. Remark. When we consider a linear connection, that is to say whenwe take E = T, then a may also be interpreted as a bilinear map TOT T,and the above equality is then the following:

B(a(X, Y), Z) = -B(a(X, Z), Y).

Our generalisation consists in taking a subgroup ft of OL(T) insteadof GL(T). We will denote by H' its image modulo scalars and by H itsintersection with the gauge group. We would like to investigate the questionof existence of a torsion free connection invariant under the H-action, justas we did above.

Again this depends on the affine action of the group k =ft * 7G2 on C,where the second factor represents the action by T. Exactly as before we havethe homomorphism p of this group into the group of linear transformationsof (T* ®T® (9 T)(M), determined by the action p of OL(T) and the aboveaction of r.

If N is the kernel of p, then it acts on C by translations. If we denoteby for the ]1-linear map N/ [N, N] ® R -* (T* ® T* ® T) (M), then we havethe following conclusion, exactly as above.

4.4. Proposition. The vanishing of the R-linear map for : N --> T* ®T* ®T(M) is a necessary condition for the existence of a torsion free connectionwhich is H-invariant. If H is finite, it is also sufficient.

Proof. The only thing that needs checking is that the linear action of H*Tfactors through a finite group. It is clear that if b is an element of H whichis not in H, then we may use it to identify T* with T. Then all the elementsin H are in H modulo b, and we see that the image action factors throughH x H x H, the permutation group in 3 letters, and possibly ±1.

4.5. Torsion free connections and bilinear forms.We will now apply the above considerations to determine a necessary

and sufficient condition for the existence of a torsion free connection leavinga given nondegenerate bilinear form B : T x T -- A invariant. Treat B asan element b of OL(T).

We will assume that b is either symmetric or skew-symmetric. If we usethe bilinear form to identify T with T*, then the action of p(b) on T*®T*®T*is by x(23), where (23) denotes the transposition of the second and the thirdfactors. Thus we get the image under the linear action of H*Z/2 = Z/2*Z/2


to be {±1} x S3 or 53, according as b is symmetric or skew-symmetric. Thisis because T acts as (12) and b as (23) in the latter case, and so the groupgenerated is S3. When b is symmetric, we note that the group contains-(123), and raising it to the third power, that it contains - Id. The imagebeing of order 12 (resp. 6) when b is symmetric (resp. alternating), it followsthat the kernel is generated by (br)6 (resp. (br)3).

So we will compute how (br)3 acts on connections.

4.6. Computation. Assume that B(X, Y) = ±B(Y, X). Then we have

B(X, ((b-r)V)yZ) = YB(X, Z) - B((TV)yX, Z)= YB(X, Z) + B([X,Y], Z) - B(VXY, Z)= YB(X, Z) + B([X, Y], Z) + B(Z, (VXY)).

Repeating this three times, we get the equality

B(X, ((brr)3V)y, Z) = YB(X, Z) + XB(Z, Y) + ZB(Y, X) + B([X, Y], Z)

+ B([Z, X], Y) + B([Y, Z], X) + B(X, VyZ).

Thus if B is symmetric, we repeat this once more to conclude that theaction by (br)6 on C is always trivial. Hence the group fixes a connection.Moreover, if two connections are fixed, their difference is also fixed, but sincethe image of p contains - Id in this case, the two connections are the same.

Thus we have proved the following important fact.

4.7. Theorem If g is a pseudo-Riemannian structure, then there exists aunique torsion free linear connection which leaves g invariant.

4.8. Definition. The unique connection as in Theorem 4.7 above is calledthe Levi-Civita or the Riemannian connection.

Hence any connection which is fixed under H and T is also fixed under(br)3 and so such a connection is given by the formula

2B(VXY, Z) = XB(Y, Z) +YB(Z, X) - ZB(X, Y)B(X, [Y, Z]) + B(Y, [Z, X]) + B(Z, [X, Y]).

In local coordinates, if we write as usual, V a ai _ >k I' 9, thenU.i

the above formula gives-99 'k 099i4.9. ErzSkl = 2( i2lc + - -

Riemannian manifolds have been studied intensely for over a century.In view of their importance, we will return to their study in Section 5, andprove some of their properties.


4.10. Exercises.

1) Assume that a tangent bundle is the direct sum of subbundles Ti. Wesay that a linear connection leaves this decomposition invariant ifOXY is a section of Ti whenever Y is. Apply the above criterion (us-ing the group generated by the involutions of T having Ti and (Dj#i Tjas eigenspaces with eigenvalues 1 and -1 respectively) and show thatthere exists a torsion free connection leaving the decomposition in-variant if and only if all the subbundles are involutive.

2) Assume that g is a Riemannian structure on M. Show that the in-volutions corresponding to the above decomposition commute withg in OL(V) if and only if the decomposition is orthogonal for theRiemannian structure.

4.11. Symplectic structure.Now suppose B is alternating. Then the formula gives

B(((br)3V - V)XY, Z) _ >(XB(Y, Z) + B(X, [Y, Z])).

In this case, treating B as a 2-form, this computes the torsion tensor to bethe exterior derivative dB of B. Hence we have

4.12. Theorem. The necessary and sufficient condition for the existenceof a torsion free linear connection leaving a symplectic form w invariant isthat it be closed.

4.13. Remark. A theorem of Darboux states that any symplectic formw which is closed can be written locally in a suitable coordinate system(qi, , gn,P1, , pn) as > dpi A dqi. Note that this is exactly how thesymplectic form looks on the cotangent bundle of any bundle, in the naturalcoordinate system on it (Example 1.11, 2)).

In a symplectic manifold (M, w) one can define a Lie bracket on thespace of functions as follows. If f is a function, then using the isomorphismof T with T* given by the symplectic form, we can identify the differentialform df with a vector field X f. The inner product using X f will be denotedif. Then we define the Poisson bracket {f, g} of f, g E .4(M), to be thefunction if i9w. It is therefore given by the equation w(X f, X9) = -df (X9) _-X9 (f) = X f (g). In the Darboux coordinates, Xf = > ai api + bi aqi is

) = ai. Hencecomputed by the equation w(X f, api) = bi and w(X f, Fq-i

(df) (j) = bi and df ( q) _ -ai. In other words, bi = a and ai = - ,

determining Xf to be E(ap Tq-i - aq - ). This also computes the Poissonbracket in local coordinates. It is given by X Xg) e --w ( f > - Bpi a4i Oqi api


From this we also conclude that [X f, X9] = X{f,9}. Thus we have an exactsequence of Lie algebras:

0 --* IR --- A --> T (M).

The image in T(M) can also be checked to be the set of all vector fieldsX such that L(X) (w) = 0. Such vector fields are called locally Hamiltonian.These notions are fundamental in Hamiltonian mechanics.

4.14. Kahler structure.Suppose we are given an almost complex structure as well as a Hermit-

ian structure. The symmetry of g is equivalent to g2 = 1 in OL(V) and theHermitian condition states that J and g commute. The element -Jg rep-resents the alternating 2-form w : (X, Y) -+ g(X, JY). So for the existenceof a torsion free connection that leaves J as well as g invariant, two neces-sary conditions are that the torsion tensor associated to the almost complexstructure J is zero, and that dw = 0. One can show that these are alsosufficient, again by determining the kernel N. But since we know that theRiemannian connection is unique, we have only to check that dw = 0 impliesthat w is invariant under this connection. Thus we have the following result.

4.15. Proposition. Let an almost complex structure J and a Hermitianstructure g with respect to it, be given. A necessary and sufficient conditionfor the existence of a torsion free connection leaving J and g invariant, isthat the torsion tensor of J should vanish and that dw = 0.

In particular, such a manifold is a complex manifold as well as a sym-plectic manifold.

4.16. Definition. A complex manifold with an almost Hermitian structuresuch that the associated symplectic form is closed, is called a Kdhler mani-fold. The corresponding real cohomology class in H2(M) is called its Kahlerclass.

4.17. Examples.

1) We have computed the Chern form of the Hopf bundle on the complexprojective space in [Ch. 5, 6.2]. It is a purely imaginary form. It iseasily verified that it is the Kahler form of a Hermitian form on theprojective space. In other words the topological Chern class of thedual of the Hopf bundle is a Kahler class.

2) Any closed complex submanifold of a Kahler manifold is also Kahler. Inparticular, all closed submanifolds of the projective space are Kahler.

5. Riemannian Geometry 207

3) Any one-dimensional complex manifold is Kahler. In fact, we can takeany Hermitian structure. The corresponding 2-form is closed sincethere are no nonzero 3-forms on the manifold.

4) Any complex torus A, namely the quotient of C9 by a lattice r, isKahler. This is because firstly C9 is Kahler under the standard Her-mitian metric. Secondly, the metric is invariant under translation byelements of r and induces a Hermitian structure on A. Hence theform on A associated to the Hermitian structure on it, has closedinverse image on C.Q. Therefore, it is itself closed.

5. Riemannian Geometry

We will use notions pertaining to a linear connection on a (pseudo-) Rie-mannian manifold, such as exponential mapping, geodesics, curvature, ...,implicitly assuming that the linear connection intended is the Levi-Civitaconnection, namely the unique torsion free connection with respect to whichthe Riemannian metric tensor is invariant.

5.1. Geodesics on a Riemannian manifold.A Riemannian manifold (M, g), considered as a topological space, can

be provided with a metric space structure. In fact, if p, q are in M and'Y : [a, b] -> M is a piecewise differentiable path connecting p and q, then wedefine the length 1(-y) of the path y by the formula

b

l(y) = f g(y'(t),'Y'(t))dta

where -/'(t) is the image of dt under the differential of y. One defines thedistance d(p, q) to be the infimum of l(y) where -y runs through all pathsconnecting p and q. Note that we are not claiming that this infimum isattained.

5.2. Remark. If M is the configuration space, then a curve describes thedynamics of the system and the vector -y'(t) may be thought of as the velocityof the system. From the point of view of physics, the number E(y) definedby the formula

b

E('Y) =2

jg(-y'(t),-y'(t))dt

is equally important and is known as the energy of the path. The evolutionof the path ensures that the energy is minimal.

If y is a geodesic, then the length of the velocity vector is the same at allpoints (since the velocity vectors at any two points are obtained by paralleltranslation along the curve on the one hand [Ch. 6, Remark 1.4, 2)], and


the metric tensor field is invariant under parallel translation on the other).Hence the length of the curve from 0 to t is simply (l.t) where 1 is the normof the velocity vector. Thus, if we wish, we can re-parametrise the geodesicby requiring this length to be t, that is to say, parametrise it by its length.

5.3. Example. If M is the Euclidean space R'h provided with the Eu-clidean Riemannian structure, then this definition gives a distance functionon R. This is of course the usual Euclidean distance. To check this wemay as well assume that p is the origin and that y(1) = q. If y = (yi) isa path connecting p and q, then l(-y) is f0 'yZ'(t)2dt in this case. Nowwe have d ( yi(t)2) = E-ri(t)yz(t) and the latter is at most E'y (t)2,dt (t)2

by Schwarz's inequality. Hence we have the inequality IIg1I = E'yi(1)2 =f0 dt( 'ryi(t)2)dtj G fo -yz(t)2dt = 1(7). In this case, of course, the

infimum is attained when we take for y the straight line segment connectingp and q. For the line segment o- is given by ai(t) = txi where xi are thecoordinates of q. Now wi(t) = xi and l(a) = f0 Exidt = 11g11.

5.4. Proposition. The distance function on a Riemannian manifold Mas defined above, gives rise to a metric space structure on the set M. Thetopology of this metric structure is the same as that of the manifold.

Proof. We will first check that the distance function that we defined aboveis indeed a metric. By definition, the distance function is symmetric in p, q.Also if p, q, r E M, then the distances d(p, q) and d(q, r) can be approximatedby l(yi) and 1 (y2) where yl and 72 are two paths connecting p to q, and qto r respectively. The composite path from p to r has length l(yl) + l(y2)which approximates d(p, q) + d(q, r). Hence d(p, r) < d(p, q) + d(q, r). Ifp = q, we take the constant path y : [0, 1] - M taking all t to p, and notethat in that case, y' (t) = 0 for all t and so 1(-y) = 0. Hence d(p,p) = 0.Conversely, assume that p 54 q. Choose a coordinate system (U, x) aroundp such that q does not belong to U. Let Ba, be the closed ball around pobtained by transferring to U the closed ball in Il around the origin withradius a. Now any path y : [0, 1] --> M connecting p to q, has to intersectthe boundary Sa of Ba,. For otherwise, {t : y(t) is in the open ball} is openand closed in [0, 1], contradicting the assumption that p belongs to this setand q does not.

Take the least s such that y(s) E Sa. Then the path 7 is the compositeof the part 71 [0, s] and the part yI [s, 1]. The length of y is the sum of thelengths of these two parts and so the length of ry is at least the length of thefirst part. This first part has the property that the initial point is p, the endpoint is in Sa, and all other points are in the open ball. Its length is given

5. Riemannian Geometry 209

by AS g(y'(t), y'(t))dt. Now the following simple remark completes theproof, by comparing the distances in the Euclidean metric and the inducedRiemannian metric on the open ball.

5.5. Lemma. Let gl, 92 be two Riemannian metrics on a domain D in R'and K a compact subset of D. Then there exist positive constants k, k' suchthat

k92(v,v) C 91(v,v) k'g2(v,v)for all tangent vectors v at all points of K.

Proof. From the bilinearity of gi and 92, we see that it is enough to provesuch an inequality for v in the unit sphere, say for the metric 92. The spaceof all tangent vectors v at all points of K with g2(v) = 1 is a compact subsetof the tangent bundle. The function v -4 g2(v)/gl(v) is a strictly positivefunction on it and hence is bounded above and below by positive constants.

Completion of the proof of 5.4. From Lemma 5.5, applied to the givenRiemannian metric g and the Euclidean metric on the compact set Ba, weconclude on integration that the length of y over [0, s] in the given metricis at least (resp. at most) k times (resp. k' times) its length with respectto the Euclidean metric. This implies that d(p, q) > ka 0 and concludesour proof that the distance function defines a metric space structure onM. At the same time it also shows that that the metric space induced bythe Riemannian structure is equivalent to the Euclidean metric, and so theinduced topology is the same as the Euclidean one, on the coordinate openset.

Let m be a point of M and assume that the exponential map is a dif-feomorphism of an open set W around 0 in V = Tm(M) onto an openset U containing m. The tangent bundle on W has a natural trivialisa-tion, identifying all tangent spaces with V. Hence the tangent space at anyp = exp(q) E U can be identified (by the differential of the exponential mapat m) with V. But there are two Riemannian metrics on W. On the onehand, V has the Riemannian metric given by the symmetric form g on thetangent space at m and W inherits it. On the other hand the exponentialdiffeomorphism transports the Riemannian metric from U to W. These twoare of course in general different, one of them being the flat metric. We willdistinguish between the two by denoting them g fI and g.

We wish to restrict ourselves to W \ {0}. On it we can introduce polarcoordinates, and consequently we have a vector field X = 9, which hasnorm 1 at all points. Hence using the metric, we can decompose the tangentspace at all points as the direct sum of the trivial bundle given by the abovevector field and its orthogonal complement. The question arises which of the


two metrics do we use for this decomposition. Actually, it does not matter!Firstly, any radial vector v of norm 1 at x E W is mapped by the exponentialmap into the tangent to the geodesic t - p exp(tx) at exp(x). It is thereforethe parallel translate, along the geodesic, of v E T,,,(M). Hence its normis 1 in the metric on Texp(v) (M) as well. In other words the tangent vectorv at x has norm 1 in both metrics. Our claim above is that its orthogonalcomplement for g f1 is also orthogonal for g. This is known as Gauss' lemma.

5.6. Lemma (Gauss). The orthogonal complement of the radial vector forthe flat metric at a point x E W \ {0} is also orthogonal for g transportedfrom Texp(x) (M) by the exponential map.

Proof. Note that since the radial vector field X of norm 1 is invariant underparallel transport for g, it follows that VX(X) = 0. If v is any tangent vectorat x orthogonal to X, then one might as well confine oneself to the two-dimensional subspace spanned by x and v. Let then Y be the angular vectorfield a. Then Yx is a multiple of v. Now Y(X, X) = 0. Since the metric isinvariant under V, this is 2(VyX, X). Since the torsion is zero, this impliesthat (V XY, X) = 0. Again since V XX = 0, we get X (Y, X) = 0. In otherwords, (Xv, Yv) is invariant along any radius. But since the vectors Xx andYx are orthogonal for the flat metric, they are zero for g at T,(M). Thisproves the assertion that Yx is orthogonal to Xx for g.

5.7. Geodesics as minimising distances.From Gauss' lemma, we will deduce that if c(t) is any curve in W join-

ing 0 to x E W, then the length of exp(c(t)), namely fo g(c'(t), c'(t))1/2dt,is at least b = l x 1 i . We can and will assume that for all t 0, c(t) isnot the zero vector. We write c(t) E W as r(t)v(t) where r(t) = I1c(t)JI,and g(v(t), v(t)) = 1. Then c(t) = r(t)v'(t) + r'(t)v(t). Since we have as-sumed that v(t) is of norm 1, it follows that v(t) and v'(t) are orthogonalfor g fl. Hence they are also orthogonal for g in Texp(c(t)) . In particular,g(c'(t), c'(t)) > Ir'(t)l. Hence the length of c(t) from 0 to 1 is at leastfa r'(t)dt = b. But the length of the geodesic joining m to exp(x) is b. Thusthe geodesic has the shortest length among curves joining m and any pointin U.

5.8. Definition. A Riemannian manifold is said to be complete if it is com-plete as a metric space, the metric being that induced from the Riemannianstructure.

5.9. Remark. It can be shown that in a complete Riemannian manifold,any two points can be connected by a geodesic.

6. Riemannian Curvature Tensor 211

6. Riemannian Curvature Tensor

The curvature of the Levi-Civita connection on a Riemannian manifold iscalled the Riemannian curvature tensor. It is a 2-form R(X, Y) with valuesin End T.

6.1. Local expression.We computed the Levi-Civita connection in local coordinates in 4.9.

From this we also get the following expression for the Riemannian curvaturetensor. We will write Rijkl = (R(aai

, aa) aa,-- , axe) . Then we have

Rijkl =

T krirngrl +m,r

Z m m

CR(a, aa)(aak),

Vaoa as - oava aa,axa_;, axe j a_Xj a--i x

V m a a (v m a a\ axi (rjk ax,, ax,) - a (rik axTR, )' 0x1

8 m r a-VrI',.,,,a,- 9mlm

Here r are determined by the equation23

m,r m3

a 099klrij = 2 9jk + 9ki -9ij)We, however, warn the reader that the traditional notation Rijkl stands for(R(aak, a ti), which differs from the above by a factor of -1. See[10, p. 21].

We may try to get some idea of the Riemannian curvature tensor byfinding the consequence of its vanishing. Recall that we defined a connectionto be flat, if the curvature form is zero. So we call a (pseudo-) Riemannianmanifold flat if the Riemannian curvature is zero. Notice first that locallythis implies that the tangent bundle can be trivialised together with thelinear connection. In other words, around any point m E M, there exista coordinate neighbourhood and a basis (Xi) for vector fields (over thealgebra of functions) such that VyXi = 0 for all vector fields Y. Since theconnection is torsion free, it follows that

VX,XX -VX,Xi - A, X31 = 0.Since the first two terms are zero, we have

[Xi, Xj] = 0 for all i, j.

Hence we can introduce a coordinate system (U, x) such that Xi = fori

all i. Finally, the invariance of the metric tensor implies thata a_ a a a


In other words, g(-, ak ) is a constant function. Now we can change thecoordinate system by a real linear transformation so that the metric formis simply E +(dxi)2. In particular, we deduce that if a Riemannian metricis flat, then there is a coordinate diffeomorphism of U with an open set in1187E that takes the given metric to the Euclidean metric. This explains theorigin of the name `flat' for curvature free connections.

6.2. The space of curvature tensors.The Riemannian curvature tensor has many symmetries. Firstly, since

the linear connection is actually one on the reduced orthogonal bundle, ittakes values in the adjoint bundle associated to this principal 0(n)-bundle.In other words, it can be considered as a 2-form with values in Skew(T), thebundle of skew-symmetric endomorphisms of T (with respect to the metricform g). In any case, one can directly verify that the invariance under V ofg, namely

Xg(Y, Z) = g(VxY, Z) + g(Y, VxZ),implies that the curvature form R satisfies

g(R(X, Y)Z, T) + g(Z, R(X, Y)T) = 0.

It satisfies other identities as well. Consider the multilinear form in fourvariables on the tangent space, given by (X, Y, Z, T) H R(X, Y, Z, T) _g(R(X, Y) Z, T). It satisfies the following identities:

i) R(X, Y, Z, T) = -R(Y, X, Z, T).This states that R is an alternating form in X, Y.

ii) R(X, Y, Z, T) = -R(X, Y, T, Z).This reflects the remark above that R(X, Y) is a skew-symmetric en-domorphism with respect to g.

iii) R(X, Y, Z, T) + R(Y, Z, X, T) + R(Z, X, Y, T) = 0.This is the first Bianchi identity, valid for any torsion free linear con-nection [Ch. 6, 2.8].

As a consequence, one can also derive the following identity:

iv) R(X, Y, Z, T) = R(Z, T, X, Y).

To see this, write out the Bianchi identity for (X, Y, Z, T), (Y, Z, T, X),(Z, T, X, Y) and (T, X, Y, Z), add the first two and subtract the other two,to get

R(X, Y, Z, T) - R(Z, T, X, Y) = 0.Since all these identities are tensorial identities, namely identities on thetangent space at every point of M, it makes sense to consider the followingpurely linear algebraic setup. Let V be a (finite-dimensional) vector spacewith a nondegenerate quadratic form g on it. Consider the space RC(V) of


multilinear forms on V in four variables, satisfying the identities i), ii) andiii) above (where X, Y, Z, T are now vectors in V) - Any R E RC can also beconsidered as an alternating 2-form in the variables X, Y E V with values inthe space of skew symmetric endomorphisms of V. Let us call this space theRiemannian curvature space. In what follows we will denote, for convenienceof notation, the inner product given by g on V by (X, Y) H (X, Y).

If M is a Riemannian manifold and m E M, then we are interestedin taking V = T,,,, (M) and g to be the metric on the tangent space. Infact, there is a vector bundle on M (which we will denote by RC(M))whose fibre at m E M can be identified with RC(T,,) for all m. TheRiemannian curvature tensor R is then a tensor field whose value at mbelongs to RC(T,), that is to say, a section of RC(M). It contains a greatdeal of geometric information. While this tensor is easier to handle fromthe point of view of algebraic manipulations, there is an equivalent notionwhich is more geometric even at the outset.

6.3. Sectional curvatureLet R E RC. Consider the function Q : (X, Y) H R(X, Y, X, Y). Then

R can be recovered by polarisation from Q. In order to prove this, sinceR is a multilinear form in 4 variables, we may as well assume that V is4-dimensional. Now R defines (and is determined by) a quadratic formon A2(V). Note that Q is essentially the restriction of R to the space ofdecomposable tensors, i.e. tensors of the form X A Y. If R and R' aretwo elements of RC which coincide with Q on decomposable tensors, thenR - R' vanishes on decomposable tensors. But then any quadratic formwhich vanishes on decomposable tensors in the second exterior power of a4-dimensional space, is unique up to a scalar multiple. This form can in factbe described as the wedge product S2(A2(W)) -i A4 (W). So we have onlyto convince ourselves that this form does not belong to RC. As a 4-formit maps (X, Y, Z, T) to X A Y A Z A T. If T is fixed and we take the cyclicsum in (X, Y, Z), we get a nonzero multiple of the same form, proving thatit does not satisfy iii) in the definition of RC.

6.4. Exercise. Follow the above logic and actually compute R in terms ofQ.

On the other hand, the square of the area of the surface determinedby X and Y is defined to be JXUU2IJYII2 - (X,Y)2. Note that the givenquadratic form on V induces a quadratic form on A2 (V) and the above isjust JIX AYII2. If g is positive definite, so is the induced quadratic form, andif X and Y are linearly independent, this number is nonzero. The sectionalcurvature of a two-dimensional subspace W spanned by the vectors X, Y, is


defined to be the real numberQ(X,Y) - - (R(X,Y)X,Y)

6.5. Sec(W) = -IlxnYllIIxAYII

It depends only on W and not on X, Y. In the general case of a nonde-generate (not necessarily positive definite) quadratic form, it is well definedonly when the quadratic form restricts to the two-dimensional subspace as anondegenerate form (which is always the case, if g is positive definite). Wehave thus proved the following fact.

6.6. Proposition. The Riemannian curvature is determined by the sec-tional curvature and conversely. In particular, the sectional curvature iszero if and only if the Riemannian curvature is zero.

If M is a Riemannian manifold of dimension 2, that is to say, a surface,and m E M is any point, and we take W = Tm(M), then the number givenby the sectional curvature of W is called the Gaussian curvature c,,,, at thepoint m.

6.7. Curvature of a hypersurface in R.Suppose a differential manifold M is imbedded in ]Rtm, and we take the

induced metric on it. The induced Riemannian tensor is called the firstfundamental form. We therefore get a natural linear connection on M,namely the Levi-Civita connection of the induced Riemannian structure.

Consider the exact sequence

0->TM -+A7->Nor (M,R )-0.There is a natural splitting of this sequence given by the Riemannian metric.According to [Ch. 6, 2.13] this gives rise to a linear connection V' on M.Indeed it simply amounts to the following. Given two vector fields X and Yon M, extend both to R' and consider the Euclidean connection VXY andthen project its restriction to M orthogonally to the tangent bundle of M.Let X, Y be vector fields tangential to M at points of M. Then V' Y-V' Xdiffers from OxY - VyX by a normal field on M. Hence we see that thetorsion tensor of V is zero. On the other hand, if X, Y, Z are vector fieldson I[8' which are tangential to M at points of M, then g(OXY, Z) is thesame as g(VxY, Z) and so we conclude that the Riemannian tensor is leftinvariant. Hence the induced linear connection is the same as the Levi-Civitaconnection of the induced metric.

Now we also have another form, namely the second fundamental form Sas defined in [Ch. 5, 4.11]. This associates to vector fields X, Y on M thenormal field given by projecting VxY. In other words, we have V Y =VXY + S(X, Y).


We have remarked that a hypersurface in R'z is oriented and that itsnormal bundle is trivial. Let v be a section of the normal bundle, giving theunit normal at all points. Then we can use the connection V to define anendomorphism of the tangent bundle as follows. Notice first that (V XV, v) +(v, Vxv) = X (v, v) = 0. This implies that (V v, v) = 0, and hence thatOXv is tangential to M. The map X - Vxv gives an endomorphism wof the tangent bundle, called the Weingarten map. All our computationscan be made in terms of w. Firstly, let X, Y be tangential to M. Then(VxY, v) = X (Y, v) - (Y Vxv). But Y is orthogonal to v so that we have(V XY, v) _ - (Y, w (X)). This means that the normal component of OXYis - (Y, w (X)) v. The tangential component is V' Y by definition. Thus wehave

6.8. V, Y = VXY+ (w(X),Y)v.

In particular, the second fundamental form is given by (X, Y) H (w(X ), Y) v.Thus the data of the second fundamental form and the Weingarten endo-morphism are the same.

Interchanging X and Y in equation 6.8 and subtracting we get

(w(X),Y) = (X,w(1')),that is to say, w is a symmetric endomorphism of TM.

Let X, Y, Z be tangential to M. Then we will compute R(X, Y)Z whereR is the curvature tensor of V. We have

V V Z = Vx(VyZ) + (w(X),V, Z)v= OX(VyZ + (w(Y), Z)v) + (w(X), VyZ)v.

Interchanging X and Y and subtracting, and noting that R(X,Y)Z is tan-gential to M, we get the equality

6.9. R(X, Y)Z = (w(Y), Z)w(X) - (w(X), Z)w(Y).

An equivalent form of the above is

Q(X,Y) = (w(Y),X)(w(X),Y) - (w(X),X)(w(Y),Y).

Since our definition of the Riemannian curvature is intrinsic, this ex-pression, which involves the Weingarten map or what is the same the sec-ond fundamental form, is in fact independent of the imbedding. Originallythis formula was the definition of curvature of an imbedded hypersurface.So the formula has, as a corollary, the fact that the curvature so definedis independent of the imbedding. In other words, if the same hypersurfaceis isometrically imbedded in some other manner, the curvatures defined in


terms of the second fundamental form for the two surfaces are the same.Gauss, who discovered this fact, termed it `the most excellent theorem', andtherefore this theorem is known as the `theorema egregium'.

We have derived the consequence of the vanishing of the sectional cur-vature. A weakening of this drastic assumption is to say that the sectionalcurvature is a constant. This means that the sectional curvature at anypoint is the same for all two-dimensional subspaces of the tangent space atthat point. A priori, there are two notions possible here. One is that thisconstant depends on the point m E M, and the other is that it is in additionthe same at all points. The first condition is always satisfied for surfaces,while the second is not generally true. But the two notions coincide forhigher-dimensional manifolds.

6.10. Theorem (Schur). If the sectional curvature is a constant at ev-ery point of a connected Riemannian manifold of dimension at least 3, thisconstant is the same at all points.

Proof. We would like to convert the assumption into one on the Riemann-ian curvature tensor. Define a tensor s by the formula s(X, Y, Z, T) _g(Y, Z)g(X,T) - g(X, Z)g(Y,T). Then our assumption says that

R(X,Y,X,Y) = C(g(X,Y)2 -g(X,X)g(Y,Y)) = Cs(X,Y,X,Y).

Thus we conclude that R = Cs, for some function C. Now we will usethe second Bianchi identity, namely (VXR) (Y, Z) = 0. In our case, thissimplifies to

xc.s(Y, z) + C E(Vxs)(Y, Z) = 0,

where the summation is cyclic and X, Y, Z are any three vector fields, and sis considered as a bilinear form with values in End(T). From the definitionof s and the invariance of g, it follows that Vs 0. Hence the cyclicsum E (X C) s (Y, Z) vanishes. Moreover, it is easily seen that if X, Y, Z arelinearly independent at a point p, then s(Y, Z)p, s(Z, X)p and s(X, Y)p areall linearly independent. Hence the above equation implies that XXC = 0.This proves that C is a constant, and consequently our assertion.

6.11. Example. Consider the unit sphere Sn = {(x) E Rn+1 : E xi = 1}with the Riemannian metric induced from the Euclidean metric. We willcompute the curvature tensor at the point P = (0, ... , 0, 1). A coordi-nate neighbourhood is given by {(x) E S' : x,,,+l 0}. The local coor-dinates are simply xi, i < n. In this coordinate system the metric ten-sor g is given by E? 1(dxi)2 + (d(1 - Ez1 X?). This simplifies to


En>1(1 +x2

2)(dxi)2 + E 1 x x dxidxj. In other words, the metric ten-sor is given by E gijdxidxj with

bij + xixjZZr - 1 _

x2

Notice that in this expression, all the functions gij have the value bij at Pand their first partial derivatives at P are all zero. It follows that I' (P)are zero as well. Now using the determination 6.1 in local coordinates, ofthe Riemannian curvature tensor, we conclude that

l l

(Rijkl)(P) = axi (P) a- (P)

j ik _ il jk.

2COxiaxk +

Ox ) (P)7Oxl Oxiaxl

Incidentally, our calculation above is quite general, and is valid in thefollowing generality.

6.12. Lemma. If g = > gijdxidxj is a Riemannian metric on a domainin Rn with gij - bij and a vanishing at a point P, then the Riemannian

OXkcurvature tensor is given at the point P by the expression

1 a2gjl + a2gik - a2gl - a2gjk (P)2 (Oxiaxk Oxjaxl Oxjaxk axiaxl)

Going back to our example, we conclude that the Riemannian curvaturetensor on Sn is given at the point (0, ... , 0,1) by

Rijkl (P) = bjkbil - bik6jl.

Invariantly expressed, this means that

R(X, Y, Z, T) = (Y, Z) (X, T) - (X, Z) (Y, T)

for any four tangent vectors X, Y, Z, T at P, since it is true for the basicvectors axti at the point. Hence the sectional curvature corresponding tothe plane X, Y in Tp is given by R 'YX -7 which is 1, independent

IIxII IlYli (X,Y)of the plane. By Schur's theorem it is also independent of the point. Inother words, the Riemannian structure on the unit sphere Sn is of constantsectional curvature 1. Incidentally, this explains why in the definition ofsectional curvature, we introduced a negative sign. We preferred Sn to havepositive curvature. We may also see the constancy of sectional curvatureby noting that the orthogonal group acts transitively on Sn and that theisotropy group at P is the orthogonal group O(n - 1) which acts transi-tively on all the two-planes on the tangent space at that point. Since the

2g a2g a2gl a2g


Riemannian metric on Sn is invariant under O(n + 1), it follows that thesectional curvature is also invariant under the action and hence independentof the plane and also of the point.

6.13. Example. Consider the closed submanifold M' of R'+1 given byn

(y1, ... Yn+1) y2i - yn2+1 = -1i=1

This is a hyperboloid of two sheets, and is not connected, since the contin-uous function yn+l on it has image R \ (-1, 1). Consider the component Mgiven by yn+l > 1. The map which sends (yl, ) yn+i) to (y1i ... , yn) inIIBn is in fact a homeomorphism of M with R, the map Rn -> M given by(yi, ... , yn) ti (yi, ... , yn, (1 + En1 y?)1/2) being its inverse (see Remark1.6, 2)). Taking this isomorphism as a (global) coordinate system, we mayexpress the Riemannian metric induced on M from the pseudo-Riemannianmetric En

i=1 dyZ - dyn2+1 on II87'+1 in terms of these coordinates. It is givenby the form

n 1,2\ 2J(dxi)2 - (d (1+ x?) I

i=1 \ //

This Riemannian manifold is called the hyperbolic space. As in the firstexample, this leads to the determination of the Riemannian metric on M asEgijdxidxj, with

E xixjgij - bij - (1 + E x?)112I

Thanks to Lemma 6.12, which is applicable to this example also, we haveat the point P = (0, ... , 0,1) in M,

Rijkl = -(bjkail - &iksjl)

Hence in this case the sectional curvature turns out to be -1. Of course, asa consequence of Schur's theorem this implies that the sectional curvatureis -1 at all points. The orthogonal group of the quadratic form yl + . +yn - yn2+1, namely O(n, 1), acts on M' transitively. In fact, even SO(n, 1)acts transitively on it. Although this group is not connected, the connectedcomponent of 1, which is denoted SO°(n,1), acts transitively on M, implyingagain the constancy of the sectional curvature.

6.14. Definition. A connected Riemannian manifold is said to be a spaceform if it has constant sectional curvature. It is said to be of spherical, flator hyperbolic type according as the constant curvature is positive, zero ornegative.


7. Ricci, Scalar and Weyl Curvature Tensors

We will now decompose the curvature space RC(V) defined in 6.2 for avector space provided with a nondegenerate quadratic form, into a directsum of three subspaces all of which are invariant under the natural actionon RC(V) of the orthogonal group 0(g) of g.

Firstly, we have the following homomorphism of S2(V*) into RC. Startwith a symmetric bilinear form b on V. Then we can define an element p(b)in RC by setting

p(b) (X, Y, Z, W) = b(X, Z) (Y, W) - b(Y, Z) (X, W)

+ b(Y, W) (X, Z) - b(X, W) (Y, Z)

for all elements X, Y, Z, W E V. The first two (resp. the last two) termstaken together are clearly alternating in the variables X and Y. Again thefirst and last terms (resp. the second and third terms) taken together, arealternating in the variables Z and W. We will verify 6.2, iii). On taking thesum cyclically permuting X, Y, Z, the first two terms cancel out, thanks tothe symmetry of b and the inner product. The same is true of the last twoterms. Thus p(b) belongs to RC.

We also have a linear map in the reverse direction. To see this wewill interpret the elements of RC as maps V x V -+ End(V). For anyF E RC, consider the bilinear form on V which associates to (X, Y), thetrace of Z F(X, Z)Y. From 6.2, iv) we conclude that this is a symmetricform, thereby yielding a linear map RC --+ S2(V*). The symmetric formcorresponding to F will be denoted Ric(F). The composite of Ric and pis not the identity on S2 (V*), but is nevertheless an automorphism (whenn = dim(V) is at least 3). It is easy to compute this composite.

Let (ei) be a basis of V and (ei) the dual basis with respect to the innerproduct. We then have

Ric(p(b)) (X, Y) = trace of Z H p(b)(X, Z)Y

(p(b) (X, ei)Y, ez)

b(X, Y) (ei, ez) - b(ei, Y) (X, ez)

+1: b(ei, ei)(X, Y) - E b(X, ez)(ei, Y)

nb(X, Y) - (X, E b(ei, Y)ei) + tr(b) (X, Y)

-(E b(X, ez)ei,Y)(n - 2)b(X, Y) + (tr b) (X, Y).

In other words, Ric(p(b)) = (n - 2)b + tr(b)g. If n > 3, the map b HRic(p(b)) is easily seen to be an isomorphism.


We will call the kernel of Ric the Weyl subspace of RC and denote it byWeyl(V). From our remarks above, we have the following conclusion.

7.1. Proposition. Assume that the rank of the vector space V is at least3. Then the curvature space RC(V) is a direct sum of the kernel Weyl(V)of Ric : RC --> S2(V*) and the image of p : S2(V*) --+ RC.

We can further decompose p(S2(V*)). Let So(V*) consist of trace freesymmetric endomorphisms of V. Consider the following canonical element sin RC. To X, Y E V associate the endomorphism s(X, Y) of V which takesZ to (X, Z)Y - (Y, Z)X. Clearly it is the same as Zp(g). We will call s thescalar element.

7.2. Proposition. The image of p is the direct sum of the one-dimensionalspace spanned by the scalar element s, and the image of p : So (V*) -* RC.

Proof. The space S2(V*) is the direct sum of the one-dimensional spacespanned by g and So (V*). From our computation above, it follows thatpg = 2s.

We can also get an explicit splitting of the map Ric : RC(V) -* S2(V*).Let us denote the automorphism of S2(V*) which takes b to (n-2)b+tr(b)g,by a. Then we just checked that Ric op = a. So, in order to get a splittingof Ric, all we have to do is to replace p by po (a)-1. Now one checks directlythat a-1 is given by b '->b22 - (n-1)(n-2)9'

We will summarise our conclusions in the following proposition.

7.3. Proposition. Let V be a vector space of dimension n > 3 andg = ( , ), a nondegenerate symmetric bilinear form on it. There is a natu-ral surjection Ric : RC(V) -> S2 (V*), defined by (Ric(R)) (X, Y) = trace ofZ -> R(X, Z)Y. A splitting of Ric is given by b i-

b) - s,n(n-1 (n-2where p(b) is the element of RC(V) determined by (p(b)(X,Y)Z,W) =b(X,Z)(Y,W) - b(Y,Z)(X,W) +b(Y,W)(X,Z) - b(X, W) (Y, Z), and s =(1/2)p(g)

Our interest in this linear algebraic computation is its application to theRiemannian curvature R of a pseudo-Riemannian manifold.

7.4. Definition. Let M be a Riemannian manifold and R the Riemanniancurvature tensor field. The section of S2(T*) given by

(X, Y) -> -(trace of the endomorphism Z -* R(X, Z)Y)

is called the Ricci curvature tensor field. The corresponding endomorphismof T is called the Ricci endomorphism. This can also be defined as theimage, under the composition map End(T*) ® End(T*) -> End(T*), of R


regarded as a section of End(T*) ® End(T*). The component of R in thespace Weyl(T,,,,) at each point m E M gives another tensor field called Weyl'sconformal curvature tensor field or Weyl curvature.

7.5. Conformal structure.Let V be a vector space and g, g' two metrics on it. We say that they are

conformally equivalent if there exists a positive scalar a such that g = ag'.

7.6. Definition. A conformal structure on a differential manifold is an as-signment of a (conformal) equivalence class of metrics on the tangent spaceat every point.

The set of all linear transformations T of V which satisfy g(Tv) = Ag(v)for some A 0 0 form a group called the conformal group. The group GL(V)acts on the set of all conformal equivalence classes of metrics on V transi-tively. The isotropy at a given conformal structure is the conformal group.Hence one concludes that the data of a conformal structure on M is justa reduction of the structure group of the tangent bundle to the conformalgroup.

7.7. Definition. Two Riemannian metrics g, g' on a differential manifoldare said to be conformally equivalent if there exists an everywhere positivefunction cp such that g = cpg'.

7.8. Exercise. Show that any conformal structure comes from a Riemann-ian metric.

A conformal structure on a manifold is nothing but the data of a Rie-mannian metric, conformally equivalent metrics being considered the same.

7.9. Remark. If g, g' are conformally equivalent metrics, a routine, if long,computation can be made to express the Riemannian curvature of g' in termsof that of g. From this one can conclude that the Weyl curvature is the samefor g and g'. This is the reason for calling it Weyl's conformal curvaturetensor field. In particular, the vanishing of this tensor is equivalent to say-ing that there exist coordinate neighbourhoods in which g is conformallyequivalent to the Euclidean metric E dxi .

Since the Weyl curvature is a conformal invariant, one may wonder ifthere is a direct definition of this tensor starting with a conformal structureon M. In fact, one can show that there exists a unique Cartan connec-tion based on [Ch. 6, Example 2.16, 2)] such that the curvature is in theWeyl space, and that explains the invariance of the Weyl curvature underconformal equivalence.


7.10. Riemannian density.If V is a vector space over 1[8 of dimension n and g is a nondegenerate

quadratic form on it, then there is a natural nondegenerate quadratic formon all the spaces AP (V) as well. For, g gives an isomorphism between V andV* and hence a natural isomorphism of AP (V) with AP (V *) . Explicitly, thisassociates to the element (vi,. .. , vp), the p-form given by (wl,... , wP) Hdet(g(vi,wj)). In particular there is also a canonical quadratic form on theone-dimensional space AT(V), that is to say an element of ®2(An(V*)). Itis called the discriminant of g.

We are interested in the case when V is the tangent space at a point m ofa pseudo-Riemannian manifold M. Thus the pseudo-Riemannian structuregives a section of ®2 (An (T *)) = K2. If g has the expression > gig dxidxj ina local coordinate system (U, x), then the above section of K2 is obviouslygiven by det(gij)(dxl A A dxn,)2. If we take the sheaf S = IC ® OR ofdensities of M, and if dx is the Lebesgue measure in the coordinate system,then this element is given by det (gig) (dx) 2.

Let us now assume that g is actually a Riemannian metric, that is tosay positive definite. Then one can actually find a section of S itself, whosesquare is the above. It has the local expression det(g%3)dx. Since it isuniquely characterised as the positive measure whose square is the abovesection of S2, it is determined globally as a section of S.

7.11. Definition. The unique positive density whose square is the discrim-inant section of S2 is called the Riemannian density.

The Riemannian density, being nonzero everywhere, trivialises the sheafS. In the case of a Riemannian manifold, we will always trivialise S in thisfashion. In the case of an oriented Riemannian manifold, K is the same as Sand is therefore trivialised. It is obvious that its square as a section of K2 isinvariant under the Levi-Civita connection, and therefore the Riemanniandensity itself is invariant under the connection.

7.12. The star operator.Let V be an n-dimensional vector space. There is then a canonical

bilinear map AP (V) x A'-P(V) --* An (V), given by the wedge product. Thismap gives rise to an isomorphism of AP(V) with (An-P(V))* 0 A'(V). Inthe case of differential manifolds, this leads to an isomorphism of bundles

AP(T*) - (An-P(T*))* ®K.

Tensoring with the orientation system OR, we get an isomorphism

(AP(T*)) ® OR --> (An-P(T'*))* ® S.


If M is Riemannian, then the bundle S is canonically trivial on the onehand, and the bundles AP(T*) are self-dual on the other. Hence we obtainan isomorphism of AP(T*) with An-P(T*) ® OR.

7.13. Definition. The linear map * : AP(T*) -* An'-P(T*) ®OR defined asabove, using a Riemannian metric g, is called the star operator.

By definition, the star operator is determined by the equality

f g(*a, /3)dm = faAfiwhere a (resp. /3) is a (resp. twisted) differential form of degree p (resp.n - p), and dm is the Riemannian density.

7.14. Computation. *2 = (- 1)P(--P) Id on AP (V).

Proof. For this computation we may use a local coordinate neighbourhoodand use the orientation given locally. Let V be the tangent space of anypoint in this neighbourhood and (el, ... , en) an orthonormal basis. Thenthe star operator on AP(V) can be computed to be *p(ei1 A ... A eip) =e(o)e,j, n ... A ejn_p where jl, ... , jn_, are the complementary indices inincreasing order and a is the permutation of {1, ... , n} which is defined bya(r) = it if r < p and o-(r) = .2r-p for r > p + 1. On the other hand,*n_p(ej, n ... A ein_p) = e(r)eil n .. A ei, where T is the permutation givenby r(r) = j, for all r < n - p and r(r) = i,,-(n_p) for all r > n - p + 1.But then it is clear that o- o a = r where a is the permutation given bya(r)=p+rfor r<n-pand a(r)=r-(n-p) for allr>n-p+1.Hence we conclude that *n_P o *P(ei1 A . A eip) = e(a)eil n A eip. Thisproves our assertion.

In [Ch. 3, 3.8, Example 1)], we saw that the adjoint of the exteriorderivation di : Ai(T*) -+ Ai+1(T*) is (-1)i+ldn-i-1 : An-i-1 (T*) ® S -+An-i(T*) ® S. This uses the canonical isomorphism between (Ai (T*)) * andAn-j(T*) ®1C.

On the other hand, in the case of Riemannian manifolds, we have just re-marked that the Riemannian form g gives rise to a nondegenerate quadraticform on Ai(T*). So in this case, there is a natural adjoint Ai+l(T*)Ai(T*). The canonical duality transforms the adjoint to (-1)i+ld. Since *jis the composite of the canonical duality and the self-duality given by themetric on Aj, it follows that the adjoint operator is the transform by * of(-1)i+ld, namely (-1)i+l *d*-l. Substituting for the inverse of*, we finally get the adjoint operator to be (-1)ni+1 * d * . This differentialoperator from Ai+l(T*) to Ai(T*) is denoted 8i+i In other words, 8 on Aiis defined to be (-1)nj+n+l *n-'j+l odn_i 0 *i.


We will now compute it when M = R. On functions it is zero, bydefinition. We will compute it on 1-forms. Let a = E fidxi be a differential1-form. Then *a = E (-1)i-1 fidxl A A dxi A A dxi,,. Therefore d* a =

a)dxi A .. A dxn,. Finally as = - .

7.15. Proposition. The lift of the symbol T* ® Ai(T*) --+ Ai-1(T*) givenby (X, a) H -ixa is a.

Proof. Since * and the Riemannian density are invariant under the Levi-Civita connection, it is clear that a is also a lift of its symbol. The adjointsymbol of d assigns to v E T* the negative of the transpose of the symbolof d. In computing the transpose we need to use the metric on the exteriorpowers. If o- is the symbol, then we have (o-(a), d) = (a, v A 0). Takingdecomposable tensors for a and ,Q, we easily conclude that o- is -iv.

7.16. Definition. The operator A = (da+ad) from the sheaf of differentialforms (resp. twisted differential forms) of degree i into itself is called theLaplacian of the Riemannian manifold. Any (possibly twisted) differentialform a such that Da = 0 is said to be a harmonic form.

From the above computation of a on 1-forms on 1R we see that A(f) _8df = a(E adx

2

i) . This is the negative of the usual Laplaceoperator on the Euclidean space. For this reason A is sometimes defined tobe -(da + ad).

8. Clifford Structures and the Dirac Operator

8.1. Clifford algebra. Let V be a real vector space of dimension n, andq a quadratic form on it. We will denote the associated symmetric bilinearform by b. The quotient of the tensor algebra T(V) of V, by the two-sidedideal generated by v ® v - q(v).1, v E V, is called the Clifford algebra C(q)of the quadratic form. The inclusion of V in T (V) gives a canonical linearmap V -> C(q). The algebra C(q) is characterised by the universal propertythat any linear map f of V into any algebra A satisfying f (v)2 = q(v).1 forall v E V has a unique extension f as an algebra homomorphism of C(q)into A. Every v gives rise to two linear endomorphisms of the exterioralgebra, namely a) the wedge product A : A -1 V A'V, and b) the innerproduct c A''V --+ A'-1V given by (vi,... , v,.) H E(-1)i+lb(v, vi)vl A

A vi A A v,.. Then one easily verifies that ) = 0, 0, andc o av + Av o t = q(v). Id. As a consequence the map f : v H av + c ofV into End(A(V)) satisfies (f (v))2 = q(v).1. By the universal property thisgives an algebra homomorphism f : C(q) -* End(A(V)). Note that f (v)

8. Clifford Structures and the Dirac Operator 225

acts on the scalars in A(V) as a H av. In particular, f is injective on V.This proves also that the canonical map V - C(q) is injective.

In fact we have more. Consider the map x H I (x)(1) of C(q) intoA(V). The algebra C(q) has a natural filtration and a Z/2-gradation as well(coming from those of T(V)). The above map respects the filtration andthe Z/2-gradation on A(V) induced by its gradation. Therefore there is acompanion homomorphism of the associated graded algebras. Now Gr(C(q))comes with a linear map V -a F1(C(q))/Fo(C(q)) and all elements in theimage have square 0. By the universal property of the exterior algebrawe have therefore an algebra homomorphism of A(V) into Gr(C(q)). Themap v --i (1(v))(1) of C(q) given in the first paragraph of this section alsorespects the filtration and induces a linear map GrC(q) -- A(V). It is atrivial computation to see that this provides an inverse.

8.2. Theorem. The Clifford algebra of a quadratic form q on V is a Z/2-graded filtered algebra whose associated graded algebra is canonically isomor-phic to the exterior algebra A(V). There is also a canonical linear isomor-phism of A(V) with C(q) preserving the Z/2-gradation and filtration, whosecompanion map at the associated graded level gives the inverse of the abovemap.

8.3. Structure of C(q).We will hereafter assume that q is a nondegenerate form and recall some

of the properties of C(q). They are quite easy to prove, and one may consult[12] for details. The algebra C(q) itself is a central simple algebra whenn = 2m is even. Hence it is isomorphic to the (2', 2') matrix algebraover Il8 or a (2m-1, 2m-1) matrix algebra over the quaternion algebra. Theeven part C+ (q) has a nontrivial centre and is either isomorphic to a matrixalgebra over C or breaks up into the product of two simple algebras, eachisomorphic to a matrix algebra as above over the reals or the quaternions.

The Clifford group r(q) is the group of invertible elements x in theClifford algebra which satisfy xVx-1 C V. The group r+(q) of elements inC+ (q) which satisfy the same condition is called the even Clifford group.

If n is even, there are up to isomorphism two complex irreducible C+(q)-modules denoted Spin+ and Spin-. In particular, the group r+ has rep-resentation in these spaces. These are irreducible representations, calledthe half-spin representations. The group r+ has also a representation on Vwhich is called the vector representation. This is simply the action p(x)v =xvx-1. Since q(xvx-1) = (xvx_1)2 = xv2x-1 = x.q(v).x-1 = q(v), it fol-lows that the induced automorphisms p(x) preserve the quadratic form. Inother words, we have a natural homomorphism r -> O(q) and r+ , SO (q).The kernel of this representation is C" or RX x Rx. The representations of


I'(q) on Spin+ and Spin- do not go down to representations of the orthog-onal or the special orthogonal groups.

Now if we start with a pseudo-Riemannian manifold of dimension n =2m, we get in the above manner a bundle called the Clifford bundle. Butthere are in general no bundles corresponding to the two half-spin represen-tations.

8.4. Definition. A pseudo-Riemannian manifold M of signature (p, q) issaid to have a Clifford structure if the structure group of its tangent bundlecan be lifted from O(p, q) to the Clifford group I'(p, q). If M is oriented, thestructure can then be lifted to the group r+(p, q).

8.5. Remark. Here we are interested only in the Riemannian case. Eventhere we actually have two choices. We may use the Riemannian metric g orits negative -g and then take the Clifford group. It does make a difference.It is usually taken to be the latter.

Assume given a Clifford structure. Associated to the lifted principalbundle E, are the bundles Spin (M), associated to the spin representationsSpin+ and Spin- of r+ (n). The C(q)-module structure on Spin+ ® Spingives linear maps V ® Spin+ -* Spin- and V ® Spin ---> Spin+. Theseare clearly r+(q)-homomorphisms. Hence they induce vector bundle homo-morphisms T ® Spin+ (M) --+ Spin- and T ®Spin- (M) -+ Spin (M).Since T is naturally isomorphic to T*, these are potential symbols of differ-ential operators of order 1 from one half-spin bundle into another.

In order to lift these into differential operators, what we need, accordingto the prescription in Chapter 5, is a connection in the r+(q)-bundle. Ac-cordingly we assume given a connection on this principal bundle which onextension of the structure group to O (n) gives the Levi-Civita connection.

8.6. Definition. A Clifford structure on an even-dimensional oriented Rie-mannian manifold is a principal r+-bundle, together with a connectionwhich induces under the orthogonal representation of r, the tangent bundlewith the Riemannian connection. The differential operators of order 1 tak-ing one half-spin bundle into another on a Clifford manifold which lift theabove symbols are called Dirac operators.

There is a canonical antiautomorphism ,Q of C(q) which is the identity onV. It is simply the extension of the natural inclusion of V in C(q) consideredas a linear map into C(q)°PP as an algebra homomorphism C(q) -+ C(q)°PP.Then one can show that a/(a) is a scalar for every a E r(q). Moreover themap a a/(a) is a homomorphism called the norm of a. The kernel of thenorm homomorphism of r+ (q) into III" is called the Spin-group of q.

8. Clifford Structures and the Dirac Operator 227

The restriction of the vector representation of r(q) to Spin(q) gives ahomomorphism Spin(q) -> SO(q). Any v E V \ {0} is obviously invertiblein C(q),

e(v)being its inverse. Moreover vwv-1 =vwv/q(v) = b(v, w)v -

wv2lq(v) = b(v, w)v - w for all w c V. Hence by definition v belongs toI'(q). Also, in the vector representation v is represented by the reflectionin the hyperplane orthogonal to v up to sign. From this it follows thatSpin(q) ---> SO(q) is surjective. Moreover the kernel consists of nonzeroscalars of norm 1, namely ±1. Let v, w be two vectors of norm 1. Theelement vw in C(q) acts as the composite of two reflections, which is actuallya rotation in the two-dimensional space spanned by v and w. Assuming thatv, w are orthogonal, consider the one-parameter group y in Spin(q) takingt to cos(t) + sin(t)vw. Its image in SO(q) is the one-parameter group 77taking t to the rotation through an angle of 2t in the plane of v and w. Theinverse image of 77 contains -y, and since the map rr : Spin(q) -). SO(q) istwo-sheeted, and ly -* 77 is already two-sheeted, we deduce that ry is the totalinverse image of 77 and that Spin(q) is connected.

The tangent vector at 1 to -y is the element vw in the Clifford alge-bra. It maps to 2E,,,,, where E,,,,1 is the skew-symmetric endomorphismv F-> w, w t-> -w and 0 on vectors orthogonal to both v and w. Now theSpin representation is obtained as the restriction of an even Clifford modulestructure. Hence the element Ezj in the Lie algebra of SO(q) acts in theSpin representation through the element (1/2)vw in C(q). Elements of theform EzJ,,, generate the Lie algebra and hence we have identified its actionon the Spin representation.

8.7. Definition. An oriented Riemannian manifold is said to be a Spin-manifold if a lifting of the structure group of the tangent bundle from thespecial orthogonal group to the Spin-group is given.

8.8. Remark. There is not much difference between the groups r+ andSpin, since the latter is the kernel of a homomorphism into R+. Moreoverr+ is not compact. However, one can concoct another group, which ismore interesting. This is called Spin,(q). Take the product of Spin(q) andU(1) = S' and take the quotient of this group by the imbedding of Z/2 inthe product, by mapping the nontrivial element to (c, -1) where c is thenontrivial element in the kernel Spin(q) -* SO(q). We still have a vectorrepresentation of this group given by the vector representation of Spin(q)and the trivial representation of S1. A reduction (or lift) of the structuregroup of an oriented Riemannian manifold to this group is called a Spin,structure. The group Spin, acts on the two-spin representations, whereSpin(q) acts via the Clifford algebra, while S' acts by multiplication by


the corresponding complex scalars. The corresponding bundles will still bedenoted Spin+ and Spin.

Chapter 8

Local Analysis ofElliptic Operators

In this chapter, we give a quick account of the L2 properties of an ellipticoperator. Basic to this is the theory of Fourier transforms, Schwartz spaces,Sobolev spaces and the like. The main aim is to prove the theorems ofSobolev and Rellich, and the interior regularity of elliptic operators.

1. Regularisation

The notion of a kernel function K(x, y) in two (sets of) variables is usedfor transforming functions f of y into those of x by the prescription f Hf K(x, y) f (y)dy. If the kernel is nice, then since we are sort of averagingthe values of f with weights coming from K, the resulting function is wellbehaved. Convolution with functions is one such operation. If cp is a nicefunction, we take the kernel function K(x, y) = cp(x - y). We will nowexplain how this procedure leads to what is called regularisation.

Let co be any (infinitely) differentiable function in Rn with compactsupport. Then for any locally summable function f, the function

1.1. X H f cp(x - y) f (y)dy

229

230 8. Local Analysis of Elliptic Operators

is well defined. It is said to be the convolution product cp * f of cp and f.Then we claim that cp * f is differentiable. In fact, for any v E IEBn, consider

('P*f)(x+tv) - ((P*f)(x) - (aoo*f)(x)t lf((x+tv_Y)_(x_Y) _ Y)) f (y)dyt

Since go has compact support, the integration (for every fixed x, v andbounded t) needs to be performed only over a compact set K. We mayassume that f does not vanish on K. The differentiability of go implies thatfor any e > 0, we have the inequality

cp(x+tv-y)-g0(x-y) E

t fKIflfor small enough t and all y in the compact set K. This shows thatav (go * f) exists and is in fact equal to (0 go) * f . By iteration, we con-clude the following.

1.2. Proposition. Let go be any (infinitely) differentiable function withcompact support. Then for any locally summable function f, the functioncp * f is also (infinitely) differentiable. For any differential operator D of theform F, a« a a with a, constants, we have D(cp * f) = Dcp * f .

1.3. Remarks.

1) Notice that if both cp and f have compact supports, say Kl and K2,the convolution product cp * f also has compact support contained inthe sum Kl + K2 of the two supports, namely,

{x : there exist a E Kl and b E K2 such that x = a + b}.

2) The convolution product is defined whenever the integral in questionmakes sense, and is often useful in this greater generality. For exam-ple, if co and f are square summable, this product makes sense.

Suppose we can choose a sequence of differentiable functions gPk withcompact support such that cpk * f converges to f. Then we will have provedthat f can be approximated by differentiable functions. How do we findsuch a sequence? As a matter of fact, it turns out to be very easy. Take anydifferentiable function co with values in [0, 1], which is 1 in a neighbourhoodof 0, has compact support and is such that f cp(x)dx = 1. We wish toconstruct a sequence of such functions with supports in smaller and smallerneighbourhoods of 0 but with the same properties. For example, we maycut down the support to half its size, not affecting the integral, by definingcp2(x) = 2ncp(2x). Now it is clear how to define Wk. Take cpk = kncp(kx).The idea of choosing such a sequence is the following. The integrand in

1. Regularisation 231

Wk * f has the factor cok (x - y) and so for large k, the integration needs to bedone only over y very close to x. If f were continuous, the other factor f (y)will be close to f (x). Hence the integral tends to f cpk(x - Y) f (x)dy = f (x).

It remains to verify that our idea works. Assume that f is continuous.We have

(Wk * MX) = f kncp(k(x - y))f (y)dy

Substitute y' = k(x - y) to get (Wk * f)(x) = f So (y') f (x -k

)dy'. For all y'in the support of cp, the integrand tends to cp(y') f (x) uniformly as k tendsto infinity. Hence the integral tends to f (x). From the same considerationit is clear that the convergence is uniform on compact sets. Also if a func-tion has support K, and K' is any compact neighbourhood of K, then theapproximating functions cpk * f have support contained in K' for large k by(Remark 1.3, 1). We have thus shown

1.4. Proposition. The space of differentiable functions with compact sup-port in an open domain of IRn is dense in the space of continuous functionswith compact support provided with the topology of uniform convergence.

1.5. Corollary. Any linear form µ on the space of differentiable func-tions with compact support which is continuous, in the sense that if { fk}is a sequence with support contained in a compact set K that tends to zerouniformly, then {µ(fk)} tends to zero, can be extended uniquely to a linearform on the space of continuous functions with compact support with thesame continuity property.

This reconciles the definition of measure we gave in Chapter 3 with theusual one.

Let us now start with a square summable function f (i.e. f E L2) andsee where the regularisation procedure leads us.

1.6. Remark. If f E L2 and cp is one of the functions cpk as above, then

I(W*f)(x)I < fIx_Yh/2.Iw(x_y)ih/2fYIdY

-< IIW(x - y)112f(y)II

(fcox_YdY)

I I so(x -Y) 112f

(y) II

Hence f I (cp * f) (x) I2dx < f cp(x - y) I f (y) I2dxdy. Setting (x', y') = (x - y, y)the latter integral becomes f cp(x')I f (y')I2dx'dy', which by Fubini's theorem


equals 11f Ill. This shows that cp *f is actually in L2, and that I I cp * f I I C I I f I I.

Either by definition of L2 or as a standard theorem, we know that con-tinuous functions with compact support form a dense subspace of L2. Ap-proximating any f E L2 by a sequence of continuous functions with compactsupport, we get on regularisation, the following

1.7. Proposition. In any domain in R' , differentiable functions with com-pact support form a dense subspace of L2.

2. A Characterisation of Densities

2.1. Theorem. If i is a differentiable measure, that is to say, a (Borel)measure in a domain in R' such that the Lie derivatives of all orders existas Borel measures, then it is of the form f dx where f is Lebesgue summableand dx is the Lebesgue measure.

Proof. In view of the local nature of the problem, it is enough to provethe theorem for µ with compact support. We show that it is absolutelycontinuous with respect to the Lebesgue measure. In other words, if S is aset of Lebesgue measure 0, then p(S) = 0 as well. We will use the following.

2.2. Lemma. If S is a set of Lebesgue measure 0, then there exists asequence (xk) of points in R1, tending to zero as k tends to infinity, suchthat Tk(S) fl s = 0 for all k. Here Tk denotes translation by xk.

Proof. This is simply a reformulation of the following.

2.3. Characterisation of sets of positive Lebesgue measure. A sub-set of W' is of positive Lebesgue measure if and only if the set

{x E Il8n : there exist a, b E S such that x = a - b}

contains an open neighbourhood of {0}.

If µ is not absolutely continuous with respect to the Lebesgue measure,then by the Lebesgue-Nikodym decomposition theorem, there exists a setS of Lebesgue measure zero such that µ(1Rn \ S) = 0. We may assume forour purposes that S is bounded by replacing it by its intersection with thesupport of A. Let (xk) be a sequence of points as in Lemma 2.2. We mayassume, by passing to a subsequence, that there exist tk > 0 such that {X}has a limit, say v, in Sn-1. Note that {tk} is a sequence tending to zero.Since Tk(S) C Il8n \ S, we have µ(TkS) = 0 for all k. Hence I (Tk(µ) - )c)(S)is independent of k and the sequence (TkWtk µ)(`s) cannot have a limit. On


the other hand, in view of the supposed differentiability of p, we see that(Tk(l')-A)(s) does have a finite limit, namely lim (Ttk'(µ)-'u)(s) =This proves by contradiction that y is absolutely continuous with respectto the Lebesgue measure.

2.4. Remark. By Lebesgue's theorem, there exists a Lebesgue summablefunction f such that p. = f dx. The assumption that p admits iterated Liederivatives can be translated into the existence of a measurable function gsuch that f f a«co/axadx = f gcpdx for all differentiable functions cp withcompact support. Under such a condition one can show that f is itselfdifferentiable. This is known as Sobolev's theorem, and a version of it willbe proved in 6.6.

3. Schwartz Space of Functions and Densities

Let V be a real vector space of dimension n. In this section we wish todefine an algebra SF(V) of functions on V as well as a module SD over it.Any choice of a Lebesgue measure will provide an isomorphism of the latterwith SF(V) as an SF(V)-module.

There is of course no canonical choice of Lebesgue measure. Howevernotice that on V x V*, there is a canonical nondegenerate alternating bilinearform, namely ((v, f ), (w, g)) H (w, f) - (v, g). This bilinear form may alsobe considered as a translation invariant exterior 2-form on the differentialmanifold V x V*. With respect to a linear coordinate system (x1, ... , X')in V, and the dual coordinate system (e1i... , fin,) in V*, this form is givenby EZ 1 dx2 A dez. Its nth exterior power gives a top exterior form whichis translation invariant. In the above coordinate system this form has theexpression n!dxl Adl;1 A .. Adx,, Ad1;,,. Thus we have a canonical translationinvariant 2n-form which has the local expression dxl A dal A ... A dx,z A

Hence there is also a canonically defined positive, translation invariantmeasure o, on V x V*. Whenever we fix a Lebesgue measure dx on V thereis a Lebesgue measure dy on V* such that o = dx ® dy.

3.1. Definition of SF and SD.Any v E V gives rise to a vector field av on V treated as a differential

manifold. These are the translation invariant vector fields on V. Any two ofthese vector fields commute and hence by repeated operation of these we getan isomorphism Q H aQ of the symmetric algebra S(V) with the algebra oftranslation invariant differential operators on V. These may also be calleddifferential operators with constant coefficients. On the other hand, we canregard elements of S(V*) as polynomial functions on V. Thus we get a mapof S(V*) ® S(V) into the space of differential operators in V, induced by


(P, Q) H PBQ. It is easily seen that this is an injective linear map. Elementsin the image are differential operators with polynomial coefficients and willbe called simply polynomial differential operators. The set of polynomialdifferential operators form a subalgebra P(V) of the algebra of differentialoperators.

3.2. Definition. A differentiable function f such that for every D E P(V),D f is bounded, is said to be a Schwartz function. The space of Schwartzfunctions will be denoted SF(V) or simply SF, called the Schwartz spaceof functions.

3.3. Exercise. Show that Schwartz functions tend to zero at infinity.

3.4. Example. Let q be a positive definite quadratic form on V. Then thefunction e-&) is a Schwartz function. In fact, if we apply any polynomialdifferential operator to this function, the result is a finite linear combinationof functions of the type p(v)e_q( ), where p(v) is a polynomial function. Sowe have only to check that p(v)e-q(") is bounded, which is obvious.

The set SF(V) is obviously a vector space closed under multiplicationand is thus a commutative algebra. Note however that it does not have anidentity element, since the constant function 1 is not a Schwartz function.(Why not?) Clearly differentiable functions with compact support form anideal in SF.

3.5. Remark. Let q be as in the above example. We will show that(1+e z

)(n+1)/2 is integrable. Let Q be the quadric {v E V : q(v) = 1}.We identify R+ x Q with V \ {0} by restricting the map (t, v) H tv ofR+ x V -> V. The differential of the above map takes a tangent vector(A, x) E T(t v) to the vector \v + tx E TtT,. Its nth exterior power takesat A a + b with a E An-1 and b E An, to to-lv A a + tnb. From this we con-clude that the pull-back of a translation invariant n-form w on V to IRB+ x Vat the point (t,v) is tnp2(w) + t1-1dt A p2(ivw). Its restriction to R+ x Qis therefore tn-Ii,J(w)ITT(Q). This means that the image under the productmap of the product measure of dt on I8+ and to-lµ on Q is the restrictionof the Lebesgue measure on V to V \ {0}, where t is the measure given bythe (n - 1)-form on Q which associates to v the (n - 1)-differential izJ(w).This is clearly the Riemannian measure on Q. This measure is referred toas `the Lebesgue measure in spherical polar coordinates'.

The pull-back of the function (1+a(v )(n+1)/2 to R+ x Q is the function

(t, v) (1 )(n+1)/2 This function is summable with respect to the above

measure if and only if (,+t2)(,+,)/, is integrable on R+ with respect to the


measure dt. It is clearly integrable in {t < 1}, since it extends to 0 asa continuous function. On the other hand, substituting s = 1/t one seesthat the transformed differential is -1/(1 + s2)ds, which is also integrablein {s<1}.

Write any f E SF as f.(1 +q(x))I'(1+e(- )I\', and note that the functionf.(1 + q(x))N belongs to SF for large N and is therefore bounded. TakingN = x1, we conclude that f is integrable. Applying the same to any powerof f, we deduce that f is in LP for all p.

3.6. Definition. If µ is a density on V such that Dµ is bounded for everyD E P(V), then it is called a Schwartz density. The set of all Schwartzdensities is denoted SD(V) and is called the Schwartz space of densities.

The set SD, besides being a vector space, is also an SF-module. In fact,if f E SF and It E SD, the density f µ is again easily seen to be in SD.

3.7. Remark. We have seen that any density µ is of the form f dx, wheredx is a Lebesgue measure. Since the Lie derivatives L(v) of dx are all zero,it follows that L(v) (f dx) = 8v (f )dx. Hence µ is a Schwartz density ifand only if it is of the form f dx where f is a Schwartz function. In otherwords, the identification of Schwartz densities and Schwartz functions iscompatible with the action by polynomial differential operators. This showsthat although there is no canonical identification, the two SF-modules SFand SD are isomorphic.

3.8. Convolution of densities.

We will now define the convolution product of two densities. Let µ, v betwo densities. Then we define their convolution µ * v to be the image of theproduct measure µ ® v on V x V under the addition map V x V -> V. Moreconcretely, it is given by the prescription (µ * v) (f) = f f (x + y)dµ(x)dv(y),for every f E CC°. This makes sense, if we assume that one of µ, v hascompact support, or that both of them are bounded (i.e. of finite measure).In particular, if both µ and v are in SD, then µ * v is well defined and isactually in SD. In fact, if we show that on applying any polynomial differ-ential operator, the resulting measure is a linear combination of measuresof the same form, our assertion will follow. For any v E V, the linear formf -(µ * v) (8v f) is simply f H ((8vµ) * v) (f) from the definition. On the


other hand, if 1 is a linear function on V, then we have

(µ * v) (l f) = J I (x + y) f (x + y)dµ(x)dv(y)

= f f (x + y)l(x)dµ(x)dv(y)

+ f f(x + y)dµ(x)l(y)dµ(y)

In other words, (µ * v) (l f) = (lµ * v) (f) + (µ * lv) (f) for all f. This impliesthat l(µ * v) = (lµ) * (v) + µ * (lv). We deduce that µ * v E SD. Since itis obvious that the map (µ, v) H µ * v is bilinear, we have thus defined alinear map SD ® SD -> SD.

This multiplication is commutative and associative and makes the spaceof densities a commutative algebra. If µ E SD(V) and f E SF(V), wecan define a convolution product µ * f E SF(V) to be the function x Hf f (x - y)dµ(y). This makes SF(V) a module over SD(V).

If we identify Schwartz densities with Schwartz functions (which we cando on fixing a Lebesgue measure dx on V), then the space of Schwartz func-tions also acquires the structure of a commutative algebra under convolution.However, it is already an algebra under the usual product of functions. Inorder to distinguish the two structures, even when we fix a Lebesgue mea-sure on V and identify densities with functions, we will continue to denotethe algebra under convolution by SD(V) and under ordinary product bySF(V). We will now compute the formula for this product for the convolu-tion of two functions after identification of SF with SD. Let f, g, h E SF.We identify f and g with the measures f dx and gdx respectively. Then theconvolved measure (f dx) *f(gdx) gives on h the value

Jf(y)g(z)h(y + z)dydz.

Substituting (y', z') = (y + z, z), we see that the integral is the same as

f f (y z')g(z')h(y')dy'dz' = J(f * g) (y)h(y)dy

where (f * g) (x) = f f (x - y)g(y)dy. In other words, we have

3.9. Proposition. If f, g are in SF, then the convolution of the measuresf dx and gdx is the measure (f * g)dx, where f * g is the function xf f(x - y)g(y)dy.

3.10. Remark. This formula is in conformity with the convolution productof two functions which we defined in 1.1 in connection with regularisation.Note that unlike the convolution of measures or that of a measure and a


function, which does not require any Lebesgue measure on V, the convolu-tion of functions does depend on such a choice.

3.11. Topology on SF and SD.We will now topologise SF. There is a unique topology on it in which

a sequence { fk} tends to f if and only if for every polynomial differentialoperator P, the sequence {P fk} tends to P f uniformly. If Q is any polyno-mial differential operator, it induces a map Q : SF --* SF, and the abovetopology is designed to make these maps continuous. In fact we have onlyto check that for every polynomial differential operator P : SF -; SF thecomposite P o Q is continuous into SF provided with the uniform topology.This follows from the definition since P o Q is also a polynomial differentialoperator. Moreover, it is clear that multiplication of functions is continuousin this topology. We have only to show that if (fk) (resp. (gk)) tends to f(resp. g) in SF and P is a polynomial differential operator, then (P(fkgk))tends to P(fg) uniformly. But P(fkgk) is a finite linear combination ofterms of the form Pi(fk)Qi(9k) where Pi,Qi are polynomial differential op-erators. Now (Pi(fk)Qi(gk)) tends uniformly to Pi(f)Qi(g). This proves ourassertion.

3.12. Approximate identity.

We remarked that the algebra SF does not have an identity element.With the introduction of this topology, we can make good this deficiencysomewhat, by the notion of an approximate identity, namely a sequence { fk}such that { fk.g} -+ g in the above topology, for every g E SF. In orderto construct such a sequence, we start with a function f E SF with valuesin [0, 1] which restricts to the constant function 1 on the unit ball and hassupport inside a ball of radius 2. Then we define fk(x) = f (x/2k). It is clearthat for any g, and any compact set K, we have (fk.g)IK = gjK for largeenough k. For any e > 0, IgI is smaller than e outside some compact set K,and supKI fk.g - g1 can be made arbitrarily small by taking large enough k.Thus the sequence { fk} constitutes an approximate identity.

It is clear that SD can also be provided with a topology in a similar way.The algebra structure on SD (under the convolution product) also has anapproximate identity. Take any density p in SD such that the total measureis 1. Define µk by uk(f) = p(fk), where fk(x) = f (x/2 k) for any f E C°°.Then (µk * v)(f) = f f (x + y)dµk(x)dv(y) = f f (2-kx + y)dp(x)dv(y).As k tends to infinity, the integrand tends to f (y) uniformly (having sup-port contained in a fixed compact set) and therefore the integral tends tof f (y)dµ(x)dv(y) = f f (y)dv(y) = v(f ). Hence µk * v - v as k -+ oo.


If we fix a Lebesgue measure dx and identify SD with the space of func-tions, and take µ to be cp(x)dx, then the measure µk is given by µk(f) _f f (2-kx)c,o(x)dx = f (y)cp(2ky)2kdy on substituting y = 2-kx. Otherwisestated, the approximate identity µk is simply f (k) dx, where f (k) = 2k f (2kx).

In addition to all these structures, there is also the action of the groupV on SF by translation, that is to say, if f E SF and V E V, then wedefine Tv (f) to be the function taking w E V to f (w - v). Clearly the latterfunction also belongs to SF. In a similar way we can also make the groupGL(V) act on SF.

4. Fourier Transforms

Let V be a finite-dimensional real vector space and V* its dual. The Fouriertransform is defined as explained at the beginning of this chapter, using thekernel K(x, y) = e-i(x,y) as a function on V x V*. This function is calledthe Fourier kernel.

Fourier transform of Schwartz densities.

4.1. Definition. The Fourier transform of a Schwartz density on V, namelyan element µ of SD (V) , is the function µ on V * given by the formula

Je-z(x,1)d1-t(x)

Since any bounded measurable function is integrable with respect to µ,our definition makes sense. It is also clear that the function µ is bounded(by the bound of the measure µ). We will now prove that µ is actually aSchwartz function on V*. Since P(V*) is generated by operators of the form,91, 1 E V* and multiplication by linear functions, namely elements v E V, itwill be enough to show that 8lµ and vµ are again of the same form.

4.2. Lemma. If µ E SD(V) and v e V, then the Fourier transform of 8vµis the function iv.µ.

Proof. By definition, 8µ is given byff J

8v(e-'*1))dµ(x)

= fwhere v is considered as a linear function on V*. The right side is clearly(ivµ) ()


4.3. Exercise. Note that the equality cp(x)d(aµ)(x) = - f 8 cp(x)dp(x)is the definition of 8,,, but one had to take cp in C°° for this. We have usedit for the function e-i(x,e). Justify it.

4.4. Lemma. If µ E SD(V) and rl E V*, then the Fourier transform of 77pis ia,,p (where 77 is regarded as a function on V).

Proof. We have f e-i(x,0r7(x)dp(x) -if a,a(e-i*0)da(x) _i f e-'(x,0 d(a,,A)(x) as claimed.

These two simple lemmas show that the image of SD (V) under Fouriertransform is contained in SF(V*) for all µ E SD(V).

We will see next how Fourier transform behaves with respect to thevarious structures on the Schwartz spaces that we defined above. To startwith, it is obvious that it is (C-linear.

4.5. Lemma. If µ and v are two elements of SD(V), then the Fouriertransform of the convolution µ * v is the product of the functions % and v.

Proof. We have µ * v* f e-i(x+V,)dp(x)dv(y). This simplifies, thanksto the theorem of Fubini, to the product of f e-i(x,)dµ(x) = andf e-i(Y,)dv(y) = v( )

In other words, µ --> µ is a homomorphism of the algebra SD (V) intothe algebra SF (V *) .

4.6. Remark. If we fix a Lebesgue measure dx on V and identify SD withthe space of Schwartz functions, then we get (f (x) dx) f e-i(x,t) f (x) dx.Hence we may define the Fourier transform of a Schwartz function f on Vto be the Schwartz function f on V* given by

f fe)f(x)dx.

Fourier transform of Schwartz functions.

4.7. Definition. The Fourier transform from SF(V) into SD(V*) asso-ciates to any f E SF(V), the linear form f on defined by g Hf f (x)g(e)da. The measure used on the product is the canonicalLebesgue measure on V x V*.

It is clear that since f and g are integrable, the above integral exists.Moreover it it easy to check that the linear form is continuous for the usualtopology on C,°. Let us choose Lebesgue measures dx, dt; on V , V* respec-tively in such a way that the product measure is canonical. Then we will


show (Remark 4.9) that for f c SF(V), the measure f as defined in 4.7 isthe same as f d in the notation of 4.6. This shows that the measure factually belongs to SD(V*). Finally it is also true that

*f =µf4.8. Exercise. Prove the above identity. Thus the SD(V)-module struc-ture of SF(V) is taken by the Fourier transform to the SF(V*)-modulestructure of SD(V*).

4.9. Remark. If we identify a function f in SF(V) with the density f (x)dx,then f (x)dx is a function in SF(V*). Choosing the Lebesgue measure don V* such that dx ® d is canonical, we can identify it with a density inV*. Let us now compute this density. Firstly, f (x)dx(e) = f e-i(x'C) f (x)dx.Hence the corresponding density is the linear functional which takes cp(e) tof f (x) dxd. This is simply f (go) by definition. Thus the Fouriertransforms SD(V) --j SF(V*) and SF(V) --* SD(V*) are the same if weuse dx to identify SD(V) with SF(V) and simultaneously SD(V*) withSF(V*) using

As far as the topological structure is concerned, we have the following

4.10. Lemma. The Fourier transform is a continuous map from SD(V)(resp. SF(V)) into SF(V*) (resp. SD(V*)).

Proof. In view of Lemmas 4.2 and 4.4, we need only prove that the mapis continuous when the target is provided with the topology of uniformconvergence. Now if µ E SD(V), then f Ie-i(x,0IdIyI = IµI(RI)Hence if {µk} --> µ, we see that sup I/-Ik - Al can be made arbitrarily smallfor large k. The proof that SF(V) --> SD (V *) is continuous is similar.

4.11. Remark. Note that since the Fourier transform is an algebra ho-momorphism, the above lemma implies that any approximate identity forthe usual multiplication of functions is taken to an approximate identity ofthe convolution algebra and conversely. Indeed, for the former, we took theapproximate identity f k(x) = f (x/2k), for a suitable f. It has as Fouriertransform, the density

g'-' fef(x/2k)g(v)d(x,v) = fe_i()f(y)g(w/2k)du(y,w)

on setting (y, w) = (x/2k, 2kv). Hence it is yk(g) with the notation of 3.12,where µ is the density f .


4.12. Lemma. For f E SF(V) and v E V, the Fourier transform of thetranslate T (f) is given by e-iv f where v is considered a linear function onV*.

Proof. In fact, T, f (cp) = f e-i(x,") f (x - v)cp(t;)do, (x, e). On substitut-ing (y, rl) _ (x - v, ) this becomes f e-i(y,n) f (y)e-i(",'1)co(r))dv(y, rl)

(e-2v f) (p)

4.13. Theorem (Fourier Inversion Formula). The composite of thetwo Fourier transforms SF(V) -3 SD(V*) and SD(V*) -* SF(V) is themap f (x) F-* f (-x).

Proof. Let f E SF(V). We will first show that for any g E SF(V*), wehave g f (x) = f * g, where f is the function x H f (-x). In any case, if theassertion of the theorem were true, this would be an obvious consequence.Conversely, if this is proved, then we can apply it to an approximate identity(9k) in place of g, and since (gk) is an approximate identity in SD (4.8), thetheorem would follow on letting k tend to infinity.

By definition we have

fed(gf)()9f (x) =

= fe-i(x, )glS)e-i(y,)

On the other hand, we have

f * 9(x) = ff(x_Y)d(y)

= f f (-x + y) f e)

= f f W) g W) e-'(x+y',E') do, (y', ')

on substituting (y', (-x + y, 6).

4.14. Corollary. Fourier transform is a linear homeomorphism (as well asan algebra isomorphism) of SD(V) with SF(V*) and SF(V) with SD(V*).


Proof. The map F' : M A(-x) composed with the Fourier transformSD (V) --* SF (V *) is Id on SD (V) by the above theorem. In view of Remark4.6, the composite of the Fourier transform SF(V*) --f SD(V) with F' isalso the identity on SF(V*).

The expression for the Fourier transform of a function, namely f (e) _f e-i(1°0f (x)dx, makes sense even if f is just summable. This satisfies theidentity f f gdx = f j gds for all g E SF. In fact, since g is also summable,the function f is summable with respect to the measure v =dxd4. By Fubini's theorem, we see that the above expressions are the same.

To make things more symmetric, we may look at the pairing betweenSF(V) and SF(V*), given by .F(f, g) = f (g) = f e-i(x,E) f (x)g(6)do-(x, 6).It is clear that this is a symmetric pairing. In other words, we have

4.15. Proposition. If f E SF(V) and g E SF(V*), then f f dg = f gd f .

Let us fix a Lebesgue measure dx and take g(x) = h(-x) in the aboveproposition. Then we get f f (x)h(x)dx = f h(-x) f (x)dx. Now the Fouriertransform of h is given by f e-i*g) h(x)dx = f ei(x,e) h(x)dx = Interms of the L2 scalar product, the above formula therefore reduces to(f , h) = (f , h) on replacing h by T. We have thus proved

4.16. Theorem (Plancherel). The Fourier transform respects the L2 scalarproduct on Schwartz functions.

The Fourier transform can consequently be extended to a unitary iso-morphism of L2 (V, dx) into L2 (V*, d is the measure Notehowever that if f is square summable, then the expression f e-i*0 f (x) dxdoes not make sense, since the integrand is not summable in general. Forthat we need f itself to be summable. If f is summable as well as squaresummable, then we thus have two notions of Fourier transform, namely theexpression given above, and the extension of the map SF -* SD that wehave given, using Plancherel theorem. But we know that in this case, f sat-isfies cp), which is also satisfied by the L2-extension of Fouriertransform. Hence the two extensions coincide.

5. Distributions

Let U be a domain in V. Consider the space C°° of differentiable functionson U with compact support. We say that a sequence { fk} in this space tendsto f if all fk have support contained in a fixed compact set K C U, and forevery differential operator D with constant coefficients, the sequence {D fk}tends to D f uniformly. A similar definition can be given also for densities

5. Distributions 243

with compact support, at least by identifying densities with functions andchecking that the topology is independent of the Lebesgue measure chosen.We will call this space D°°.

5.1. Remark. If we fix a compact subset K, then the space CK of differ-entiable functions with support in K may be topologised by defining theneighbourhood filter at 0 to be the one generated by the sets of the formUD,,, = { f : sup ID f I < a} where D is a differential operator with constantcoefficients and a a positive real number. This makes CK a topologicalvector space.

5.2. Definition. A distribution (resp. an n-current) in U is a continuouslinear form T on D°° (resp. C°°), that is to say, if {µk = fkdx} -+ y = fdx(resp. { fk} --4 f) in the above sense, then {Tµk} -i T p (resp. {T fk} -Tf).

In view of the remark above, a distribution is a linear form T on D°°such that for every compact subset K, there is an inequality of the form

5.3. supITf1 <CDsupIDfI

for every differential operator D with constant coefficients and for all f EK,

If we fix a Lebesgue measure, we may identify n-currents and distribu-tions.

5.4. Examples.

1) It is clear that if µ is any measure, it can be considered a current aswell. In particular, for any v E V, the measure f H f (v) is a current,known as the Dirac current at v.

2) If f is a measurable function defined on U, we say that it is locally squaresummable if for any compact subset K of U, it is square summableon K. If f is locally square summable, then it defines a distributionTf. Indeed, for every µ E D°° we define Tf(p) = f f (x)dl-t. One caneasily check that the map f T f is an injective linear map. Hencewe may identify such functions with the distributions they define.

3) Let f be as above. For any v E V and µ E DC°, define T(µ) _- f f (x) (,9,, A) (x). It is easy to check that this is also a distribution.Moreover, if 8, f exists as a function which is locally square summable,then T is the same as the distribution defined by 8y f . Therefore itis apt to define the distributional derivative with respect to v of alocally square summable function f (whether 8,J f exists or not, as a


function) to be the distribution defined above. A good notation forthis is therefore a, (T f) or even aT, (f ).

In fact, we will generally define the derivative of a distribution as follows.

5.5. Definition. If T is a distribution (resp. current), then we define itsderivative av with respect to v E V to be the distribution (resp. current)given by (0,,T) (IL) = -T(& ,p), p E Dc° (resp. (a T)(f) _ -T(a f ), f ECC°).

When we say a distribution is a function, we mean that it is the distri-bution associated to a function.

5.6. Definition. Any continuous linear functional on SD(V) (resp. SF(V))is called a tempered twisted distribution or tempered current. The space oftempered distributions will be denoted by ED(V).

From the fact that D°° -+ SD is continuous, it follows that a tempereddistribution gives rise to a distribution. Moreover, since the image of D°°in SD is dense, we see that tempered distributions form a subspace of thespace of distributions.

The significance of tempered distributions is in the following. We haveseen that the Fourier transform defines an isomorphism of SD(V) withSF(V*) and of SF(V) with SD(V*). But Schwartz functions are too re-strictive, and we would like to extend the notion of Fourier transform to alarge class of functions. Explicitly, we have the following definition.

5.7. Definition. The Fourier transform t of a tempered distribution (resp.tempered current) T on V is the tempered current (resp. tempered distri-bution) on V* defined by the prescription t(g) = T(g) for all g E SF(V*)(resp. g E SD(V*)).

Since the Fourier transform is a homeomorphism, it is obvious that 1' istempered when T is. If f E SF(V) then t f is defined therefore by g H T f (g)for g E SF (V *) . But T f (g) = f f dg = f gd f by Proposition 4.12. Thuswe get t f (g) = T f (g), or what is the same, T f = T1. This shows that ourdefinition of the Fourier transform of tempered distributions is compatiblewith the definition of the Fourier transform of Schwartz functions.

Since the Fourier transform of the Schwartz spaces is a continuous linearisomorphism, it also defines by transposition, a linear isomorphism of thespace ED(V) with EF(V*) and of EF(V) with ED(V*). The point of thisis that large classes of functions can be considered as subspaces of EF andso Fourier transform theory becomes applicable to them.


Let f be a square summable function on V. The distribution it givesrise to is a tempered distribution. In fact, to every µ E SD, associate f f dµ.Since µ is of the form gdx with g E SF, it is the same as f f gdx. We haveseen in 3.5 that g is also square summable. Hence the integral makes sense.We have thus given a linear functional on SD (V) .

It remains to show that it is continuous. In other words, if (gk) is asequence in SF(V) such that (Dgk) tends to 0 uniformly for all polynomialdifferential operators D, then we have to show that (f fgkdx) tends to 0.Notice first that by assumption, for every N, the sequence {(1+q(x))Ngk} isuniformly bounded. Choosing N so large that (l+qx)is square summable,we conclude that (9k) is uniformly majorised by a square summable function.Hence (9k) tends to 0 in L2 by the dominated convergence theorem. Sincef is also in L2, we conclude that f fgkdx tends to zero.

Thus we have given an injective linear map of L2 into EF. In particularthe Fourier transform of a square summable function makes sense. If (fk)is a sequence of functions in CC° tending to f E L2, then we have alreadyremarked that the Plancherel theorem ensures that (fk) tends to a functioncp in L2. Hence for any g E L2, the sequence { (fk, g) } tends to (cp, g). Inparticular, the sequence of distributions T-k tends to T. It is obvious thatT-k = Tfk . Thus we conclude that T f = T. In other words, the continuousextension to L2 of the Fourier transform defined on Schwartz functions, coin-cides with the restriction of the Fourier transform on tempered distributionsto L2.

5.8. Remark. If P is a polynomial differential operator and T a tempereddistribution, then PT is also a tempered distribution. Formulas 4.2, 4.4 arevalid for the extended Fourier transform as well.

6. Theorem of Sobolev

We will hereafter fix a positive invariant measure on ll8n.

Let r be any fixed nonnegative integer. Then one defines the Sobolevspace of order r to be the space 7-lr of functions f on U which are squaresummable so that the distributions D f are square summable functions, forall differential operators D with constant coefficients of order < r. Themeaning of the local Sobolev spaces 7-1;00 is therefore clear.

Sobolev spaces come equipped with Hilbert space structures, with thescalar product

aj<r


6.1. Remark. We saw in 1.7 that differentiable functions with compactsupport form a dense subspace of L2. One can actually show that they alsoform a dense subspace of 7 ,,z in R', for all m. In other words, any function in7-G,,,, can be approximated by differentiable functions with compact support.However this is not true for functions in an arbitrary domain U in W'.If a function on U belonging to 7-t,,,, has compact support K, then it canbe extended to the whole of W by setting it to be zero outside U. It canthen be approximated by differentiable functions, with support still compact(which may be slightly bigger than K) and contained in U. This reducesthe problem to the case of R'. The proof is, as in 1.1, by regularisation.

We will use the same notation and show that if f belongs to 7-tl forexample, then for all v E 118', the sequence {8 (A * f )} tends to 8 f in L2.Note that by definition, 8 f belongs to L2 and hence so does cpk * 8 f. Asone might expect (in view of 1.2), it is indeed true that 8 (cp * f) = cp * 8 fas distributions. For any g E C°°, we have

(av(W * f))(g) f (co *

f f

f (x - y)co(y)g(x)dxdy

f (cP * af)(x)g(x)dx

(cp * avf)(g)

Now replacing cp by cpk and letting k tend to infinity, we deduce that Of isthe limit in L2 of cpk * 8 f =a, (cOk* f ). In other words, cok * f tends to fin7-11. It is clear by induction that if f E 7-1m, then the convergence is actuallyin 7-,n. Thus we have proved

6.2. Theorem. Differentiable functions with compact support form a densesubspace of 7tm in Rn.

The above argument proves the following stronger statement.

6.3. Proposition. If u is in 7.1,,,, and such that Du E L2 for some differ-ential operator D with constant coefficients, then there is a sequence { fk}of differentiable functions with compact support, which tends to u in 7-t,n sothat {D fk} tends to Du in L2 as well.


Proof. We only have to verify that D(cp * u) = cp * Du and repeat theargument above. But this follows from the assumption that D has constantcoefficients.

6.4. Remark. Actually it will turn out that we can even take any differ-ential operator D which has constant coefficients outside a compact set, inthe above proposition. The proof is more involved since it is no longer truethat D(cp * u) = cp * Du. We will now discuss this in greater detail.

Let u c 7-1,,,, such that Du E L2. We will approximate u by convolvingwith the functions cok. Clearly then Wk * u are all differentiable, and we haveseen that the sequence {cpk * u} tends to u in 'H,,,,. Since Du belongs to L2by assumption, the sequence {cal, * Du} converges to Du in L2. But we wishto show that D(cck * u) converges to Du in L2. We are therefore naturallyled to considering the difference J = D(cp * u) - cp * Du.

If we can show that the map 7-1,,,, --- L2 which takes u to 6,u is continuousand has uniformly bounded norm for all cpk, then we will be through. Forif u is infinitely differentiable, then {cpk * u} tends to u in Hr (where r isthe order of D) as well, so that {D(cpk * u)} tends to Du in L2. Hence thedifference D(cpk * u) - Wk * Du tends to 0 in L2. Let u be any element of1H,,,, , and {uk} a sequence of differentiable functions with compact support,tending to u in 7-1,,,,. For any e > 0, there exists R such that IIU -ulIm < 2ntfor all r > R, where M is a uniform bound for the norms of all the operators5Wk. Hence (uR - u) 11 < e/2 for all k. But we have just noted thatsince UR is differentiable, we have JIS,k (uR) II < e/2 for large enough k. Thisimplies that J I SWk (u) I I < e, showing that the sequence {D (Wk * u) - Wk * Du}tends to zero in L2, as required.

Notice that by assumption, the coefficient functions in the expressionfor D are constant outside a compact set. By subtracting the correspondingoperator with constant coefficients, we get an operator whose coefficientshave compact support. This does not change the commutator that we areconsidering since any operator with constant coefficients commutes withconvolution with Wk. Thus we need to prove only the following.

6.5. Lemma. Let D be a differential operator with compact support. Thecontinuous linear map f,,,, -> L2 given by u H D(cpk * u) - Wk * Du hasbounded norm, independent of k.

Proof. Since differentiable functions with compact support are dense inHm, we may compute norms by restricting the operators to C°°. Any dif-ferential operator D is given by the expression E as (x) 8. To prove thelemma, we may assume that D is just a« (x) . We will estimate the L2-norm of D(cpk * u) - cpk * Du by applying the Fourier transform and using


the Plancherel theorem. Then we are led to the following computation:

jD(SOk * u) - ('k * Du)I = laa(S) * (CaOk(C)u(C)) - k(S)aa(S) * (Sau(S))I .

=J

aa(S -

cOk(0 f aa( - n)nu(n)dn

= J as ( - ?7)?7au(?7) (4(77) -

We need to get an L2 estimate for the above function of . Before we setout to do that, we make a few comments. Since aa(x) has compact support,it and its Fourier transform are Schwartz functions and so remain boundedon multiplication by (1 + )N for any positive integer N. We will replacena by E (p) (n - The powers of 77 - can then be clubbed withC ( - n). We will denote by ba the functions Therefore we wishto estimate functions of the type

fba( - 77)u(77) (0(n) - (P(6))dn.

We are looking for an L2-estimate in terms of the norm of u in H,,,,. We willmultiply 6,3 by a suitable power of (1+q( , so that the product is inOnce that is done, we need only to bound the integral

f b( - n)u(n) (1 + 0(n))dn

where b is a product of as and a polynomial. The term causing difficulty inthe estimate is the factor (1 + If we had an inequality of the typeq( - n) > then we would have (1 + q( - 77)/,\)L andso we may replace b by a similar term b' and get the estimate. Accordingly,we split the integral as a sum Il + I2 of the integrals over two subsets of R,namely, one in which q( - n) > and the other in which the oppositeinequality holds, and estimate each separately. In the former case, the term10(n) - cp(s) I can be majorised by C11 ,q - III where C = sup I a cpl , whichis finite since cp is a Schwartz function. The integral of b'( - 77)u can bebounded in terms of the L2-norm of u or, what is the same, that of u, sincethe term b' is a Schwartz function. This estimate is valid for all cpk.

As for 12, we note as above that 0(77)I = ( whereC is some point in the segment n]. Since 9,0 is a Schwartz function, wehave the inequality, for any N,

v(C) - (n) <_ CZ (1II+q(C7711

))N


(where C2 of course depends on N). This is not good enough for us yet,since we do not have (1 + )N in the denominator. However, we notethat if A is small, the ball {v : q(v) < with the origin as centre andthe ball {v : q(Z; - v) < with as centre do not intersect. Since wehave assumed that rl belongs to the second of these, the segment [C771 iscontained in it and in particular ( belongs to it. Thus we have q(() > Aq(t;)and consequently i+q

C) < i+a4(g) This implies that (1+q«»L < (-1)L andwe are done. Thus we have shown

6.6. Lemma (Friedrichs). If u E flm is such that Du E L2 for somedifferential operator D whose coefficients are constant outside a compactset, then there exists a sequence { fk} of differentiable functions with compactsupport, which tends to u in 'H, so that Dfk tends to Du in L2.

There are also other spaces which are of interest to us. Consider thespace cm consisting of all measurable functions f such that P f is in L2 forall polynomials P of degree < m. It is obvious that the Fourier transformtakes 7-1,,,, bijectively onto cm. It is a simple matter to see that the norm in'H,,, is equivalent to the L2-norm of f (1 + q(x))m/2. It is this latter normthat we will use below.

Similarly we can also provide c,,,, with the norm given by 11f 1122 _E Ixa f 112. From the Plancherel theorem and Lemma 4.2, it follows thatthe Fourier transform is actually a metric equivalence of 'H n with c,,,,.

6.7. Theorem (Sobolev). Suppose a function belongs to all the spacesThen it is differentiable.

Proof. The assumption and conclusion are both local. Therefore it isenough to show that the function is differentiable in a neighbourhood of0. We may multiply the function by a differentiable function with compactsupport which is 1 in a neighbourhood of 0 and obtain a function f whichbelongs to 7-1m for all m. It is enough to show that f is differentiable underthis hypothesis.

We will first illustrate how to prove this by showing that f is continuous.Applying the Fourier transform to f, we see that it satisfies P j E L2 forall polynomials P. In particular, f (1 + q(x))N E L2 for all positive integersN. Now f (x) = f f (-) )Ndl;. Take N large enough(for example N > n/2) so that (1+e )N is square summable. Then bythe Schwarz inequality, we get I f (x)I < Cjj f (1 + I, for some constantC. Approximating f in f-1,,,, by a sequence of differentiable functions with


support in a compact neighbourhood K' of K, and applying the above in-equality to the Cauchy differences, we see that the convergence is actuallyuniform. Hence f is continuous.

As for the general case, following the same procedure, we get the in-equality II0«f II <_ CII f (1 + if Ial < r. Now if m > N + r, thenI I f (1 +

C I If IIm. It follows that the sequence fk E C°° which

tends to fin ?-1,,,, for large m satisfies sup I a«fk - afi I CII fk - ft II m Thisshows that fk tends to an r times continuously differentiable function.

6.8. Remark. In the above argument we could have taken N to be anyinteger greater than n/2. Thus we have actually shown the following moreprecise result.

6.9. Proposition. If f and all its derivatives of order < m are locallyin L2, and m > n/2, then f is continuously differentiable r times wherer <m-n/2.

We also showed above that under the same hypothesis, we have theinequality (for any compact set K and for all r < m - n/2)

6.10. sup a6x« f K<CKIIfllm

for all a of order < r.From this we derive the following.

6.11. Corollary. If T is any distribution, then its restriction to CK, whereK is compact, extends to a continuous linear form on hr, for sufficientlylarge r.

Proof. In fact, this follows from the above inequality and 5.3.

This can again be reformulated as follows. Denote the dual of fr by'H-r. Choose a Lebesgue measure and introduce the L2-metric in ?1 = ?-lo.We will identify the dual of 'H with itself. Then it is clear that we havecontinuous inclusions

... C H-(r-1) C?'l_rC ..C?-lC...C7{rC'Hr+1C...,

This is called the Sobolev chain. We will fix a compact set K and consider theabove chain taking spaces of functions with support in K. For some integerm, a distribution T satisfies (by 5.3) the inequality I T f I< C sup I D f I fordifferential operators of order < m for all fin ?-lm, in the closure CK. Thisimplies the inequality I T f I < C'1 I f IIr by 6.10. Hence T belongs to one ofthe spaces in the Sobolev chain. We may paraphrase this by saying that


the intersection of all the spaces in the local Sobolev chain is the space ofdifferentiable functions, and the union is the space of distributions.

7. Interior Regularity of Elliptic Solutions

7.1. Definition. Let L be a matrix of differential operators of order r, ina domain in 118'x. We say that it is elliptic if its rth symbol, interpreted as amatrix of polynomial functions in R1, is invertible, for every r E 118 \ {0}.

One of the most important properties of elliptic operators is the follow-ing.

7.2. Theorem. Let u be a distribution solution of an elliptic equationLu = f, with f differentiable. Then u is differentiable.

Proof. For notational simplicity, we consider L to be an operator on func-tions and not a matrix of operators, but the general case is identical. Tostart with, we know that locally u belongs to some Sobolev space H.. andthat it is enough to show that it belongs to every Sobolev space, in view ofthe theorem of Sobolev.

We observe that the operator 1 + A = 1 - a actually gives anz

isomorphism of 1 k with xk-2. In fact, we apply Fourier transform onboth sides, use the Plancherel theorem and reduce to proving that the map9k = xk -' 9k-2 = xk-2 given by multiplication by (1 + E x2) is anisomorphism. But this is obvious.

Going back to our question, let u E be a solution of Lu = f . Weclaim that we may assume that m > 0. If m is negative, consider s E f-,,,,such that (1 + 0)'s = u. Now the differential operator L(1 + 0)m isobviously elliptic, since its symbol is the composite of the (elliptic) symbolof L and (E x?)m. The element s is in 9-1-m and satisfies L(1 + 0)ms = f .

Now we will show by induction that s is in all xk. Thus we have reducedthe problem to showing the following.

7.3. Proposition. If U E f,,,,_1 for some m > 1, and Lu E fm-r for someelliptic operator L of order < r, then u E Rm.

Proof. Suppose that u is, a priori, a differentiable function. Then we willprove the following inequality.

7.4. A priori inequality. Let u be a differentiable function with supportin a compact set K, and L an elliptic operator of order r in a compactneighbourhood K' of K. Then we have the inequality

IjulIm <- C(I Lullm-r + jjujjm-1)


for some constant C. Here the norm on the left side is the norm in theSobolev space H,,,, with respect to the set K, and the ones on the right sideare the norms in Sobolev spaces with respect to K'.

7.5. Derivation of regularity from the a priori inequality.Note that u can be approximated in 'H,,,,-i by a sequence {uk} of differ-

entiable functions with support in K', ensuring at the same time (Lemma6.6) that LUk tends to Lu = fin Nm_r. Now IILuk - Luj IIm,-r is arbitrarily.small for large k and 1. On the other hand, Iluk - ui1lm-1 is also small forlarge enough k and 1. Hence, from the above inequality we conclude that{uk} is a Cauchy sequence in xm as well. It is clear that its limit in R. isu, since xm --> x-1,,,,_1 is a continuous inclusion.

7.6. Proof of the a priori inequality.We write L = L1 + L2, where L1 is the top homogeneous term of the

operator L and L2 is an operator of lower order. Then L2 gives a continu-ous map from H,-, to xm-r. Hence there exists a constant C1 such thatIIL2uIIm-r <- CiIIUIIm_1. If we show that the inequality is valid for homoge-neous operators (in particular for L1), then there exists a constant C2 suchthat IIuIIm < C2(IILiuII', -r + IIuIIm-1) Hence we obtain

IIuIIm C2(IIL1uIIm-r+IIuIIm-1)< L 2(IlLullm-r + IIL2UI4m-r + IIuIIm-1)

< C2(IILuIIm_r + (Ci + 1) IIuIIm-1)< C2(C1+1)(IILuIIm-r+IIuIIm-1)

Thus it is enough to prove 7.4 for homogeneous operators. We will proveit first for homogeneous operators with constant coefficients and then deducethe general case.

7.7. Proposition. Let L be a homogeneous elliptic operator of order r,with constant coefficients. Then there is some a > 0 such that, for everydifferentiable function u whose support is contained in a closed ball of radiusa, we have the inequality

IIuIIm < C(IILuIIm-r + IIuIIm-1)

Proof. If L is the differential operator Elal=r a« y and P the polynomialE where as are (r, r) matrices, then the ellipticity assumption impliesthat is nonsingular for all real nonzero vectors t;. If C is the infimum


of the norms of P(6)-1,11611 = 1, we have the obvious inequality IIP(6)2IIC I I6I I2r for all 1; . If I I I I

1 , then we derive the inequality

(1 + C(1 + IIP(6)II2r)

for a suitable C'. If 1, we have (1 + (2IICII2)r <-So we have the following inequality, valid for all l:

7.8. (1 + gII2)r <_ C(1 +

We now prove the proposition by multiplying 7.8 by (1 +and applying the Fourier transform. We then get, thanks to the Planchereltheorem, the following inequality:

IIuIIm < C(Ilullm-r + IILuII n_r)

where u =,b. But we have IIuIIm-r C IIuIIm-1 so that we may as well writethis inequality as

Ilull . < C(II Lull r2n_r + IIuIIm-1)

Clearly this implies the proposition.

Let L be any homogeneous elliptic operator and Lo the operator withconstant coefficients, which equals the operator L evaluated at a point, say0. Both these operators are elliptic, and we wish to compare them. Noticefirst that the difference L - Lo is an operator which vanishes at 0, and hencecan be written in the form E xiDi, where Di are homogeneous differentialoperators of order r. Now we make the following observation.

7.9. Lemma. Let Ba, be the closed ball of radius a and D a homogeneousoperator of the form E xiDi of order r. Denoting the closure of CC° in 7-lkby 7-lk,a, the norm of the induced continuous map 7-lk+r,a --+ 7-lk,a tends tozero as a tends to zero.

Proof. It is only a matter of checking the statement for an operator xiD.It is clear that the norm of this operator is the product of the norm of Dand the norm of the operator of multiplication by xi in 7 k,a -->'Hk,a Thelatter clearly tends to zero as a tends to zero.

Let Lo be the system with constant coefficients, obtained by evaluatingthe coefficients of L at 0. Then Lo is also elliptic and we have the inequality

IIuIIm < C(IILOUIIm-r + IIuIIm_i).

for all functions u E By Lemma 7.7 there exists a > 0 such thatfor all u with support in the ball of radius a, we have II (L - Lo)ullm_r <


(1/2C)IIuII,,,,. Then for all such u we get the inequality

IIuIIm < C(jjLOUIIm-r + IIuIIm-1)

< C(I LuIIm-r + II (L - Lo)UII'm-r +IIuIIm-1)< C(II LuII m-r + (1/2C) IIuIIm + IIuIIm-1)

This implies the inequality in 7.4 with the constant 2C instead of C for allfunctions in 7lm,a

Finally, given any compact subset K in U, we can use the above observa-tion and cover U with open balls in which the assertion is valid. Since K iscompact, finitely many of these balls cover K. Hence the a priori inequalitypersists for all u E C°° proving 7.4. As a consequence the regularity theoremis proved as well.

7.10. Rellich's theoremWe recall first a standard theorem in analysis and sketch its proof.

7.11. Theorem (Ascoli). If { fk} is a sequence of differentiable functionswhich, together with their first order derivatives, are uniformly bounded, thenthere is a subsequence which converges in the topology of uniform conver-gence on compact sets.

Proof. We remark first that there is a subsequence which converges point-wise (using Tychonoff's theorem to the effect that the product of a family ofcompact spaces is compact). If we show that this sequence is equicontinuouson compact subsets, it will follow that the convergence is actually uniformon compact sets. But equicontinuity follows from the assumption of uniformboundedness of first derivatives, thanks to the mean value theorem.

Rellich's theorem is similar in spirit.

7.12. Theorem (Rellich). If { fk} is a sequence of functions in a domainU with support contained in a compact set K C U such that they and theirfirst derivatives belong to L2 and are bounded in the L2-norm, then there isa subsequence which converges in the L2-norm.

Proof. We remark that we may extend the functions to the whole of Rn bysetting them equal to 0 in the complement of U. By regularising and replac-ing K by a bigger compact set if necessary, we may also assume without lossin generality that fk are all differentiable. By the Plancherel theorem, thesequences { fk} and {(1 + q(x))1/2 fk} are bounded in L2. From the equality

fK fk(x)e-i(x, >dx, we deduce that

Supfkl < IIfkllvol(K)


from Schwarz's inequality. Similarly, we get

SUP Iaj fkI <- IIfkIIIIxjilK

for any j. From this we conclude that fk and aj fk are uniformly bounded.This implies, in view of the Ascoli theorem, that after replacing { fk} by asubsequence, we may assume that the sequence { fk} is uniformly convergenton compact sets.

So far we used only that the sequence { fk} is bounded in L2. Using thefact that (1 + q )1/2fk is also bounded in L2 we wish to conclude that {fk}is a Cauchy sequence in the sup-topology and hence in L2. This will proveour assertion.

Thanks to the Fourier inversion formula, we have

(fk - A) (X) =

Given any e > 0, we need to show that for large enough k and 1, the absolutevalue of the above integral is less than e. Let M be a positive real numberand let Sl = q-1 [0, M] and S2 = q-1 [M, oc]. Then we write the expressionabove as

1(1k - fa)(-0 e2(x,C)d

< f I (fk - fa) (-e) I de

+J

(Ifkl + IfiD)(-S)(1 + q(S))1/2(1 +S2

The second sum is smaller than II(fk()I+Ift()I)(1+q()1/2II (1+M)-1/2.Therefore, we may choose M so large that the second term is less thane/2. Having chosen such an M, we may choose p large enough so thatIfk - ft 15 e/2 vol(Si) if k and 1 are greater than p. Then the first term isalso less than e/2. This completes the proof.

Chapter 9

Vanishing Theoremsand Applications

Many operators which occur naturally on differential manifolds (with addi-tional structures perhaps) are elliptic. The local theory that we have devel-oped in Chapter 8 lead to global results on solutions of elliptic equations oncompact manifolds. We will see here how the theory of lifting symbols leadsto a criterion for vanishing of solutions of elliptic equations. We will givespecific examples and some applications.

1. Elliptic Operators on Differential Manifolds

1.1. Definition. A differential operator D from a vector bundle E to F ona differential manifold M is said to be elliptic if its symbol, interpreted tobe a polynomial map of the vector bundle T* into Hom(E, F), assigns toany nonzero vector in T,*,,,, an isomorphism of the fibre of E with that of F,for any m E M.

Clearly this definition is compatible with the definition of an ellipticoperator on open domains in R'a given in [Ch. 8, 7.1].

1.2. Remark. From the definition, it is clear that if there exists an ellipticoperator from E to F, then the ranks of E and F are the same.

1.3. Examples.

1) Let M = R. Then the operator dx on real- or complex-valued functionsis elliptic. Its symbol associates to any element a of T,,;,, (which is

257

258 9. Vanishing Theorems and Applications

canonically isomorphic to R), multiplication by a. Hence it is anisomorphism for nonzero a.

2) Take a second order operator on 1182 of the form f a +gaeav + h ,

where f, g, h are real-valued functions. Its symbol at any point mis given by v = (a, b) f (m)a2 + g(m)ab + h(m)b2. It is thereforeelliptic if and only if the above quadratic form is positive definite ornegative definite at all points.

More generally any second order operator on R with real-valuedfunctions as coefficients, is elliptic if and only if the quadratic formwhich its symbol at any point represents, is positive or negative def-inite. Such polynomials are elliptic, i.e. have ellipsoids as level sur-faces, and this is the origin of the term `elliptic operators'.

An operator of order k with real coefficients on IR is elliptic if andonly if its symbol at any point of 1R' (which may be looked upon as ahomogeneous polynomial of degree k in n variables) has no nontrivialreal zeros. In particular, if n > 2 and the operator has real coefficients,it has to be of even order.

3) Let M be any Riemannian manifold. Then the Laplacian [Ch. 7, 7.16]A : Ai(T*) --+ Ai(T*) is elliptic. In fact, its symbol is a quadraticmap of T* into End(Ai(T*)) given by v H -g(v). Id, where g is thequadratic form on T* given rise to by the metric. Hence it is anisomorphism for nonzero v. Notice that in the case of a pseudo-Riemannian manifold, this is no longer so.

4) If M is in addition a Spines manifold [Ch. 7, 8.8], then we recall thatthere are three associated vector bundles in that case. These arethe bundles associated to the vector representation and the half-spin representations of Spire,. These are the tangent bundle andtwo other bundles Spin+(T) and Spin-(T). Moreover, the Clif-ford module structure on the Spin representation space gives a mapT ® Spin+(T) --> Spin-(T). Since there is a metric on T, we mapreplace T by T* and treat the above as a first order symbol. This canbe lifted into an operator from Spin+ to Spin-. This is the Diracoperator. Since any nonzero v is invertible in the Clifford algebra,this operator is elliptic.

From the definition and the formula in [Ch. 2, 7.23] for the symbolof a composite of two operators, it follows that the composite of an ellipticoperator from E into F and another from F into G is again elliptic. Moreoverthe adjoint of an elliptic operator is elliptic, in view of the symbol calculationof the adjoint in [Ch. 3, 3.10].

1. Elliptic Operators on Differential Manifolds 259

We will now give global analogues of the results of Chapter 8 and deriveconsequences for general elliptic operators on manifolds. In order to do this,we need to set up the machinery of Sobolev spaces in the context of sectionsof vector bundles.

We first define distribution sections of a vector bundle E. Consider thespace r° (E) of differentiable sections with compact support, of a vectorbundle E. If rK (E) denotes the subspace of sections with support in afixed compact set K, then we can define a topology on it by saying that asequence {sn} tends to zero if the sequence {Dsn} of functions tends to zerouniformly, for every differential operator D from £ to A. Now we considerthe adjoint bundle adj(E) = E* ® S and define a distribution section of Eto be a linear form on F°°(adj(E)) whose restriction to any rK(adj(E)) iscontinuous in the above topology. A differential operator from E to F hasan adjoint adj(F) --> adj(E). It induces a continuous map of FK(adj(F))into rK (adj (E)) and hence also a linear map of the space of distributionsections of E into that of F.

1.4. Remark. According to our definition in [Ch. 8, 5.2], a distributionon M is simply a distribution section of A. Similarly an n-current is adistribution section of S.

Next we would like to define the Sobolev space of sections. The firststep is to define square summable sections. We may fix a global, everywherepositive density p on a differential manifold M and a Hermitian metric alongthe fibres of E. Then for any measurable section s of E, the meaning ofII s I I2 as a function on M is clear, and we require that this be integrable withrespect to the fixed density. We can introduce the norm IIsilm by settingI I s I I nM = f I I s I 12dµ. We will denote this space by L2 (E) . If M is compact, thenorms corresponding to different choices of the metric and of the density areclearly equivalent. Square summable sections of E (after making the abovechoices) can be considered distribution sections, since they can be pairedwith sections with compact support of the adjoint bundle adj(E).

1.5. Definition. Let E be a vector bundle over a compact differential man-ifold M. Elements u of L2 (E) with the property that Du is a distribution inL2 for every differential operator £ - A of order < m, form a vector space7-lm,(E), called the Sobolev space of sections of E.

If M is compact, we can cover M by finitely many coordinate opensets and choose a partition of unity with respect to the covering. It isthen easy to check that u belongs to 1-lm, if and only if cpzu belongs to thecorresponding Sobolev space in the domain in Rn given by the coordinatechart, for all i.


Thus we can imbed the Sobolev space ?-1,,,,(E) in the product of theSobolev spaces ?-1,,,, in domains in the Euclidean space by the map s --* (cpti.s).We can therefore provide ?-1,,,, with the induced Hilbert space structure.It is again clear that this norm is, up to equivalence, independent of thecoordinate charts chosen.

Since the assumption and the conclusion of the regularity theorem areboth of a local nature, it implies the following.

1.6. Regularity Theorem. Let L : E -* F be an elliptic operator onvector bundles over a differential manifold. If s is a distribution section ofE such that Ls is a differentiable section (in particular, if Ls = 0), then sis actually a differentiable section.

We have the following global analogue of Rellich's theorem.

1.7. Theorem. Let E be a vector bundle on a compact differentiable man-ifold. The inclusion of 7-1m(E) in ?-lm,_1(E) takes bounded subsets into rel-atively compact subsets.

We also have the following globalisation of the inequality in [Ch. 8, 7.4].

1.8. Theorem. Let L : E -> F be an elliptic operator of order r on acompact differential manifold. Then for every m, there exists a constantC > 0 such that

IISIIm G C(IILsIIm-r + 11816-1)-

The map u H (Lu, u) of ?-lm(E) into ?-lm-r(F) x 11m_1(E) is a homeomor-phism onto a closed subset. In particular, if s satisfies Ls = 0, then we haveIISIIm < Cllsllm-1

Proof. Since the map u H (Lu, u) is injective, and is a continuous linear op-erator, the inequality implies that the map is actually a metric equivalence.In particular, the image is complete, and hence closed.

From these, we are in a position to deduce the following finiteness state-ment.

1.9. Finiteness Theorem. If M is a compact differential manifold,and L : E --+ F an elliptic operator, then the space S of solutions, i. e.

{s E £(M) : Ls = 0}, is finite-dimensional.

Proof. Consider the continuous map of ?-1,,,, (E) into ?-1,,,,_r (F) induced by L(denoting by r the order of the operator). By Theorem 1.6, its kernel is thesame as S. In view of Theorem 1.8, the kernel is mapped homeomorphicallyonto the subspace S of ?-lm,_1(E) by the restriction to S of the inclusionof ?-1,,,, in ?-l,r,,-1. By Rellich's theorem, the latter takes bounded sets to


compact sets. In particular, the closed unit ball in S is compact. Hence theHilbert space S cannot admit an infinite orthonormal set, proving that it isfinite-dimensional.

We also deduce the following important fact about elliptic operators.

1.10. Theorem. Let L : E --> F be an elliptic operator on a compactmanifold. Then the image of Nm(E) in 7-1m_r(F) is closed.

Proof. Let S-' be the orthogonal complement of S in lm,(E). Clearly, therestriction T of L to S-L is injective and the images of 7-1,m(E) and S1 inhm_r(F) are the same. In order to prove that the image is closed, it isenough to show that the inverse T-1 is a bounded operator from T(S--)to S1. In other words, we must show that there exists C > 0 such thatIlvllm < C for all v E S' with IITvII,,,,,_r = 1. If this were not true, therewould exist a sequence {vk} in S-L with IITVkIIm-r = 1 and llvkllm > k. LetXk = Vk/IIVkHIm. Then we have

I xkllm = 1 and IITxkIIm-r =1/IVklm < 1/k.Thus {xk} is a sequence in S1 such that IIxkIIm, = 1 and {Txk} tends to zeroin 7-lm-r. The inclusion of 7i,,, in 7-lm-1 takes {xk} to a relatively compactsubset. Therefore we can assume (on passing to a subsequence) that {xk}has a limit in 7-1,,,,_1. Since {Txk} tends to 0 in 7tm_r and {xk} has a limitin 7-Gm-1, it follows from Theorem 1.8 that {xk} has a limit in ? 1m as well.This limit clearly belongs to S1 and is in fact zero since T is injective. Butthis is in contradiction to the assumption that IIxkIIm = 1.

2. Elliptic Complexes

2.1. Definition. Suppose we have a sequence of vector bundles Ei anddifferential operators d' : Ei --+ Ei+1 Assume that di o di-1 = 0 for all i. Inother words, E° is a complex of vector bundles where the differentials aregiven by differential operators. Then we say that it is an elliptic complexif for every nonzero cotangent vector v at any point m E M, the symbol

sequence E" Ol2(v)E2. Pi(vi) Ei+l is exact.

We will generally assume that the Ei's are zero for all but finitely manyi's.

2.2. Remarks.1) From the definition, it follows that if a single operator is considered to

be a complex consisting of only two vector bundles, then the complexis elliptic if and only if the operator is elliptic.

2) If E° is a complex

-a E''-1 d-1>EiEi+1...


with differentials given by differential operators, then we have a sim-ilar complex, namely the adjoint complex adj(E°), given by

adj(Ei+1) adj(E') awl, adj(EZ-1) ...

where ai is the adjoint of di. For any nonzero element v E T, thesymbols of the adjoint complex give, up to sign of differentials, thesequence

... -, adj(Ez+1) W M) *) adj(Ez) (- l(v»*) adj(Ei-1) --+ ... .

Hence the adjoint complex adj(E°) is elliptic if and only if E° is.

2.3. Examples.

1) The de Rham complex of a differential manifold is elliptic. In fact, tosee this we need to check the exactness of the symbol sequence for allv E T,4,, \ {0}. This is a fibrewise check and amounts to the following.If V is a vector space, and v E V \ {0}, then the sequence

... -, A'-'(V) -> Ai(V) --> A'+1(V) --* ...

is exact, where the maps are given by wedging with v. In fact, we mayassume that v = el is the first member of a basis (ei). Any elementw of A' (V) can then be written as el A a +,3 where the expressions interms of the basis for a and 3, do not involve el. If el A w = 0, thenel A ,d = 0 which implies that 0 itself is zero. In other words, w is ofthe form el A a with a E A'-' (V). This proves our assertion.

2) The Dolbeault complex

... -, AP,q-1(T*) -3 AP>q(T*) , AP,q+l(T*) ...

of a complex manifold is elliptic. Recall that the Dolbeault differentialon AP,q(T*) is obtained by composing the exterior derivative with theprojection to the AP ,9+'(T*). Hence its symbol for any v E T, isgiven by wedging with its (0, 1) component. So the exactness of thesymbol sequence for any nonzero v follows from the above observationin which we take V = (T,;,°'1))* and tensor the wedging sequence by

Letdi-1 d°Ei-1 _3 Es - - Ez+1 _, .. .

be an elliptic complex. We choose Hermitian metrics along the fibres of thebundles and a positive density on the manifold. For convenience, we willdenote all the di's by d. Then the adjoint bundles adj(Ei) can be identified


with E' and we will denote all the adjoint operators 82 : Ei+1 -+ EE by 8. Itis obvious that this complex

... - Ei+1 Ei a Ei-1 -+ ... ,

which, by abuse of language, will also be called the adjoint complex, iselliptic.

We are interested in the cohomology spaces of an elliptic complex. Tostart with, we will assume that all the operators in the complex are of thesame order r.

2.4. Definition. The operator A: Ei --* E' defined by A = 8idi+d''-182-1is called the Laplacian of the elliptic complex. Sections s of Ei which satisfyAs = 0 are called harmonic sections.

2.5. Theorem. The Laplacians of an elliptic complex are elliptic.

Proof. We take a nonzero cotangent vector v at a point m E M and consider

the induced symbol maps (denoting ai(v) by si), (P-1)m ss-1 (EZ).,,, sue.

(Ei+l),,,,. Using the computation of the symbol of the adjoint in [Ch. 3,3.10] and that of a composite in [Ch. 2, 7.22], we compute the symbolof A to be the linear endomorphism (-1)r((si)* o si + si-1 0 (s2-1)*) of(E')m. Here s* denotes the usual linear adjoint defined by (x, s*y) = (sx, y)with ( , ) representing the Hermitian metric. Let T be the map EZ -a Ei

given by (-1)' times the symbol of 0 evaluated on v. Then for any x inthe fibre of Ei at m, we have (Tx, x) = (((si)* o si + si-1 o (si-1)*)x, x) =(six, six) +((sz-1)*x, (si-1)*x). Hence if Tx = 0, then in view of the positivedefiniteness of the scalar product, we conclude that six = 0 and (s2-1)*x = 0.From the exactness of the symbol sequence, there exists y E E,,,1 such thatx = sz-1y. But then (x, x) = (x, sx-1y) = ((si-l)*x, y) = 0. It followsthat x = 0. In other words, the linear map 7 is injective and hence anisomorphism. This shows that A is elliptic.

2.6. Theorem. Let M be a compact, connected manifold and E° an ellipticcomplex on it. A section s E £z(M) is harmonic if and only if it satisfiesds = 0 and 8s = 0. The natural map of the space V of harmonic sections tothe cohomology space H'(£°(M)) of the induced complex

£i-1(M) d-" £2(M) ', £i+1(M) ...

is an isomorphism.


Proof. We will use the assumption that M is compact in choosing a finitepositive density on M which determines the scalar product on sections. Lets be any section of £i. Then we have ((da + ad)s, s) = (as, as) + (ds, ds).Hence we conclude that s is harmonic if and only if ds = as = 0. Inparticular we have a natural linear map of the space of harmonic sectionsinto H i (£° (M) ). If a harmonic section s represents the trivial cohomologyclass, it is of the form dt where t E £i-1(M). But then since s is harmonic,it follows that adt = as = 0. Consequently, (adt, t) = (dt, dt) = 0, whichimplies that s = dt = 0. This shows that the map from V to Hi is injective.

We have finally to show that every cohomology class is represented bya harmonic section. We claim that any element x in V is orthogonal to Ayfor all y. For, (x, (da + ad)y) = (dx, dy) + (ax, ay). Since x is harmonic,both dx and ax are zero and our assertion is proved.

The image of A : 7-12r -> L2 (E) is closed by Theorem 1.10, and Vis finite-dimensional by 1.9. We easily conclude from this that the spacespanned by V and Im(A) is also closed. We claim that it is the whole ofL2(E). In fact, if v E L2(E) is orthogonal to Im(A), we have (v, Ow) = 0for all w E £(M). But this is the same as saying that (Av, w) = 0. HenceOv = 0, i.e. v belongs to V. If it is also orthogonal to V then it is zero.Thus V ®Im(A) = L2 (E).

Given any element x of £(M) with dx = 0, note that x belongs to L2(E),and take the harmonic projection h(x) of x in V. The difference x - h(x)belongs to Im(A), say x - h(x) = Ay. Here y belongs, a priori, to 'H2,. Butwe started out with a differentiable section x, and hx, being harmonic, isalso differentiable. Thus Ay is differentiable. From Theorem 1.6, it followsthat y is also differentiable. Moreover, since dx is zero by assumption anddh(x) is also zero, we have d/y = dady = 0. Taking scalar product withdy we get ady = 0. Again taking scalar product with y, we conclude thatdy = 0. Hence we have Ay = day. Thus x - h(x) = day with y E £(M),proving that x and hx represent the same cohomology class.

In the course of the proof, we also showed the following decomposition.

2.7. Harmonic decomposition. The L2-space of sections of E is theorthogonal direct sum of S and the image of A. Ifs E £(M), then it can bewritten as t + Au with t, u E £(M), t harmonic.

2.8. Corollary. The cohomology spaces of an elliptic complex on a compactmanifold are all finite-dimensional.

Proof. We assumed that all the operators of the complex are of the sameorder. In that case, the above assertion is a consequence of Theorem 2.6 andFiniteness Theorem 1.9. In the general case, we may modify the sequence


Ei-1 --p Ei -* Ei+1 by composing the differential Ei --* Ei+l (or E'-1 --f E)with an elliptic operator Ei+1 -, Ei+1 (or Ei--' -} Ei-1) whose order is thedifference between the orders of di and di-1 and which induces isomorphismon sections. In order to find one such operator, notice first that we cantake any elliptic symbol and lift it to an operator to get an elliptic operator.Adding to this a large positive multiple of the identity, we get the requiredoperator. This reduces the case to one in which all operators are elliptic ofthe same order.

In particular, applying the above to the standard complexes, we get thefollowing corollaries.

1) The cohomology spaces with values in R or C (or even in a local sys-tem of vector spaces) of a compact differentiable manifold are finite-dimensional.

Let E be a holomorphic bundle over a compact complex manifold X.Consider the Dolbeault resolution [Ch. 7, 3.7-3.10]

0--). £h-3£®A---). AO,1(T*)®£--> --+AO,n(T*)®£-->0.

This is a soft resolution [Ch. 7, Proposition 3.8] of the sheaf of holomor-phic sections of E. Hence Hi(£h) can be computed by taking global sectionsof the Dolbeault complex, which is elliptic. Thus we conclude

2) The cohomology spaces Hi(X, £h) of a holomorphic bundle E on a com-pact complex manifold are finite-dimensional.

3) The solution space of the Dirac operator of a compact Spin, manifoldis finite-dimensional.

We also have the following duality statement.

2.9. Theorem. Let E° be an elliptic complex on a compact connectedmanifold. Then the ith cohomology of E° and the ith cohomology of theadjoint complex adj(E)° are canonically dual to each other.

Proof. If s E r(E), and t E I'(adj(E)), then their pointwise pairing givesa density on M which, on integration, gives a complex number (s, t). Ifu E I'(Ei) and S E r(adj(E)i+i), this pairing satisfies (diu, s) = (u, 8is).Hence if du = 0, then u is orthogonal to the image of 8i. Restrict thispairing to cycles s, namely, s such that 8is = 0. If u = d'-'v, we have(u, s) = (di-1v, s) = (v, 9i s) = 0. Thus the pairing (u, s) (u, s) goes downto a canonical pairing between Hi(E°) and H'(adj(E°)). It is this pairingthat gives the duality asserted in the theorem. In order to prove that thisis a perfect pairing, we may choose Hermitian metrics on E' and a positivedensity on M and use the harmonic decomposition. We will now show that


if x is any cohomology class such that (x, y) = 0 for all cohomology classesy E Hi(adj(E°)), then x = 0. Let hx be the harmonic representative of x.Then we have (hx, s) = 0 for all sections s such that ds = 0. But hx is alsoorthogonal to the image of A, since (hx, At) = (A(hx), t) = 0. Hence hxis orthogonal to the whole of F(adj(E2)) and is consequently 0. This showsthat the induced map of Hi(E°) into the dual of Ht(adj(E°)) is injective.In particular, dimHi(E°) < dimHt(adj(E°)). By symmetry the oppositeinequality is also valid. Hence the two dimensions are the same and theabove injective map is also an isomorphism.

2.10. Examples.1) If L : E -> F is any elliptic operator, then the cokernel of the map

E(M) -* Y(M) (which is also finite-dimensional) is dual to the kernelof the adjoint operator.

2) We have seen in [Ch. 3, Example 3.8, 1)] that the adjoint of the deRham complex can be identified with the de Rham complex tensoredwith the orientation local system OR(M) with reindexing. Hence wehave

2.11. Poincare duality. The ith de Rham cohomology group Hi(M,R) ofa compact differential manifold M is canonically dual to H'-i(M, OR). Asa consequence, we see that if M is a compact connected differential man-ifold, then Hn(M, OR) = R. In particular, if M is also oriented, thenHn (M, R) = R.

Slightly more generally, we have a conjugate duality between Hi(M,L)and Hn-i (M, adj (L)) where L is a local system of complex vector spaces, andadj (L) is the tensor product of OR and the complex conjugate of the dual ofL. If L admits a (constant) unitary metric along the fibres, then it gives anisomorphism of (L)* and L and hence there is a Hermitian duality betweenHi (M, L) and Hn-i(M, L (9 OR) in the case of unitary local systems.

3) Similarly we have the following theorem.

2.12. Serre duality. The ith cohomology group Ht(X, Eh) of the sheafof holomorphic sections of a holomorphic vector bundle E over a compact,connected, complex manifold X of complex dimension n, is canonically dualto Hn-i(M, Kh (9 Eh), where K is the canonical line bundle, namely the nthexterior power of the holomorphic cotangent bundle.

Proof. Since the cohomology spaces of Eh are identified with the cohomol-ogy spaces of the Dolbeault complex of E, we only need to apply the dualitytheorem above to the Dolbeault complex. The constituents of the adjointcomplex are (A(O,i) (T*))* 0 S ®E*.


Since the manifold X is oriented, the sheaf S is simply A2n(T*)A(n,n) (T*).

On the other hand, the pairing AP(T*) ®A2n-P(T*) -> A2n(T*) restrictsto a perfect pairing A(°,') (T*)* ® A(n,n-i) (T*) -), An,n) (T*). Hence we havethe isomorphisms A(°,'> (T*)* ® S ® £* -- A(n,n-i) (T*) ® £* -- A(°,n-i)(T*) ®1C®£*. The differentials are easily identified with the Dolbeault differentials,again up to sign. This proves the duality claimed.

2.13. Index of an elliptic operator.Given an elliptic operator on a compact manifold, one would like to know

if it has any solution at all. In other words, one looks for criteria to concludethat the space V of harmonic sections is zero. More ambitiously, one mightwish to compute the dimension of V. By computing, we mean an integralformula, namely, that the integral of a suitable density on M (determinedby the operator) gives the dimension of V. Such a formula would imply thata (continuous) perturbation of the equation does not change the number. Inother words, if we have a continuous family of elliptic operators, then theassociated densities will also vary continuously and so the integral (beingan integer) will be independent of the operator. This is actually false. Wemay take the Laplacian on a compact Riemannian manifold for which thereare nonzero harmonic sections (for example, on functions, constants are har-monic), and perturb it by adding a scalar times the identity homomorphism.The latter does not have any solutions if the scalar is a large positive num-ber. We may summarise this by saying that the dimension of the space ofsolutions is not deformation invariant. However, it is easy enough to showthat the difference between the dimension of the solution space of an ellipticoperator and that of the adjoint, is deformation invariant.

2.14. Definition. If E° is a finite elliptic complex on a compact manifold,then the integer E(-1)i dimHi(E°) is defined to be its index. In particular,the index of an elliptic operator L : E -* F on a compact manifold is theinteger given by dimker(L : £(M) -+ .F(M)) - dimcoker(L : £(M) --F(M))-

One can indeed give an integral formula for computing this index asdescribed above. It is called the Atiyah-Singer index formula. This formulainvolves the Pontrjagin classes of the manifold and also classes associated tothe elliptic operator, or complex. Thus one defines a density on M which,when integrated, gives the index. In particular, this also gives a formula forthe index of the Dolbeault complex of a holomorphic bundle on a compactcomplex manifold. This then computes X(E) = E(-1)i dimH2(E) by anintegral formula which is known as the Hirzebruch-Riemann-Roch theorem.


We will not prove the index formula here, but content ourselves with statingthe Riemann-Roch theorem.

Firstly, we need a piece of notation. Suppose M is a compact orienteddifferential manifold and E a vector bundle on it of rank r. Let P be apower series in r variables Xi, which is symmetric in the variables. Then wecan express it as a power series in the elementary symmetric functions of theXi's. In this, substitute for the ith elementary symmetric function, the ithChern class of the bundle. Since the cohomologies of the manifold vanishbeyond its dimension, this makes perfect sense as an element of the totalcohomology of M. We will denote this by P. There are two ways in whichthese kinds of classes arise in the Riemann-Roch theorem. Firstly, startwith a power series Q in one variable. Then we can take for P the powerseries E Q (X,) and we call the resulting cohomology class, the characterclass ch(Q) (E).

When P = ex this is called the Chern character and denoted ch(E).On the other hand, we may take P to be the product IIQ(Xi) and define

the corresponding cohomology class to be the Hirzebruch class h(Q)(E).When we take P to be the power series 11-X we call the resulting Hirze-bruch class the Todd class and denote it td. Finally if x is an element of thetotal cohomology class we denote by f x the element of 118 obtained by takingthe top component and taking the corresponding element of R obtained byintegrating the top form. Then the Riemann-Roch formula is the following.

2.15. Theorem. For any holomorphic vector bundle E on a compact com-plex manifold, we have

x(E) _ 1: (-1)' dimH'(E) = fch(E).td(TM).

2.16. Remark. When E is a line bundle L we may write simply ec(L) forch(E), so that the formula becomes f E c(L)2 (td(TM))(n-i) In particular,if M is of dimension 1, the integrand is td(TM)1 + c(L).

The original version of the proof of the index formula may be found in[15]. A later proof, using the heat kernel, was found by Atiyah, Bott andPatodi [1]. The Hirzebruch Riemann-Roch theorem for projective manifoldsappeared in [8]. A relative version was given by Grothendieck in a far-reaching generalization. Other proofs are due to Fulton [4] and Nori [14].

2.17. Riemann-Roch theorem for a compact Riemann surface.We wish to see the Riemann-Roch formula for a compact Riemann sur-

face C in a user-friendly form. Looking at the formula, one realises thatthe first invariant to understand is the class td(TC). Fortunately, for thiswe need only the first coefficient of the power series (1_e_y) , which is x/2.


Hence the Todd class is in this case 1 + cl(Tc)/2. Now there are severalinstructive ways of computing cl for a Riemann surface. For example, onenotes that it is the negative of the Euler class of the tangent bundle, treatingit as a real vector bundle [Ch. 5, Remark 6.15, 2)]. But the Euler class is anelement of H2(C) and can be thought of as a number. We have remarked[Ch. 5, Remark 6.15, 2)] that it is the Euler-Poincare characteristic. Thisnumber is 2 - 2g where 2g is the first Betti number of C. On the other handif L is any line bundle, then ch(L) = 1 + ci(L). Here again we can thinkof cl (L) as a number identifying it with f cl (L). This number is called thedegree of the line bundle. The formula reduces then to

2.18. dimH°(L) - dimH1(L) = f (cl(L) +cl(Tc)/2) = deg(L) + 1 - g.

Take for L the trivial bundle. Then H° (L) consists of global functions,but the only global holomorphic functions on a compact connected manifoldare constants, so that its dimension is 1. Clearly deg(L) = 0 and so theRiemann-Roch theorem implies that dim H1 (C, 0) = g.

The Riemann-Roch theorem sets out to compute X(L) in terms of thetopological invariants. If we can compute X(L) by other means, it can alsoserve to compute the topological invariants.

We will denote by w the holomorphic line bundle Th and take now L = w.Then we get dim H° (C, w) - dim H'(C, w) = deg(w) + 1 - g. Noting thatby Serre's duality theorem we have dim H' (C, w) = dim H° (C, O) = 1 anddim H°(C, w) = dim H1 (C, 0) = g, this computes the degree of w:

2.19. deg(w) = 2g - 2.

We will see another example of a similar sort. Let X E C. We have seenthat the ideal sheaf is locally free of rank 1. It is in fact defined by thefollowing exact sequence:

0-*Tx->0 --*0,-*0.Here Ox is the `skyscraper sheaf' which is 0 outside x and has C for itsstalk at x. It can be defined as the direct image of the sheaf 0 for theinclusion of the single point manifold {x} in C. Now Hi(C, Ox) is easilyseen to be Hi({x}, 0) = 0 if i 0 0 and C when i = 0. Hence we haveX(Tx) = X(O) - 1 = -g. Appealing to the Riemann-Roch theorem, weconclude that

2.20. deg(Zx) = -1.

2.21. Remark. Equation 2.20 can of course be directly checked as follows.Note that Zx is trivial outside x. On the other hand it is also trivial inan open set containing x. We choose a coordinate system (U, z) at x andidentify this with a disc with 0 as centre. Note that the ideal sheaf To on the


disc is trivialised using the multiplication 0 -* 1° by the function z. Henceif we take the trivial connection on the trivialised line bundle on M \ {x},it also gives the trivial connection in U \ {0}. Under the transformation ofmultiplication by 11z the connection becomes f -+ df + f dz/z. Choose asmaller disc V around 0 with V C U. We now have a trivial bundle on U\Vwith the above connection on it. This can be extended to the whole of U byextending the differential form dzl z from V C U to a (differentiable) forma on U. Now clearly the trivial connection on M \ {V} and the connectiond + a on U coincide on the intersection, and so we get a connection on1 . We now compute its curvature form R. Since the connection is flat onM \ V, the curvature form is zero there. On the other hand it is da insideU. We need to integrate this on M to get the Chern class. Now fm R isfV da = fs a where S is any circle between the two discs. But a = dz/z inthis annulus and so we need to compute fs dz/z. This is clearly 27ri. By ourdefinition of the topological Chern class, namely (-1/2iri) times the above,we get the degree as claimed.

The dual of the line bundle Tx is usually denoted O(x). So the aboveequality is equivalent to saying that the degree of O(x) is 1.

2.22. Remark. If L is any line bundle on a compact Riemann surface,tensoring L with the exact sequence

0-*I -30--SOX --*0

one deduces that x(L) - x(L (9 0(-x)) = 1. On the other hand, we justsaw that deg(L) - deg(L (9 O(-x)) = 1. From this, one can easily prove theRiemann-Roch theorem for line bundles on a compact Riemann surface.

3. Composition Formula

3.1. Lifting of higher order symbols.We have seen that if M is a differential manifold and N a closed sub-

manifold of codimension 1, then its ideal sheaf is locally free of rank 1. Muchthe same is true for complex manifolds and for the same reason, namely thesubmanifold is locally defined by a single equation. If M is a complex man-ifold and D a closed submanifold of complex codimension 1, the ideal sheafTD of D is a holomorphic line bundle. Its dual is denoted O(D). Sectionsof TD are holomorphic functions on M which vanish on D, while those ofO(D) consist of meromorphic functions which have at most a pole on D.It is therefore clear that sections of holomorphic line bundles or of vectorbundles are of great importance in the study of complex geometry. Thefirst question that arises in this connection is how to compute dimH°(E).If we know for example that Hi (E) are zero for all positive i (where E is


a holomorphic vector bundle over a compact complex manifold), then theHirzebruch-Riemann-Roch formula mentioned above computes the dimen-sion of H°(E).

This question can be asked in the wider context of elliptic complexes,the case of the complex manifold being covered by the Dolbeault complex.We will be interested in criteria for the vanishing of the cohomology spacesHi (E°) of an elliptic complex E°. We will give a general sufficient conditionfor the vanishing of H(E°) when the differentials are given by lifting firstorder symbols using connections, according to the theory in Chapter 5.

We begin by discussing the question of lifting a higher order symbolE -p Sk (T) ® F into a differential operator E --* F of order < k. Forfirst order symbols we have seen in Chapter 5 that this is accomplished bychoosing a connection on E. We will therefore assume given a connectionon E. One would hope to get on iteration, first, an operator from E to®TT* ® E and therefore, an operator E - Sk(T*) 0 E of order k whosekth order symbol is essentially the identity map. But, in order to iteratedo, we need a connection on T* 0 E, T* 0 T* 0 E and so forth. So let usassume that a linear connection is also given. Actually it is practical to takea torsion free linear connection, as we will soon see.

We get on such iteration, an operator (&k (V) : E , (2)k T* 0 E oforder k, and hence on composing with the natural map (&k T* , Sk(T*),an operator Sk(V) : E -+ Sk(T*) ® E as well. Its symbol can be com-puted according to the recipe given in [Ch. 2, 7.23]. In fact, the symbolof ®k(V) is the image in Sk(T) ® ((gk T*) ® End(E) of the identity el-ement in (&k (T) ®k(T*) 0 End(E). Hence the symbol of Sk(V) is themap E -+ Sk(T) 0 Sk(T*) 0 E given by rl ® IdE where rl is the image inSk(T) ® Sk(T*) of the identity section of (&k (T) (&k (T*). Given a symbols : E -* Sk (T) OF, it is clear now how to lift it to an operator. We composeSk(V) : E , Sk(T*)®E and ISk(T.) ®s : Sk(T*)®E , Sk(T*)®Sk(T)®Fand contract Sk(T*) ® Sk(T) to A and get an operator E --> F.

3.2. Definition. Let s : E --* Sk(T) ® F be a given kth order symbol.Assume given a linear connection on M and a connection on E. Then thecomposite of the differential operator Sk (V) : E , Sk (T*) 0 E, the A-linearmap ISk (T.) ® s : Sk (T *) ® E --> Sk (T *) 0 Sk (T) ®F and the A-linear mapSk(T*) ® Sk(T) ® F -+ F, which is the tensor product of the contractionmap and IdF, is called the lift of the symbol s.

Starting with the above data, namely a connection on E and a linearconnection, we have given a procedure to lift all symbols E -+ Sk(T) 0 Fto differential operators. In other words, this yields splittings of the exact


sequences (for all k)

0 -+ Dk-1(E, F) --; Dk(E, F) --> Hom(E, Sk(T) (9 F) -> 0.

Operators obtained by lifting symbols are analogues of homogeneous dif-ferential operators, in the sense that any differential operator can be writtenuniquely as a linear combination of lifted differential operators. For, givenany differential operator D of order k, we may lift its kth order symbol toan operator Dk using the above splitting. Then since D and Dk have thesame kth order symbol, D - Dk is an operator of order < k - 1 and so thisproves our assertion inductively. Thus we have shown the following.

3.3. Proposition. Given a connection on a vector bundle E and a linearconnection on the manifold M, every differential operator E -- F can bewritten uniquely as a sum of differential operators E --> F obtained by liftingsymbols.

3.4. Composition of lifted operators.Lets : E -> Sk(T) ®F and t : F --> S' (T) ®G be two symbols. By

composition and multiplication in the symmetric algebra of T, we get acomposite symbol t o s : E -4 Sk+l(T) ® G of order k + 1. Assume given aconnection each on E, F and T. Using the connection on E and T we maylift s to a differential operator D, : E --+ F. Similarly, using the connectionson F and T, we get an operator Dt : F -> G with symbol t. On theother hand, using those on E and T we may also lift the symbol t o s to adifferential operator Dt05 : E --> G. Since Dt o Ds and Dtos have the samesymbol, namely t o s, their difference is an operator of order < k + l - 1.

If l = 0, then Dt = t is an A-homomorphism F -* G, and the connectionon F plays no role in the definition of Dt nor of course in Ds and Dtos. Bydefinition of the lifts, we have obviously the following result.

3.5. Proposition. If the order of the symbol t is zero, then we have Dt03 =toD,.

On the other hand, if k = 0, again we have that D3 = s is an A-linearhomomorphism E --> F, but the lift of t involves the connection on F whilethat of t o s involves the connection in E. Therefore one cannot expect sucha simple formula as above. Notice that we have (1T* (9 s o VE) and VF o s areboth first order operators from E to T* ® F. Their difference is by definitionds where d is the absolute derivative of s with respect to the connection onHom(E, F) given rise to by VE and VF. As such, ds can be regarded as amap E->T*®F. Then wehave (1T*®s)0VE-VFos=ds.Ifl=1,wedenote as usual by t the map T* ® F -+ G determined by t and conclude


that

3.6. Dtos = Dt o s + (t) o ds.

The next interesting case is when k = 1 = 1. We will obtain a formulafor the expansion in terms of what we may call V-homogeneous operators,of the composite of two V-homogeneous operators. We have seen that thetop term in the expansion is the lift of the composite symbol. Thus we needto identify the first order V-homogeneous operator and the constant termin the expansion.

In order to make the computations easy to understand, we will use anotational artifice. Whenever we have a diagram D of the following type:

P S

YR

we denote the map P -+ S obtained as f o - g o cp by OC(D) (obstructionto commutativity of Diagram D). Consider now the following diagram:

T*®T*®E,S2(T*)®E

VEE T* ® E

FThe two diamonds in the above diagram, denoted D1 and D2, are notclaimed to be commutative. We will in fact compute the obstruction tothe commutativity of the two diagrams.

3.7. Lemma Let s" : T* ® E --3 F be the map given rise to by the symbols : E -* T ® F. Then the absolute derivative dps, interpreted as a homo-morphism T* ® E -+ T* ® F, is the obstruction to the commutativity ofDiagram D1 above.


Proof. In fact, the evaluation of doss on any vector field X is by definitiongiven by (VF)X o s - (10 §) o (VT*®E)X

3.8. Lemma. The obstruction to the commutativity of D2 is the bilinearmap on T* with values in Hom(E, G) obtained by alternatising t(1T* (9 s").

Proof. Note that the composite of the natural map T* ®T* ®E -* S2 (T*) ®E and t o s : S2(T*) ® E --- G is by definition t o (10 §) o sym, wheresym : T* ® T® ® E -> T* 0 T* ® E is induced by the bilinear map (X, Y)(1/2) (X ®Y+Y®X). Hence the term OC(D2) is t(1®s) -t(109) osym =t(1®s)oalt.

Now in the above diagram, the map E -f G obtained by skirting alongthe counterclockwise (resp. clockwise) path is by definition Dt o DS (resp.Dtos). We introduce the via media V : E -i G by setting V = t o (1 0 s) oVT*®E o VE. Then write Dt o DS - Dt03 as (Dt o DS - V) + (V - Dtos).The expression in the first parenthesis is t o OC(D1) o VE, while that in thesecond parenthesis is OC(D2) o VT*®E o VE

If we plug in the values of OC from the above two lemmas, we get theformula for Dt o DS as follows:

3.9. Dt o Ds = Dtos + t(dpg) o VE +2t(1

®s) o alt OVT*®E o DE.

3.10. Composition formula. Let s : E -j T 0 F and t: F -- T 0 G betwo first order symbols. Given connections on E and F we may lift themto differential operators Ds : E -+ F and Dt : F - G respectively. Letus also fix a linear connection without torsion and lift the composite secondorder symbol t o s : E -* S2 (T) ® G (obtained by symmetrising the composite(10 t) o s : E --* T ® T 0 G) into a differential operator Dtos from E to G.Then we have

Dt o D. = Dtos + Diiodps + 2(t(1T* ®s)) o RE.

Proof. In this formula, doss is the absolute derivative of s" with respect tothe connection on Hom(T* 0 E, F) given rise to by the three connections,and is to be interpreted as a homomorphism T*®E -> T* OF so that todv.is a homomorphism T* ® E -> G. This, being a first order symbol, can belifted to an operator using the connection on E. Also, RE is the curvatureform and is a homomorphism E --> T* OT* ®E. The homomorphism (10. )tis a homomorphism T* 0 T* 0 E --} G, and the composite with RE gives an0-linear map E -* G. That is the explanation of the formula.


The composition formula is essentially a restatement of formula 3.9. Wefirst note that the term todv soVE is by definition the lift of the symbol whichwe have explained above. Now let us evaluate OT*®E o VE on vector fieldsX, Y. We obtain, for any section u of E, the expression VX(Y H Vyu) =VxVyu - VvXyu. Its composite with alt is the same as the alternatebilinear form (X, Y) -* VXVyu - VyVxu - Vv yu + Vv,xu. The lasttwo terms add up to -V[x,y]u since the torsion of the linear connectionis zero. Now the sum simplifies to RE(X,Y)u giving the constant term asclaimed.

3.11. Remark. The constant term in the above formula gives an A-homo-morphism of E into G. It can therefore be computed fibrewise. Let m E M.Choose a basis {ei} of T. We have homomorphisms (s)i : E -> F and(t)i : F -3 G defined as evaluation of the respective symbols at ei. Thenconsider the homomorphism ci7 : ti o sj - tj o si of E into G. Also write thecurvature form of E as E Rid ei A ej where Ri.7 are endomorphisms of E.Then the constant term is 2 Ei<j cii o Rii .

Let E° be an elliptic complex. Assume given Hermitian structures onthe bundles Ei and a positive density on M. Suppose also that all theEi's are provided with Hermitian connections and that the differentials ofthe complex are given by lifting first order symbols. Assume given a linearconnection on M which is torsion free and preserves the density. Finallyalso assume that the symbols of the operators di are invariant with respectto the connections on Ei, Ei+1 and T. In the following we will also assumethat the differentials are of order 1. Under these conditions, we wish toderive a formula for the Laplacian of the elliptic complex.

3.12. Definition. An elliptic complex satisfying all the above conditions iscalled a geometric complex.

Then we can trivialise S and identify Ei with adj(Ei) so that the adjointai of di may be identified with a differential operator from Ei+1 to Ei. Usingthe connection on Ei+l, we may lift the adjoint symbol Ei+1 T ® Ei toan operator Ei+1 -> Ei. From [Ch. 6, Theorem 2.12] it follows that thislifted operator coincides with ai, if the symbol of di is invariant with respectto the connections on E, F and T.

In the computation below, we will denote by adj(s) the adjoint symbol ofs, which, we recall, associates to any covector v the negative of the adjoint(i.e. conjugate transpose) of s(v). Let si be the symbol of di evaluatedat a cotangent vector v E T,;, at a point m E M. The top term of theLaplacian is of course the lift of the composite symbol, namely the negativeof the quadratic map s2 si + si_lsz_1 which is positive definite for each v E


T,n \ {0}. Since the symbols are invariant under the connections, there is nofirst order term. The constant term is the composite A o REi where RE :E --- T* ® T* ® E, or equivalently a homomorphism T ® T ® E - E, is thecurvature form of the connection on E, and A is the homomorphism Ei -->T ®T 0P associated to the alternate bilinear form (v, w) i (si(v))*si(w) -

(si(w))*si(v) + si-1(v)(si-1)*(w) - si-1(w)(si-1)*(v)

Thus we derive the following formula for the Laplacian of a geometriccomplex.

3.13. Formula for the Laplacian. The Laplacian of a geometric complexis given by the expression A = DS*3+ARE, where A is the alternating 2-formon T* taking (v, w) to 2 (sz (v) o si(w) - sz(w) o si(v) - si_1(w) o sZ-1(v) +si_1(v) o s7_1(w)) with values in End(E), RE is the curvature form as asection of A2(T*) ® End(E) and ARE is the element of End(Ei) obtainedby contraction on A2(T*) and the composition in End(E).

3.14. Remark. Choose, as in Remark 3.11, a basis ei of T , for any m E M.Then the constant term is

2

Ek<l((Sk)* o si - 8i-1 o (S'-')*) o Rkl, where s*is the conjugate transpose of s.

4. A Vanishing Theorem

Let W (resp. V) be a real (resp. complex) vector space. Assume given aHermitian positive definite metric on V. We will denote the correspondinginner product by ( , ). If q is a quadratic map from W into End(V) such thatq(w) is a Hermitian positive definite endomorphism of V for all w E W \ {0},let b be the corresponding (polarised) bilinear map of W into the space ofHermitian endomorphisms of V. Consider the map (W x V) x (W x V) --r Cgiven by (W1, v1, W2, v2) - (b(wl, w2)vl, v2). It is clearly ll8-multilinear, C-linear in vl and C-antilinear in V2. Thus we get a Hermitian form h onW ® V. Assume that this is positive definite as well.

We are interested in applying the above in the following situation. LetE be a Hermitian vector bundle and V a Hermitian connection on it. Wewish to compute the composite adj(V) o V. For this to make sense we needa density and a torsion free linear connection leaving it invariant. Besideswe also need a Hermitian metric on T* ® E. It may be thought that onemay take a Hermitian metric on T* and use it. However it is more fruitfulto use the above setup for introducing a Hermitian metric on T* ® E. Inother words we assume given a bilinear map b of T* with values in End(E)such that the above condition is satisfied on each fibre with W = T'*' andV = Em. We need also to assume that this bilinear map is invariant underthe connections in question.

4. A Vanishing Theorem 277

We will now compute the composite t o (1 (9 s), where s and t are thesymbols of the connection V and its adjoint respectively. For any v Ethe symbol of the connection V takes any x E E. to v ® x E T;; ® Em.The symbol t of its adjoint at v is determined, up to sign, by (x, t(w ® y)) =(v ® x, w (9 y) = (b(v, w)x, y). The composite can be thought of as the bilin-ear form associating to (v, w) the endomorphism tv o sw determined by theequation (x, (tv o sw) (y)) = (svx, swy) _ (v ® x, w ® y) = (b(v, w)x, y) =(x, b(v, w) y). Hence (t o (1 (9 s)) (y) = b(v, w)y. Finally this computes thecomposite as (v, w) b(v, w). Id. This being symmetric, its composite withRE is zero. Thus we have

4.1. Proposition. Let E be a Hermitian bundle. Assume given a sym-metric bilinear map b of T* into the bundle of Hermitian forms on E, suchthat the associated Hermitian form on T* ® E is positive definite. We alsoassume given a torsion free linear connection which leaves a density invari-ant and also the above bilinear map. Then the composite of a Hermitianconnection E -> T* ® E and its adjoint is a lifted operator, with symbol -b-

Proof. We are of course going to use the composition formula 3.10. Theabove computation gives the composite symbol to be -b. So the top term isas claimed. We have also assumed that b is invariant under the connections,so that the first order term in the expansion vanishes. We have alreadyremarked that the constant term is zero as well. This completes the proofof the assertion.

4.2. Corollary. Let E° be a geometric elliptic complex. If the negativesymbol b of the Laplacian at Ez satisfies the condition above, namely, theHermitian form on T* ® Ez given by (v, x, w, y) - (b(v, w)x, y) is positivedefinite, then the top homogeneous term in the decomposition of the Lapla-cian gives a nonnegative Hermitian operator on L2(Ez).

Proof. The bilinear map associated to the negative symbol of the Laplacianis obtained by symmetrising (v, w) .' (sz)v 0 (sz)w+sV i-1 0 (si-1)u,. It is clearthat for every v, w, the image is a Hermitian endomorphism of Ez (withrespect to the chosen Hermitian structure on it). Moreover, if v = w, it ispositive semidefinite, and by the ellipticity assumption, it is even positivedefinite for v 0 0. From the definition of a geometric complex, it follows thatthe connections preserve the symbols of the operators involved, which impliesthat the above bilinear form on T* with values in Hermitian endomorphismsof Ez is also V-invariant. From the above proposition, we see therefore thatthe top term in the expansion for the Laplacian is adj (V) o V, where weuse the above Hermitian form on T* ® Ez. Hence it induces a nonnegativeHermitian operator on L2(Ez).


From this we get the following sufficient condition for the vanishing ofthe ith cohomology of a geometric elliptic complex.

4.3. Theorem. Let E° be a geometric elliptic complex on a compact com-plex manifold as in .44.2. If the A-linear homomorphism Ei -> Ei givenby

o (1T. ®s) + k-' 0 (1T. ® (si-r)*)RE

is positive definite, then Hi(E°) = 0.

Proof. The cohomology space is isomorphic to the space of harmonic sec-tions by Theorem 2.6. Now use the formula for the Laplacian of a geometriccomplex given in 3.13. Finally use the fact, proved in Corollary 4.2, thatthe top term in this expansion is a nonnegative operator, to conclude thatall harmonic sections are zero under the hypothesis of the theorem.

Notice that the de Rham complex of a Riemannian manifold is a geo-metric complex. The Riemannian density is invariant under the Levi-Civitaconnection, and since there is a natural induced metric on all the bundlesAi (T*), the Laplacian of the de Rham complex is defined. We will now applythe above principle in this case.

The symbol of the exterior derivative d : Ai(T*) , A'+1 (T*) associatesto v E T* the exterior multiplication A(v) by v. The symbol of its adjoint is-iv [Ch. 7, 7.15]. Hence the symmetrised composite may be computed bytaking v = w. To this it associates -(iz,A, + Aiv) = -g(v).Id. Hence theHermitian form on T* ® Ei is (v, x, w, y) -* (g(v, w)x, y) = g(v, w) (x, y). Inother words, it is the tensor product of the metric on T* and that on Ei.In particular it is positive definite, as we have checked even generally, andis also V-invariant. We will now apply the above vanishing theorem to thiscomplex.

The alternatised composite of the symbols, associates to (v, w) the en-domorphism (1/2)(iw)tv - i,Aw + Awiv - Avav,). This is easily seen to bethe same as iwA, - ivAw, and it is a derivation (of even type) of the exte-rior algebra of T*. On T*, this action is simply the natural isomorphism ofA2 with the subspace of skew-symmetric endomorphisms. Hence for any p,the above map A2(T*) --> End(AP(T*)) is A D(A), where D(A) is thederivational extension of the corresponding skew-symmetric endomorphismof T* to the exterior algebra. The curvature form on AP(T*) is given by(X, Y) -* DP(R(X, Y)). Hence the substitution of R in the above mapyields the endomorphism of AP(T*) obtained by identifying R with an ele-ment of End (T *) 0 End (T *) and applying DO D to this to get an element ofEnd(A(T*)) ® End(A(T*)) and then composing. We denote this element by

5. Hodge Decomposition 279

D (R) . We thus have the following particular case of the vanishing theorem.

4.4. Theorem (Bochner). The ith Betti number of a compact Riemannianmanifold is zero if D'(R) is positive definite. In particular, if the Riccicurvature is positive definite, the first Betti number is zero.

4.5. Remark. The formula A = VV + Di (R) thus obtained is called theWeitzenbock formula.

5. Hodge Decomposition

We will soon look at the case of a compact Kahler manifold and the Dol-beault complex of a holomorphic bundle on it and derive a nice corollary ofour vanishing theorem. For now, we will take up a quite different applicationof the formula for the Laplacian.

5.1. Proposition. Let M be a Riemannian manifold and A : Ai(T*)Aj(T*) any homomorphism of bundles which commutes with the connectionson the two bundles induced by the Riemannian connection. Then it takesharmonic forms to harmonic forms.

Proof. It is of course sufficient to show that the map A commutes with theLaplacian. To show this we may use the above formula. Clearly A commuteswith the top term since it is the lift of the symbol by a connection whichleaves A invariant. If we show that the constant term also commutes with A,we will be through. If we consider R to be an element of End(,,,) ®End(T, *)for each m E M, then we claim that it actually belongs to W ® W, whereW is the subspace of End(TT) generated by R(vl, V2), vi E T,,,,. Since R issymmetric, this follows from the obvious fact that R belongs to End(T,,) ® Win any case.

5.2. Corollary. If M is a Riemannian manifold, then the star operatormaps harmonic forms to harmonic forms.

5.3. Remark. In particular, there is an isomorphism between H'(M, IR

and Hn-i (M, OR). We already know by the duality theorem that for acompact, connected Riemannian manifold M, the spaces Hi(M, IR) andHn-i(M, OR) are dual to each other. Using both we get a nondegener-ate bilinear form on all the cohomology spaces. To two forms ce,,3 this formassigns f a n *Q.

Suppose M is a compact Kahler manifold. Then we get the followingconsequence of Proposition 5.1.


5.4. Corollary (Hodge decomposition). Let M be a compact Kahler man-ifold. There is a canonical decomposition of H'(M, C) into a direct sum ofHA'-P(M) where HA'-P is the Dolbeault cohomology H'-P (M, AP (T*)).

Proof. This is a direct application of Proposition 5.1 after we note that Jcommutes with the Riemannian connection in a Kahler manifold. Hence theprojection to the (p, i - p) component is also an endomorphism of Ai(T*)which commutes with the Riemannian connection. According to Proposition5.1, this implies that it takes harmonic forms into harmonic forms. Thisgives the decomposition claimed. Now the cohomology classes in HA'-P arerepresented by harmonic forms of type (p, i - p). This means that bothd and 8 vanish on them. But obviously this implies that d" and 8" alsovanish on them. It follows that these are also Dolbeault harmonic forms.Since complex conjugation also takes harmonic forms to harmonic forms,we conclude that the Dolbeault harmonic forms which are d"-closed and8"-closed are also d'-closed and 8'-closed. Hence these are A-harmonic aswell. This identifies the space of harmonic forms of type (p, i - p) with thecorresponding Dolbeault cohomology spaces.

5.5. Definition. The above decomposition is called the Hodge decomposi-tion.

One can also compute the Laplacian corresponding to the Dolbeaultcomplex, using the composition formula. Again, at a (real) cotangent vectorv at a point m, the sum of the composite of the symbols of d" and 8" and thatof the symbols of 8" and d" can be computed as for the de Rham complex. Itturns out that it takes the value (1/4).g(v+iJv, v-iJv). Id = (1/2)g(v). Id,where g is the complex bilinear extension of the Riemannian metric. Thisactually proves, in view of the formula for the Laplacian, the following fact.

5.6. Theorem. Let M be a Kahler manifold. The Laplacian of the un-derlying Riemannian metric leaves all the sheaves AP,q(T*) invariant and istwice the Laplacian of the Dolbeault complex.

This proves again the Hodge decomposition of the de Rham cohomologyspaces of a compact Kahler manifold.

5.7. Remark. While the Hodge decomposition was arrived at using theKahler metric, the decomposition itself is independent of the metric. Con-sider the sheaf homomorphism given by the exterior derivative d of the sheafSZ4-1 of holomorphic (i - 1)-forms into fl'. We claim that the image is thesheaf of closed i-forms. In other words, any closed holomorphic form canbe written locally as the image by d of a holomorphic form. This is proved


exactly as in [Ch. 2, Proposition 6.14]. The kernel of this sheaf homomor-phism is the sheaf of closed (i -1)-forms. Hence we have the exact sequenceof sheaves

0 -- SZCI Qi-1 , Q > 0.The cohomology exact sequence gives

- Hi(SZ2-1) -; H3(1cl) --+ Hj+'(SZ-')

The induced map Hi (S2z-1) -+ Hi (SZ'I) is zeros, since any cohomology classcan be represented by a harmonic form for d" but then it is also harmonicfor d and consequently its image by d is zero. So the above cohomologysequence can be written as a short exact sequence

0 -* Hi(1ca) - HI (Q'-') --+ Hj+'(Q'-') --- 0.

Notice that this gives an increasing sequence of vector spaces finallyending up with Hi+j(SZ°l). But a closed holomorphic function is locallyconstant and so we have the isomorphism SZ°1 C of sheaves. Thus the spaceH'(X, C) is filtered by vector subspaces and the associatedquotients are isomorphic canonically to H'(SZT-'). This filtration does notdepend on the Kahler metric and is called the Hodge filtration. Indeed, bytaking the complex conjugation and intersecting, we can actually get backthe decomposition from the filtration.

Note that at the 0th stage, H° (M, C) as well as H° (M, 0) are constants,since holomorphic functions on a compact connected manifold are constants.Next, we will illustrate the decomposition for the first cohomology. Theabove sequence gives (on taking i = 1 and j = 0)

0 --+ H° (QC11) --> H1(O) --j 0.

Since all holomorphic forms are d"-closed and are also obviously 8"-closed,they are d"-harmonic. Hence they are also d-harmonic. This shows thatH°(1l 1) = H°(fl'). Thus the above sequence becomes

0-*H°(S21),H1(C),H1(O),0.This is the Hodge filtration in this case. Since there is a complex conjugationautomorphism in H1 (C), we may apply it to H° (SZ1) and check that it mapsisomorphically to H1(0) giving the decomposition.

6. Lefschetz Decomposition

We will give another application of a similar nature to the study of the co-homology of a compact Kahler manifold, this time depending on the metric.A map A satisfying the hypotheses of Proposition 5.1 takes harmonic formsto harmonic forms. This means that although cohomology spaces are global,they are represented by harmonic forms and as such the induced maps on


them can be understood from a fibrewise computation since A is by assump-tion 0-linear. We will now give an example of the power of this method.What follows is what happens at the fibre level, and after explaining it wewill return to the application to Kahler manifolds.

6.1. Digression on s((2)-modules.Let V be a finite-dimensional vector space over C. Assume that H, E

and F are linear transformations of V which satisfy the relations [H, E] =2E, [H, F] = -2F, and [E, F] = H. We will refer to such a triple oflinear transformations as an si(2)-module. Two such modules V and W areconsidered isomorphic if there is a linear isomorphism of V with W whichtransforms the two triples, one to the other.

6.2. Remark. The Lie algebra of SL(2, C) consists of (2, 2) matrices oftrace zero. A basis for the underlying vector space is given by the matricesh = (o °1), e = (o o ), and f = (°

o

). These have the above commutationrelations, and so V as above is simply a representation of the Lie algebra ofSL(2).

6.3. Example. The above realisation of the three elements with the givencommutation relations makes V = C2 an st(2)-module. From this we alsoget the symmetric powers of V as .51(2)-modules. These have the followingdescription too. Consider the space of homogeneous polynomials P of degreem, in two variables x and y. Define H(P) = A.P,-, - yam, E(P) = xPand F(P) = yam. Denote this module by V,,,,. Note that the binomialsvp = xpyq, p + q = m are all eigenvectors for H with eigenvalues p - q.Also E takes v,,,, to 0 and for p < m, vp to qvp+1, and F takes vo to 0 andfor p > 0, vp to pvp_1. Any nonzero sl(2)-invariant subspace contains aneigenvector for H, namely one of the vp's and applying E and F successivelywe get all the basis vectors. This shows that this subspace is the whole ofVm. In other words, V,,,, is an irreducible sl(2)-module. Finally the fact thatL = x ax + y acts as the scalar m. Id (Euler's formula) can be expressed

in terms of H, E and F. We have L2 - H2 = 4xy 92= 2(EF + FE) - 2L.

Hence the transformation H2 + 2EF + 2FE acts as the scalar m2 + 2m.

6.4. Classification of s((2)-modules.We wish to show that all s((2)-modules are essentially made up of the

examples V, above. In the analysis that follows we take the cue from theabove example.

Let V be an arbitrary sl(2)-module. We will first show that E is singular.Let v be an eigenvector for H with eigenvalue A. Then Ev, if nonzero, is alsoan eigenvector of H, with eigenvalue A + 2. For, HEv = EHv + [H, E]v =


(A + 2)Ev. (In a similar manner, we also see that Fv, if nonzero, is aneigenvector for H with eigenvalue A - 2.) Consider the sequence of vectorsv, Ev, E2v,.... All these cannot be nonzero, since they would then beeigenvectors of H with distinct eigenvalues, but H has only finitely manyeigenvalues. This proves our assertion that E has nonzero kernel.

Next note that the kernel of E is invariant under H. For if Ew = 0, thenEHw = HEw - [H, E]w = -2Ew = 0. Thus there is an eigenvector w for H(say, with eigenvalue A) such that Ew = 0. Consider the sequence of vectorsw, Fw,... , F'w. We claim that E takes Fiw to a multiple of Fi-lw. Assumeinductively that EF'-'w = ,a Fi-2w. We know that this is the case if i = 1with p, = 0. Then we have EFiw = ([E, F] + FE)Fi-'w = HFi-1w +FEF'-'w = (A - 2(i - 1))F'-'w + FEFi-lw = (A - 2(i - 1) + µi)Fi-lw.We have therefore proved our claim and in fact the scalars pi are evaluatedto be (i - 1) (A - i + 2). Again as before all the members of the sequence Fiwcannot be nonzero. Let m be the least positive integer such that F-+lw = 0.The vector space W spanned by w, Fw, ..., F'w, is invariant under H, Eand F. Moreover, H takes the diagonal form (A,,\ - 2, ... , A - 2m). SinceH is a commutator, it has trace 0. Hence we have (m + 1) A - m(m + 1) = 0,i.e. A is the nonnegative integer m. We have already computed EFiw tobe i(m - i + 1)Fi-lw. The linear map of W into V,,,, which takes FPw to(xrm,-PyP)/(m - p)! is an s[(2)-isomorphism of W with V,,.. This proves thatevery s[(2)-module V contains as a submodule, an isomorphic copy of oneof the V,,,'s.

6.5. Exercise. In any s[(2)-module, check that the element C = H2 +2EF + 2FE commutes with H, E and F. Show that C has only eigenvaluesof the type m2 + 2m, with m a positive integer.

Now we have the following classification of all s[(2)-modules.

6.6. Theorem. Let V be any finite-dimensional s[(2)-module. Then V iscanonically a direct sum of subspaces V (m) in each of which the element C =H2 + 2EF + 2FE is represented by the scalar endomorphism (m2 + 2m). Id.The subspace V (m) is itself the direct sum of modules isomorphic to V,,,namely the space of homogeneous polynomials of degree m in two variables.

Proof. We will decompose the given module into a direct sum of sl (2)-modules by taking the generalised eigenspaces of C. Since all the threeelements commute with C, these subspaces are invariant under s[(2). Callthem V (m). We will now restrict ourselves to one of these modules. Ac-cordingly we will assume that the only eigenvalue of C in this module ism2 + 2m and show that it is a direct sum of submodules isomorphic to V,,,..


As a first step, we will consider the case V = V (m) with m = 0. In thiscase, any irreducible submodule of V is of the form V,,,, for some m. But it hasto be trivial, i.e. H = E = F = 0, since C has the eigenvalue m2+2m on V,,,,which is assumed to be 0. Let W1 be the direct sum of all trivial submodulesof V. We may apply the same argument for the quotient of V by W1, andconclude that there is a filtration Wo = 0 C Wl C C W,._1 C W, = Vwith the property that Wi+1/Wi is a direct sum of trivial modules for all i.Now H, E and F take Wi to Wi_1. We need to show that r = I. If not,acting by [H, E] = HE - EH on W2 we conclude that [H, E] annihilatesit. Since [H, E] 2E, this means E is zero on W,,_2. Similarly F also actstrivially on it. Since H = [E, F], it follows that W2 is the trivial module,contradicting our assumption.

We will finally consider the case when C has only one eigenvalue m2 +2m, m 0. Let W be maximal among subspaces of V which are directsums of modules isomorphic to V,,,, as st(2)-modules. We will show thatW = V. If not, the z[(2)-module V/W contains a submodule S isomorphicto V. Consider the space L of all linear maps of V,,,, into V such that theircomposites with the natural map V -+ V/W are sl(2)-module homomor-phisms into S. The space L can be considered as an sI(2)-module by settingx f = xv o f - f o x = H, E, or F. The generalised eigenspaceL(0) of C in L for the eigenvalue 0 is, as we have seen above, a trivial repre-sentation. In other words, elements f e L(0) are st(2)-homomorphisms. Toany f E L(0) associate its composite with the natural map V --> W to geta surjective map to the one-dimensional trivial representation, namely thespace of zl(2)-homomorphisms of Vm into S. In particular, the isomorphismV,,, -> S can be lifted to an st(2)-homomorphism V,,,, -> V. Its image isa module isomorphic to V,,,, and is supplementary to W, contradicting themaximality of W.

A posteriori, we see that C, H are all semisimple in any module sincethey are so in V,'s and therefore the generalised eigenspaces V (m) of C areactually eigenspaces.

6.7. Remark. We proved in 6.4 that all irreducible modules are isomorphicto V,t for some m. In the proof of Theorem 6.6, we saw that every repre-sentation of st(2) is a direct sum of irreducible modules, using the elementC which acts on any st(2)-module.

But any zt(2)-module arises by a complex extension of a representationof the real Lie algebra su(2) consisting of skew-Hermitian matrices. Thelatter is the Lie algebra of the special unitary group SU(2) which is simplyconnected and compact. So the complete reducibility of st(2)-modules isentirely equivalent to the complete reducibility of complex representations


of the compact group SU(2). To prove this, one can use global methods,such as integration.

6.8. Exercise. Let G be a compact Lie group and V a G-module. Showby proving the existence of a Hermitian metric invariant under the groupthat V is completely reducible.

We are interested in the zt(2)-module that arises in the following manner.Let V be a real vector space of dimension 2n. Assume given a J-structureand a Hermitian metric with respect to it, that is to say, a symmetric positivedefinite 118-bilinear form satisfying g(X, JY) + g(JX, Y) = 0. Let w be thecorresponding nondegenerate alternating bilinear form, namely (X, Y) F-ag(X, JY). We can diagonalise the Hermitian form so that there exists alinearly independent set (ei), i = 1, ... , n, such that ei, fi = Jei form anorthonormal basis for g, and w has the form E e? A fi with respect to thedual basis. In the following we will denote e?, fi by ei, fi for notationalsimplicity.

Define E to be multiplication in the sense of the exterior algebra A(V) byw. Define H to be the transformation given by Ha = (i-n)a for a E Ai(V).It follows that HE(a) = H(a A w) _ (i + 2 - n)a A w. On the other handEH(a) = (i - n)a A w. Thus (HE - EH) (a) = 2a A w = 2E(a). In otherwords, [H, E] = 2E. We will now define a linear transformation F in theexterior algebra. Note that the element v = el A fl A ... A en A f,- gives anorientation on V and also the metric gives a star operator denoted *. Usethis operator to define the conjugate F = *E(*)''. Then F maps A'(V)into A''-2(V). Now 2F = 2 * E(*)-1 = *[H,E](*)-1 = [*H(*)-1,F]. But*H(*)-1 acts on Ai(V) as multiplication by 2n-i-n. Hence *H(*)-1 = -H.Thus we get [H, F] = -2F.

We will now show that EF - FE = H. We will do this by computingit on the elements of the form es A fT. Here S and T are subsets of the setN = {1, . . . , n}, with some ordering. It is convenient to split up the subsetsS and T as follows. We will denote by A, B, C, D the subsets S fl T, S \ T,T \ S, N \ (S U T) respectively. Then we will write the element as eAB A fAC

Then *(eAB A fAc) = ±eDC A fDB the sign being the same as in eAB AfAc A eDC A fDB = ±v. We will denote by the corresponding small lettersthe cardinality of the sets involved. In both EF and FE the term *2 occurstwice so that we may take +v for the volume element throughout withoutbothering to find out which, as long as we use the same volume element onboth occasions. After choosing A, B, C, D as above, we will actually takefor the volume element, the term eABCD A fABCD, which is clearly ev andit is not relevant to determine e and carry it along.


The operators EF and FE preserve each A'(V) and we will compute(E*)2 on a = eAB A fAc, noting that it differs from EF only by the factor(- 1)'+'. We have *(a) = E1(eDc AfDB), the sign being the signature of thepermutation taking ABACDCDB to ABCDABCD. This gives the parityof the sign to be c(a + c+ d) + d(a + c) + b(c+ d). When we apply E to *a,we therefore get (E*) (eAB A fAC) = E2 (/IiEA eiDC A fiDB) where E2 has theparity of c+d+E1 = c+d+ ca +c2 + cd+da+ dc+ bc+bd or, what is the same,that of d + ac + ad + be + bd. Noting that each term in the summation is ofthe same form with A, B, C, D replaced by iD, C, B and A\ {i} respectively,we may repeat the formula and get (E*)2(a) = E3(EjE{i}UD,iEAej(A\{i})BA

ej(A\{i})C) = E3(EiEA eAB AeAC + I:jED,iEA eju(A\i)B A fju(A\i)C), where E3has the parity of d+ac+ad+bc+bd+(a-1)+(d+l)b+(d+l)(a+l)+cb+c(a-1). This simplifies to b + c which means that E3 = (-1)b+c In other words,EF(a) _ (#A)a+EjED,iEA e(A\{i})UjBAf(A\{i})ujc A similar computationgives FE(a) _ (#D)a + EjED,iEA e(A\{i})UjB A fjED,iEAf(A\{i})ujc. Takingthe difference we get [E, F] (a) = (#A - #D)a. But it is clear that #A -#D = r - n, proving that [E, F] = H, as claimed.

6.9. Space of harmonic forms as an SL(2)-module.Let M be a Kdhler manifold. Then the exterior algebra of T,;,, at any

point m E M is an s f(2)-module in a canonical way as shown above. Thanksto Proposition 5.1, we conclude that the operations E, F, and H commutewith the Laplacian and hence make the space of harmonic forms an sl(2)-module.

6.10. Definition. A harmonic form as well as the cohomology class repre-sented by it, is said to be primitive if it is in the kernel of F under the aboveaction.

The module of harmonic forms can be decomposed as a direct sum ofirreducible s((2)-modules. But the action has the property that all the eigen-values of H in the kernel of F are at most 0. In other words, all primitiveforms are of degree i < n.

Moreover a harmonic form a of degree i < n is primitive if and only ifE(*a) = 0. Since *a is of degree 2n - i, we conclude that it belongs to theeigenspace of H with eigenvalue 2n - i - n = n - i. Also *a is in the kernelof E so that the SL(2)-module generated by *a is actually Vn_i. Equallya generates the same SL(2)-module. Iteration of E on a leads to *a, i.e.En-ia is a nonzero multiple of *a. To summarise, we have

6.11. Proposition. Let M be a compact Kahler manifold. The action ofEm on Hi(M, C), i < n, is injective if m < n - i and the space of primitiveforms of degree i is the kernel of En-i+1 Moreover, every harmonic form


of degree i is a sum of harmonic forms obtained as E'"wr, where Wr's areprimitive harmonic classes of degree i - 2r.

As in the case of the Hodge decomposition, this implies many interest-ing constraints on the Betti numbers of a compact Kahler manifold. Forexample, for a start, the even Betti numbers bi are never zero for i < 2n.Note that this implies that even-dimensional spheres of dimension greaterthan 2 can never be Kahler, even if they admit complex structures.

Of course we can derive many more conditions. For instance, if i < n-1,we have bi < bi+2.

7. Kodaira's Vanishing Theorem

7.1. Definition. Let M be a complex manifold and E a holomorphic prin-cipal bundle on M with a complex Lie group G as structure group. Then a(differentiable) connection on E is said to be of type (1, 0) if the connectionform, extended as a complex linear map of Te ® C with values in g, E P,is of type (1, 0).

Notice that the condition can also be stated purely in terms of the(nonextended) connection form 'y on E by saying that -y(JX) = i-y(X),where J is the (almost) complex structure on E and i is the multiplicationin the complex vector space g. In the case when G = C*, that is to say, Eis the principal bundle associated to a line bundle, the complex-valued form-y on E is of type (1, 0).

A connection on a vector bundle E is the same as an operator E -T® ® E with identity symbol. This gives two operators from E, one into(T*)(1,O) ® E and another into (T*)(o'1) ® E. It is easily verified from thedefinition that the connection is of type (1,0) if and only if the latter ofthese operators is the Dolbeault differential d" which is associated to theholomorphic bundle E.

7.2. Proposition. Given a Hermitian metric on a holomorphic line bundleon a complex manifold, there is a unique connection form of type (1, 0) onthe principal bundle, which preserves the Hermitian structure. Its curvatureis a form of type (1,1).

Proof. We can use the Hermitian metric to reduce the structure groupof the bundle to U(1). If there are two Hermitian connections, then theirdifference goes down to a forma on M with values in the Lie algebra ofU(1), namely iR. On the other hand, if the connections are of type (1, 0),then a satisfies a(JX) = ia(X), which is possible only if a = 0. This provesthe uniqueness.


Let us treat the Hermitian metric as an R-1--valued function h on theprincipal bundle P of E satisfying h(es) = for all E P and s E Cx.In other words, it is a section of the associated bundle for the action of C" onR+ given by A H multiplication by IAA-2. The invariance of the Hermitianmetric under the connection therefore gives Vxh = Xh + y(X)h = 0. Thevalue of y(X)h is, by the above computation, -2 Hence wehave Rey(X) = (1/2)Xh/h, or what is the same, Rey = (1/2)d(logh).Since -y is in addition a form of type (1,0), this implies that y = d'(logh)and that its curvature form is given by d"d'(log h).

7.3. Definition. The unique connection associated to the Hermitian struc-ture on a holomorphic line bundle is called the Chern connection. Thecorresponding curvature form is called the Chern form of the Hermitianstructure.

7.4. Remark. Compare this with the computation of curvature in the caseof the Hopf bundle in [Ch. 5, 6.2]. In retrospect, the connection which wedefined there is just the Hermitian connection, using the Hermitian metricalong the fibres induced from that of the ambient vector space.

Let L be a holomorphic line bundle provided with a Hermitian metric.The above computation shows that the Chern form is then a purely imag-inary form of type (1, 1). In particular we deduce that the Chern class ofany holomorphic line bundle is actually in H1,1 (C). In particular, we have aclass in the Dolbeault cohomology space H' (M, 521). Holomorphic line bun-dles are classified up to isomorphism by H' (X, 0'). If v is the cohomologyclass corresponding to L, then the above class is obtained as the image of vunder the map induced by the sheaf map 0" -* 521 given by f H d(log f ).Indeed this follows from the commutative diagram

0 - 27riZ -i V -- V X -+ 0

1 II 1

0 C -- 0 521 ct -> 0.

Here the lower sequence has its right arrow given by f F-+ df / f . It is exact be-cause any closed holomorphic form is locally a boundary. The top sequenceis exact since any everywhere nonzero function can be locally written as theexponential of a holomorphic function.

We will now address the question: what forms are Chern forms of aHermitian connection? As we have seen above, these have to be closedforms of type (1, 1) representing cohomology classes which are (2,7ri times)integral.


Firstly, any (27ri times) integral cohomology class, represented by aclosed form of type (1, 1), is the (geometric) Chern class of a suitable holo-morphic line bundle L. Indeed, in the above diagram, by hypothesis, weare given an element c of H2(M, C) which comes from H2(M, 27riZ), andwhose image in H2 (M, 0) is zero. It follows that it comes from an elementof H' (M, 0'). In other words, there is a holomorphic line bundle L whoseChern class is c.

Next we would like to know if any closed form a of the above typeis actually the Chern form of a holomorphic line bundle with a Hermitianstructure.

If we introduce an arbitrary Hermitian metric on a holomorphic linebundle L with this Chern class, its Chern form 0 may not coincide with a.But the difference a - 0 is an imaginary 2-form representing the trivial classand hence of the form dw for some purely imaginary 1-form w. Write w as

with l; of type (0, 1). Then we get the set of equations, a-/3 = d'C-d" C,and d" l; = 0. Let 77 be the 0"-harmonic representative of the Dolbeault classof C. Then C - 71 is of the form d' f . Since 77 is harmonic for the Laplacian0 as well, we have d'77 = 0. From this we conclude that d'l; = d'd"f andtherefore that a - a is of the form d'd"(f + 7). In other words, there is areal-valued function g such that a -0 = d"d'g. Now we can modify theHermitian metric by multiplying by exp(g) and check that the Chern formof the new Hermitian connection is actually ce.

Thus we have proved the following

7.5. Theorem. 1) Let X be a compact Kahler manifold. A cohomologyclass in H2(X, C) is the (geometric) Chern class of a holomorphic line bun-dle if and only if it is of type (1, 1) and represents (27ri times) an integralclass.

2) Any form of type (1, 1) which represents the Chern class of a holo-morphic line bundle, is the Chern form of a suitable Hermitian metric onthe line bundle.

7.6. Theorem (Kodaira). Let L be a holomorphic line bundle on a compactKahler manifold M, whose topological Chern class can be represented by aclosed (1, 1)-form which is positive definite, and E any other holomorphicvector bundle. Then the cohomology groups Hq(M, E ® L) vanish for largeenough r whenever q is at least 1.

Proof. We will apply the vanishing theorem (Theorem 4.3) to the Dolbeaultcomplex of E ® L. We note first that there exists a Hermitian structure onL such that the Chern form RL is negative definite. Extend the form RLcomplex-bilinearly to T* ® C. Since it is of type (1, 1), one sees that its


restriction to the spaces of type (1, 0) as well as type (0, 1) are zero. TheHermitian form (v, w) --> RL(V, w) on (T*)o l is positive definite by ourassumption.

The composite symbol in the formula for the Laplacian is given by(v, w) H tv oAv, + ). o tv , where we identify the real cotangent spaceT* with the complex vector space of (0, 1) differentials, and use the Her-mitian metric h on the latter, in order to define t. If a is a differential oftype (0, 1), then the composite symbol corresponding to (v, w) takes a totv (w A a) + (tv,a)v = h(v, w) a - h(v, a)w + h(w, a) v. It is easily seen thatunder the above identification of T* with (T*)o.l, the symmetrisation of hgives (1/2)g. Hence the Dolbeault complex satisfies the conditions for theformula.

We will now compute the constant term. The curvature of AAA (T*)

E ® L is the sum of three terms, the first two being independent of L. Theconstant term in the formula for the Laplacian is correspondingly the sum ofthree terms, the first two being independent of L. The last term is obtainedby substituting in the alternatised composite symbol the curvature form RL.This composite is, as in the Riemannian case, a derivation of the exterioralgebra of (T*)°,1. Its action on (T*)o.l has been computed above. Thisis a sum of two terms. The first is a scalar endomorphism, obtained bysubstituting in the alternating form w on T*, the curvature form RL, andthe second is the derivation DP(RL) given by the Hermitian endomorphismRL of (T*)°,1. The first one is zero, since the curvature form preserves J.

Finally, if we replace L by IT, the constant term in the expression ofthe corresponding Laplacian becomes a sum of terms all of which are thesame as for L, except the last one, namely DP(RL), which gets replacedby DP(RLD) = prDP(RL). Since RL is positive definite by assumption, theconstant term is positive definite for large enough r, proving the theorem.

We also obtain the following precise statement.

7.7. Corollary. If L is a line bundle with a Hermitian structure on acompact complex manifold, such that the Chern form is negative definite,then H2(K(9 L)=0 foralli>1.

Proof. Note first that the Chern form is a closed form of type (1, 1), and weare assuming that it is the imaginary part of a Hermitian positive definiteform. In particular, the manifold is compact Kahler and so Theorem 7.6 isapplicable to our situation. In the above computation we will take p = n. Ifwe replace L by K ® L in the above formula, we have cK®L = -Ricci + CLand the curvature term corresponding to det(T*)O,l is the Ricci form. Thusthe term c reduces to Dq(cL) which is positive definite for q > 1.


7.8. Remark. Using Serre's duality this can also be stated as the vanishingof Hi (X, L) for all i < n, if the geometric Chern class of L can be representedby a positive definite form.

Let us illustrate how Kodaira's theorem works in the case of a compactRiemann surface C and look at the vanishing theorem in this case.

Since the degree of We is 2g - 2 by equation 2.19, the vanishing theoremreduces to the following.

7.9. Vanishing for a Riemann surface. Let L be a line bundle of degreed on a compact Riemann surface C of genus g, with d > 2g - 2. ThenH' (C, L) = 0.

7.10. Lichnerowicz vanishing theorem.We will end this section by giving one more corollary of the vanishing

theorem, by applying it to the case of the Dirac operator of a spin manifold.The result is the following.

7.11. Theorem (Lichnerowicz). If the scalar curvature on an even-dimen-sional Spin manifold is positive at all points, then the space of harmonicspinors vanishes.

Proof. Recall that the Dirac operator (8.6, 8.7) D is obtained by lifting thesymbol T* ® Spin (M) -+ Spin-(M) induced by Clifford multiplication onthe total Spin representation, with respect to the canonical lift of the Levi-Civita connection to the principal Spin bundle. We use here -g as thequadratic form to construct the Clifford algebra. The Spin representationis unitary, and any element v such that g(v) = 1 is represented by a unitarymatrix. Hence the conjugate transpose of the action of v taking Spin+ toSpin- is given by v-1 = -v taking Spin- to Spin+. Thus the adjointsymbol is given by v again. So the composite symbol takes (v, w) to vwacting as an endomorphism of Spin+. When symmetrised, this is the mapof (v, w) to

2(vw + wv) = - I b(v, w). The symbol being invariant under the

connection, the first order term in the composition formula is zero. Finally,we will compute the constant term. If (ei) is an orthonormal basis for gin T,*,,,, the Riemannian curvature tensor is represented by Ei<j RijEij inthe Lie algebra of the orthogonal group, where Rij is a 2-form for each i, j.The element Eij is the skew symmetric endomorphism of T* given by theprescription x --j b(ei, x) ej - b(ej, x) ei. Its action on the Spin representationspace is through the element 2eiej of the Clifford algebra. (See [Ch. 7, 8.6].)Hence the constant term gives the element 4 Ei<j,k<l Rijkleiejekel in theeven Clifford algebra (acting as an endomorphism of Spin+ via the Cliffordmodule structure). Using the Bianchi identity, we see that the summation


needs to be taken only over those (i, j, k, 1) where the set {i, j, k, l} consistsof two elements. Since R is alternating in the first two and the last twoindices, we may assume that i = k and j = 1, i < j. This gives theterm 4 E Rzi2jeiejeiej in the Clifford algebra. Using the fact that e2 =e = -1 and eiej = -eje2, we conclude that the constant term reduces tomultiplication by 1/4 times the scalar curvature, proving the assertion.

There are many interesting generalisations of the Dirac operator, andsimilar vanishing theorems have been proved for them. One may consult[12] for a discussion of these questions. It will be instructive for the readerto derive these vanishing theorems also from the above principle.

8. The Imbedding Theorem

Kodaira's vanishing theorem has a beautiful application, namely a char-acterisation of those complex manifolds that can be imbedded as complexsubmanifolds of the complex projective space (CFN for some N.

8.1. Imbedding theorem (Kodaira). If a compact complex manifold ad-mits a closed form of type (1, 1) which is positive definite and represents anintegral class in H2, then it is isomorphic to a submanifold of gnN for someN.

8.2. Remark. By a positive definite form, we mean a form which is theimaginary part of a Hermitian positive definite form.

Note that the assumption ensures that the manifold with the Hermitianstructure having the above (1, 1)-form as associated alternating form (whichhas been assumed to be closed) is Kahler. Secondly, by Theorem 7.5, thereis a line bundle L whose (topological) Chern class is given by the closedform.

8.3. Definition. A line bundle whose (topological) Chern class can be rep-resented by a (1, 1)-form which is positive definite is said to be positive.

Note that if L is a line bundle with a Hermitian structure such that theChern form is negative definite, then its topological Chern class is positive.

8.4. Remark. In the case when M is a compact Riemann surface, a linebundle L is positive if and only if its degree is positive. This is only aquestion of checking that a real (1, 1) form is positive in the sense that it isthe imaginary part of a Hermitian positive definite form, if and only if itsintegral is positive.


Using the vanishing theorem one would like to construct an imbeddingof M into some complex projective space.

8.5. Maps into the projective space.We will first analyse what it means to give a holomorphic map of a

complex manifold M into the complex projective space CIPN. If z0, z1, ... , ZNare the homogeneous coordinates in ]pN, then one might try to get functionscPo, P i , . ... , cpN on M such that the given map sends any m E M to thepoint in (CpN with homogeneous coordinates cpo(m), (P1 (M), ... , WN(m). Butthen the zi are not functions on IEDN, nor are the cpi. However, the zi areholomorphic (in fact, linear) functions on CN+1 \ {0}. With respect to theaction of (C" on (CN+1 \ {0}, these functions satisfy f (vA) = f (v).A. Thismeans that these functions may be considered as sections of the line bundleassociated to the Hopf bundle by the action of C' on C in which A actsas multiplication by A-1. The 0-module associated to this line bundle istraditionally denoted by 0(1). Thus zi can be regarded as sections of 0(1).Hence the functions cpi we have in mind turn out to be sections of a linebundle L, namely the pull-back of 0(1). Note that at no point of M canall these sections vanish, since they have to be homogeneous coordinates ofthe image point in CPN. To sum up, corresponding to any map M -> CPN,we obtain the data consisting of a line bundle L and an (N + 1)-tuple ofsections such that at every point of M at least one of the sections is nonzero.

Conversely, we can start with such data, namely a line bundle L andholomorphic sections s0, S 1 , ... , sN. Then we can map any m E M to thepoint of the projective space whose homogeneous coordinates are si(m).Note that the si (m) belong to the fibre Lm and not to C. But we can identifyLm with C as a one-dimensional vector space any way we like, in order toconsider so(m), sl(m), ... , sN(m) as homogeneous coordinates of a point.The point does not depend on this choice, since a different identificationchanges all the si(m) by the same scalar factor, and therefore we can thusdefine a point in pN unambiguously. Of course we have also to make theassumption here that not all the si vanish at any point of M. We thus havethe following simple translation, into the line bundle language, of maps intoPN

8.6. Proposition. There is a natural bijection between holomorphic mapsof M into FN and data consisting of a line bundle L and an (N + 1)-tuple(so, 81, ... , SN) of holomorphic sections of L with the property that for everym E M there exists some i with si(m) 0 0. Here we identify two such dataif there exists an isomorphism of the line bundles taking the sections of theone to the corresponding sections of the other.


However, we do not just want a holomorphic map into ]F" but animbedding. So we investigate what we need to assume about our datain order to ensure that the associated map into pN is injective. In otherwords, we need that if m, m' are distinct points, then the two vectors(so(m), si (m), ... , sN(m)) and (so(m'), sl(m'), ... , sN(m')) are linearly in-dependent. This is equivalent to saying that there exists a linear form whichvanishes on one vector but not on the other. Accordingly we have

8.7. Observation. There is a bijection between the set of injective holo-morphic maps into PN and the data consisting of line bundles L and an(N + 1)-tuple (so, 3 1 ,-- . , SN) of holomorphic sections of L such that for anydistinct two points m, m', there is a linear combination of si's which is zeroat m and nonzero at m'.

Similarly, one can also check that the map has injective differential at mif for every nonzero v E Tm(M), there is a linear combination of si's whichis zero at m but its differential at m is nonzero on v.

8.8. Definition. A line bundle with enough sections to imbed M into aprojective space is said to be very ample. If a positive power of a linebundle is very ample, we say that the bundle is ample.

8.9. Remark. Using the vanishing theorem, we will show that if L is apositive line bundle, Lm has enough sections to imbed M in a projectivespace, for large m.

Firstly we note the following simple fact.

8.10. Proposition. A line bundle is positive if it is ample.

Proof. If a line bundle L is ample, then by definition some positive powerLk is very ample, and it is enough to show that very ample bundles arepositive. In other words, we can imbed M in some pN using (a basis of)sections of L and then show that the restriction of the Chern class of 0(1)to M can be represented by a positive definite form. But then it is enoughto show that the Chern class of 0(1) itself can be represented by a positivedefinite form. For if such is the case, then so is the restriction of that formto M, and it represents the Chern class of the pull-back of 0(1), namely L.Thus we only need to verify the statement for the line bundle 0(1). Thisfollows from the computation in [Ch. 5, 6.2].


8.11. Remark. If a compact Kahler manifold has second Betti number 1,then it admits a Kahler structure such that the Kahler class is integral. Forwe know that the Kahler class is nonzero, and hence the Hermitian metriccan be altered by a scalar multiple so that the Kahler class is integral.Although this is a trivial statement, it has many important consequences.Any compact Riemann surface, the projective space and many varieties likeGrassmannians, are indeed Kahler and have second Betti number 1. Hencethey can all be imbedded in the complex projective space, thanks to theimbedding theorem.

8.12. Exercise. Let P be a lattice in a complex vector space V. Showthat if there exists a Kahler structure on the quotient, representing an in-tegral class, then there exists such a structure invariant under translationsby elements of the quotient. Hence deduce that there exists a Hermitianpositive definite form on V whose imaginary part takes integral values whenrestricted to the lattice.

8.13. Imbedding of Riemann surfaces.We will first see how the imbedding theorem works in the case of a

compact Riemann surface C. Let L be a line bundle of sufficiently largedegree. Such a line bundle does exist. We can take any point p E C andconsider the line bundle O(p). It has degree 1 by equation 2.20, and thekth power of this line bundle therefore has degree k. We will denote such abundle by 0(kx).

For any x E C, consider the exact sequence

0

Tensor this with L and assume that d = deg(L) > 2g - 1. ThenHl (C, L®O(-x)) = 0 by 7.9, and hence the map H°(L) -+ H°({x}, L) = Lxis surjective. In other words, for every x E C there exists a section of L whichis not zero at x. If for the data of Proposition 8.6, we take (L, s°, ... , SN)where the si form basis for H°, we get a holomorphic map of C into CIPN.But we can do better. Suppose x, y are two distinct points. Then considerthe ideal sheaf of holomorphic functions which vanish at x and y. This sheafis in fact Z ®I, = 0(-x) ® O(-y). We will denote this by 0(-x - y). Wethen get the following exact sequence:

0-->O(-x-y)- 0-->O{x,y}- 0.

If L has degree > 2g, then L ® 0(-x - y) has degree > 2g - 2 and soagain H' (C, L ® O(-x - y)) = 0 for any two points x, y E C. Hence theevaluation map H° (L) -* L ® Ly is surjective. This shows that we can take


a basis of H°(L) and use Observation 8.7 to get an injective map of C intopN where dim H° (L) = N + 1.

Finally we also need to ensure that the differential at each point is in-jective at all points. That is taken care of by using the exact sequence

0->0(-2x)-C-- 0/1x -->0.The same argument assures us that the map H°(C, L) --> L./IL. is sur-jective. (Here Lx denotes the stalk of the associated 0-sheaf L at x.) Thisshows that there is a section of L which vanishes at x, but not with multi-plicity 2. It is easily checked that this is exactly equivalent to saying thatthe above holomorphic map into CIPN has nonzero differential at x.

This proves the following

8.14. Theorem. Any Riemann surface C can be imbedded as a submanifoldof some projective space. More precisely, any line bundle L of degree d > 2gis very ample, where the genus of C is g.

8.15. Blow-up.Let now L be a positive line bundle on a compact complex manifold

M. One tries to use the vanishing theorem in order to prove the imbeddingtheorem on exactly the same lines as for Riemann surfaces. The first problemis that if we take any point of M, the corresponding ideal sheaf is not locallyfree. Only the ideal sheaves of submanifolds of codimension 1 are locally free.So we have to find a way of replacing points by submanifolds of codimension1 (in perhaps some other manifold).

In order to execute this, we need a construction known as `blow-up'.

8.16. Example. We will start with an example. Let V be a vector space.Consider the Hopf bundle on P(V). Then the associated line bundle forthe action of C" on C, given by multiplication, can be described as follows.Consider the product V \ {0} x C and make C" act on it by A(v, a) =(Av, )-la). The quotient is the total space Q of this line bundle. We have anatural map of this complex manifold into ?(V) since it is a bundle over it,but there is also a holomorphic map 7 of Q into V which maps (v, a) to av.Consider the open set W in Q given by a ; 0. It is mapped isomorphicallyto the open set V \ {0}, the inverse being given by v --> (v, 1). The entiresubmanifold D = P(V) x {0}, namely the zero section of this bundle, ismapped to 0 in V. We may therefore think of Q as replacing the origin inV by a whole projective space, while leaving other points undisturbed.

One can check that at any point x = (v, 0) of D, the differential of themap it maps the tangent subspace T,(D) to zero of course, but it induces amap of the normal space TT(Q)lTx(D) to the one-dimensional subspace of


the tangent space T,rx(V) given by v itself. This is obvious if we interpret 7ras the inclusion of the line bundle in the trivial vector bundle P(V) x V inthe obvious way, followed by the projection to V.

Now let X be any complex manifold of dimension n and x any pointof X. Then we can construct a new complex manifold by removing x fromX and replacing it with the complex projective space of dimension n - 1as above. Take a coordinate open set U around x. Identify U with anopen neighbourhood of 0 in a vector space V. Then we can make the aboveconstruction over V and take the open submanifold U' = it-1(U) of Q. Aswe have seen, W = 7r-1 (U \ {0}) is mapped by 7r isomorphically to U \ {0}.Hence we can take the two manifolds U' and X \ {x}, and use the aboveisomorphism 7r : W _- U \ {0} to glue them together. The resulting complexmanifold Qx (X) is obviously independent of the identifications made anddefines what is called the blow-up of the manifold X at x.

8.17. Definition. The complex manifold Qx(X) defined above, togetherwith the holomorphic map 7r of Q. ,:(X) onto X, is called the blow-up of X atx. The submanifold D of Qx(X), namely 7r-1(x), is called the exceptionaldivisor.

The map 7r is an isomorphism of Qx(X) \ D onto X \ {x}. Moreover,7r-1({x}) is actually D. The differential of the map at any point of theexceptional divisor maps the tangent space of D to zero and goes down toan injection of the normal space into the tangent space at x. This gives anisomorphism of D with the projective tangent space PTx(X). Hence onemay think of the points of the exceptional divisor as tangent directions at x.Moreover, this analysis has shown that the normal bundle of the exceptionaldivisor is canonically isomorphic to the Hopf bundle O(-1) on D = IIDn-1.The ideal sheaf TD restricts to D as the normal bundle. Hence we have

8.18. Lemma. The ideal sheaf of the exceptional divisor in the blow-up ofa point restricts to the exceptional divisor, as the line bundle 0(1).

Now let L be any line bundle on X and Q(L) = -7r*L its pull-back onQx(X). Then we have

8.19. Proposition. There exists a canonical isomorphism betweenHI (Qx (X), 7r* (L)) and HO (X, L).

Proof. In fact, we have a natural linear map of the space H° (X, L) intoH°(Qx(X), Q(L)), namely pulling back by -7r. This map is injective since 7ris surjective. To prove that it is surjective, note that any section of Q(L)gives a section of L over X \ {x}. Trivialising L in a neighbourhood U ofx, we can identify its restriction to U \ {x} with a holomorphic function.


The only thing to check is that this function can be extended to the wholeof U. Since it came from an extension of the section to the pull-back, it isbounded in this punctured neighbourhood of x, and so it can be extendedto the whole neighbourhood.

8.20. Remark. Actually we can extend any holomorphic function in thecomplement of a point in any complex manifold of dimension at least 2 tothe whole manifold, thanks to a theorem of Hartogs.

Now our strategy is clear. Start with a positive line bundle L on acompact Kahler manifold. As in the example of the Riemann surface, givenany point m E M, we will show that there is a positive integer p and asection of LP which does not vanish at in.

Notice that this already implies that we can arrange things so that thesame power will do for all points. For if p is chosen as above depending onm, then it serves for all points in a neighbourhood of m as well. Coveringthe compact manifold by finitely many such open sets, we get sections siof LP such that at any point x E M at least one of them does not vanish.Now take p = fJ pi, and note that if si does not vanish at a point x, thesection of LP obtained by raising si to the power flj#i pj does not vanish atx either.

In order to show that there is a section which is nonzero at m, we willconsider the blow-up Qm,(M) and the bundle Q(L). Now this line bundle isnot positive, since it is trivial on D. But it is well behaved otherwise. Wewill explain. Choose a suitable Hermitian structure on L so that (i times)the Chern form is positive definite. Then the pulled back form a is positiveat all points of Q,(M)\D, zero on vectors tangent to D and positive definiteon the normal vectors at points of D.

There is a Hermitian structure on the line bundle ZDID = 0(1), suchthat its Chern form is a negative definite form /3. Extending this Hermitianstructure on TD from D to the whole of Q,(M), we get a global formrepresenting its Chern class. Now we can find a positive power k of Q(L)such that the Chern form of its tensor product with TD, namely ,Q + ka, isnegative definite at all points of D. Hence it is so also in a neighbourhoodV of D. Let W be an open neighbourhood of D such that W C V. Now thecomplement of W in Qm(M) is compact, and so we can find a power of Lsuch that (i times) the Chern form of its tensor product with TD is positivedefinite at all points of Q(M) \ W.

From the exact sequence

0--*T(D)--;T(M)ID-->Nor(D,M)-40

we deduce that KM restricts to D as the bundle KD ® 0(1). But Kin-, iseasily computed to be isomorphic to O(-n). Thus we see that there exists


a positive power l of L such that Q(L') ®ID ® KT,,' is positive. Hence thereexists q > 0 such that for all p > q, we have H'(Qm(M), Q(LP) (9 1D) = 0for all i > 0. This implies that the map H°(Q(M), Q(LP)) - H°(D, LP) issurjective. In view of Proposition 8.19, this only means that the evaluationmap of H° (M, LP) on the fibre at m is surjective. This proves the following.

There exists q > 0 such that for every p > q, there is a holomorphic mapof M into a suitable projective space such that the pull-back of 0(1) is LP.

It hardly needs to be said how to complete the rest of the proof. Letm, m' be any two points. Then use the blow-up Q{m,m'} (M) at both points.Let Dm, Dm be the two exceptional divisors. Exactly as before, we can finda positive integer q such that Hz(Qm,m'(M),Q(LP) ®ID (9 ID,) = 0 for allp > q and i > 0. This implies that the map H° (M, LP) --* LP,, ® L, M, issurjective and proves that the associated map into the projective space isinjective.

Finally suppose given a nonzero vector v at m. The point in PTm(M)defined by v can be interpreted as a point [v] of the exceptional divisor inQm(M). Now blow Q,(M) up again at the point [v] and argue exactly asabove to complete the proof of the imbedding theorem. We leave this asan exercise to the reader, who may wish to consult [20] or [19] for a moredetailed account of Kahler manifolds.

Finally, by a theorem of Chow, any submanifold of IPN can be definedas the set of common zeros of homogeneous polynomials. Thus the studyof complex manifolds admitting positive line bundles, can actually be ac-complished by purely algebraic machinery. This belongs to the domain ofAlgebraic Geometry.

Appendix

We gather here some of the facts and concepts used in the book.

1. Algebra

1.1. Tensor products.All rings (except when explicitly stated) are assumed to have a unit

element, denoted 1. All modules over the ring are supposed to be unitary.Let A be a ring. A balanced map b : L x M -+ N where L is a right

A-module, M a left A-module and N an abelian group, is bilinear over Z(i.e. biadditive) and satisfies b(la, m) = b(l, am) for all a E A, 1 E L andm E M. All balanced maps of M x N into an abelian group are classified byan abelian group M ® N in the sense that there is a balanced map of M x Ninto the abelian group M ® N such that any balanced map of M x N intoany abelian group L factors to a unique homomorphism of M ® N into L.

Let A be a commutative ring and L, M, N, modules over A. A bilinearmap b : L x M -+ N is a map whose restrictions to {x} x M and L x {y} areA-linear, for all x E L and y E M. Bilinear maps from M x N are classifiedby the tensor product M®N in the sense that there is a natural bilinear mapof M x N into the A-module M®N such that any bilinear map of Mx N intoany module L factors to a unique linear map of M ® N into L. The r-foldtensor product ®r M is called the rth tensor power. The direct sum T(M)of all the tensor powers (setting (&° M = A) is actually an A-algebra calledthe tensor algebra of M. The tensor algebra has also a universal property.It comes with a linear map of M into T (M) such that any linear map of Minto any A-algebra B extends to an algebra homomorphism of T (M) intoB.

301

302 Appendix

1.2. Alternating and symmetric multilinear maps.Let A be commutative. A multilinear map f of M x M x... x M into

an A-module L is said to be alternating if f takes the value 0 whenevertwo or more of the arguments are equal. A multilinear map is symmet-ric if f (x1, ... , xr) remains unchanged when any two of the arguments are

interchanged. Any alternating or symmetric map gives a linear map ofO' M --> L which annihilates respectively elements of the form xl ® . . ®

x®...®x®...®xr and xl® ®x®...®y®...®xr-x1®...®y®...®

x®...®xr.

The quotients of (D' M by the submodules generated respectively byelements of the above type are called the exterior power Ar(M) and thesymmetric power Sr(M) respectively. The direct sum A(M) of all the A-modules AT(M) (resp. Sr(M)) is an algebra called the exterior algebra (resp.the symmetric algebra) of M, denoted A(M) (resp. S(M)). The symmetricalgebra of M is universal in the sense that there is a natural linear inclusionof M into it such that any linear map of M into any commutative algebraB can be extended to an algebra homomorphism of S(M) into B. Likewise,there is an inclusion of M in A(M) such that any linear map 1 of M intoan algebra B, satisfying 1(m)2 = 0 for all m, extends to a unique algebrahomomorphism of A(M) into B.

An alternating map of M into A is called an alternating form. The directsum of all alternating forms on M forms an associate algebra. One definesthe (wedge) product a A a of an alternating p-form a and an alternatingq-form ,Q to be the alternating (p + q)-form

(ml, ... , mp1-q) E e a(ma(l), ... , ma(p)) Q(mv(p+1)) ... , mQ(p+q)),

where the sum extends over the so-called shuffle permutations and e(u) isthe signature of the permutation. A shuffle is a permutation which satisfiesa(1) < < c(p) and 1) < < Q(p -I- q). Note that if a is a linearform, then a A a is by definition zero. Therefore the inclusion of M* inthe algebra of alternating forms gives rise to an algebra homomorphism ofA(M*) into the algebra of alternating forms on M. This is easily seen to bean isomorphism.

A polynomial function on V of degree r is an element of Sr(V*) thedirect sum of which is an algebra called the algebra of polynomial functionson M.

1.3. Linear and bilinear maps of vector spaces.We will deal with a finite-dimensional vector space V over real or com-

plex numbers.

1. Algebra 303

There is a natural duality between A'(V) and A'(V *) given by

(vl,...,vr),(11i...,lr) H det(li(vj)).

There is a slightly less elegant duality between Sr (V) and Sr (V *) . Thisis given by (vi, ... , vr), (l1, ... , lr) r, E jI li(v,(j)), where the summationruns through all permutations of [1, ... , r].

A linear endomorphism of a finite-dimensional vector space is said tobe semisimple if it satisfies a separable polynomial. A semisimple endomor-phism can be diagonalised over C.

A symmetric bilinear form b on V is said to be nondegenerate if there isonly one element x E V for which b(x, y) = 0 for all y E V, namely 0. Givena symmetric bilinear form b on a vector space V, we have a decompositionV = V1 ® V2 such that b(x, y) = 0 for all x E V1 and y E V, and therestriction of b to V2 is nondegenerate. If b is a nondegenerate bilinear form,then there exists a basis (ei) such that b(ei, ej) = ±5ij. If V is a real vectorspace, then the number of positive signs in this formula is independent ofthe basis. If there are p positive and q negative signs, we say it is of type(p, q). If b satisfies b(x, x) > 0 for all nonzero x, then we say it is positivedefinite.

If V is a complex vector space and h : V x V -> C is an JR-bilinearform, then we say it is Hermitian if it is (C-linear in the first variable andantilinear in the second variable in the sense that h(x, Ay) = )h(x, y) andh(x, y) = h(y, x) for all x, y E V and,\ E C. If h is a complex Hermitian formwhich satisfies h(x, x) > 0 for all nonzero x, we say it is positive definite.Then there exists a basis (ei) such that h(ei, ej) = Sij.

1.4. Graded and filtered algebras.An algebra A over a commutative ring k is Z-graded (resp. Z/2-graded)

if it is a direct sum of k-submodules (Mi), i E Z (resp. i E Z/2) suchthat Mi.Mj C Mi+j for all i,j E Z (resp. i, j E Z/2). The modules Miare called the graded components. In the case of Z/2-gradation, the twocomponents are often written M+ and M-. Any Z-gradation gives rise toa Z/2-gradation on defining M+ = Ii even Mi and M- = Ei odd M.

The exterior algebra and symmetric algebras are examples of Z-gradedalgebras.

An algebra A is said to be a filtered algebra if it is provided with sub-modules FiA, i E Z, with FiA C FAA for all i:5 j and F2A.F3A C Fi+jAfor all i, j. If A is filtered then the direct sum Gr(A) = E FzA/Fi-1A isa graded algebra with a naturally induced multiplication. It is called theassociated graded algebra.

304 Appendix

1.5. Exact sequences.If M' -* M and M -- * M" are two linear maps, then we say that

M' -> M -+ M" is exact if the image of M' in M is the same as the kernelof the map M -* M". Clearly 0 -> M' --> M is exact if and only if M' --p Mis injective. The sequence M -> M" --3 0 is exact if and only if the mapM --> M" is surjective. A sequence of the form

1.6. M"->0

is said to be a short exact sequence if it is exact at all the three points. Thismeans that a) M' -; M is injective, b) M -> M" is surjective, and c) if weidentify M' with its image in M, then M/M' is mapped isomorphically bythe induced map, to M".

An exact sequence of the type 1.6 is said to split if there exists a linearmap M" --p M such that its composite with M -> M" is the identity. Thisis equivalent to saying that there exists a linear map M --4M' which is theidentity on M'.

1.7. 5-Lemma. Consider a commutative diagram

0 - M' -; M -p M" -* 0

1 1

0 --> N' -> N - N" -p 0

the horizontal sequences being short exact. If the first and the last downwardarrows are isomorphisms, then so is the middle downward arrow.

1.8. Simple algebras.In what follows all algebras are over a field k and are finite-dimensional

as k-vector spaces.

An algebra A is said to be central if its centre, namely {a E A : ab =ba for all b E A}, is k. A simple algebra is one which does not have anyproper two-sided ideals. An algebra is central simple if and only if it isisomorphic as a k-algebra to the matrix algebra over a division algebra overk. It has a unique simple module over the algebraic closure k of k. Theonly finite-dimensional central division algebras over IR are lib itself and thequaternion algebra. The only division algebra over an algebraically closedfield (in particular, over C ) is the field itself.

1.9. Groups.Let G be a group. It is said to act on a set S if a map G x S -> S

denoted (g, s) F-> gs is given satisfying a) g1(g2s) = (9192)s and b) 1.s = s.One can check that the relation: s a t if there exists g E G such that gs = t

2. Topology 305

is an equivalence relation. The quotient is denoted S/G. The equivalenceclasses are called orbits under the action. If there is only one orbit, that isto say, for any two elements s, t E S there exists g E G such that gs = t,then we say that the action is transitive. For any s E S, the subset of Gdefined by {g E G : gs = s} is a subgroup and is called the isotropy groupat s.

We say that a group G acts on another group A if in addition the mapsa '- ga are all automorphisms of A. If G acts on A, then we can defineanother group called the semidirect product of G by A. As a set it is simplyA x G, the group structure being given by

(a, g)(a/,

g') = g9)There is a natural surjective homomorphism of the semidirect product to G,and the kernel is canonically isomorphic to A.

The free product G1 * G2 of two groups Gl and G2 is a group containingG1 and G2 as subgroups whose union generates it as a group and satisfiesthe following universal property. If G is any group and G1 -' G and G2 - Gare two homomorphisms, then there is a unique homomorphism of G1 * G2into G, extending the given homomorphisms. Such a group exists and isunique up to isomorphism.

A subgroup of a finitely generated group of finite index, is itself finitelygenerated.

2. Topology

2.1. Coverings.We use the term `Hausdorff' for topological spaces in which any two

points can be separated by disjoint open neighbourhoods. Generally speak-ing, when we refer to locally compact, compact or paracompact spaces, theyare supposed to be Hausdorff. A compact space is a Hausdorff space in whichevery open covering admits a finite subcovering. A product of any family ofcompact spaces is itself compact. A relatively compact subset is one whoseclosure is compact in the induced topology. A locally compact space is aHausdorff space in which every point has a relatively compact open neigh-bourhood. The product of any finite family of locally compact spaces is lo-cally compact. A continuous map f of a topological space X into Y is said tobe proper if for every topological space Z, the map (f x IZ) : X X Z -p Y X Zis closed in the sense that the image of any closed set is closed. If the spacesare locally compact, it is equivalent to saying that the inverse image of anycompact set in Y is closed in X.

A normal space is a Hausdorff space in which either of the followingequivalent conditions is satisfied.

306 Appendix

a) Any two disjoint closed sets C1 and C2 can be separated by open setsin the sense that there exist disjoint open sets U1 and U2 containingrespectively C1 and C2.

b) Any two disjoint closed sets C1 and C2 can be separated by continuousfunctions in the sense that there exists a continuous function f withvalues in the closed interval [0, 1] such that f restricts to the constantfunctions 0 and 1 on the two closed sets.

Let (Ui)iE1 be an open covering of a topological space X. Then a refine-ment consists of another open covering (V3)jEJ and a map i : J -+ I suchthat Vj E U,,7(j) for all j E J. A covering (Ci) is said to be locally finite ifevery x E X has an open neighbourhood U such that {i E I : U fl Ci # O}is finite.

A paracompact space is a Hausdorff space in which every open coveringadmits a locally finite refinement. A locally compact space which has acountable base for open sets is paracompact. Every paracompact space isnormal. A shrinking of an open covering (Ui), i E I, is another open covering(Vi), i E I such that Vi C Ui for all i E I. Every open covering of a normalspace admits a shrinking.

The topological union of spaces (Xi), i E I, is the disjoint set unionUiE1 Xi with the topology in which a subset U is open if and only if U fl Xiis open in Xi for all i E I.

If X is a compact metric space, and (Ui) is an open covering, then thereexists a positive real number l such that any set whose diameter is less thanl is contained in Ui for some i E I. This number is called the Lebesguenumber of the covering.

2.2. Connectedness properties.A connected space is a topological space that is not the union of two

disjoint proper subsets which are both open.

A path ry is a continuous map of the closed interval I into X. Then -y(0)is called its origin and -y(l) its extremity. A loop is a path whose origin andextremity are the same.

A pathwise connected space is one in which any two points can be con-nected by a continuous path.

A locally connected (resp. locally pathwise connected) space is one inwhich every point admits a fundamental system of open neighbourhoodswhich are all connected (resp. pathwise connected) in the induced topology.

A locally connected (resp. locally pathwise connected) space is homeo-morphic to a topological union of subspaces each of which is connected (resp.

3. Analysis 307

pathwise connected). Such a decomposition is unique, and the subspaces arecalled connected components (resp. pathwise connected components).

A homotopy h between two continuous maps f, g : X -* Y is a continuousmap of X x [0,1] into Y such that h(x, 0) = f (x) and h(x,1) = g(x) for allxEX.

By a homotopy between two paths yl and rye with the same origin wegenerally mean a continuous map h : I x I -+ X such that all the pathsryt(x) = h(t, x) have the same origin.

A simply connected space is a pathwise connected space in which allloops with the same origin are homotopic. A locally simply connected spaceis a space in which every point has a fundamental system of neighbourhoodswhich are simply connected.

A covering space of X is a space Y and a continuous map p : Y -+ X suchthat every point of X admits an open neighbourhood U such that p-1(U)is the topological union of Uz each of which is mapped homeomorphically toU by p.

Any pathwise connected, locally simply connected space has a coveringspace cp : Y -p X with Y simply connected. A group ir acts continuouslyon Y and its action on the fibre over any point is simply transitive. Thegroup it is called the fundamental group of X, and Y is called the universalcovering space of X.

Any covering space of a locally simply connected space is obtained bytaking the quotient of the universal covering space by a subgroup of it.

3. Analysis

3.1. Measures and measure spaces.Let X be a locally compact space having a countable base for open sets.

Let S be the a-ring generated by the class of all open subsets of X. Areal- or complex-valued function is Borel measurable if it is measurable withrespect to S.

A Borel measure y is a measure defined on S such that µ(C) is finite forall compact sets C.

A Borel measure µ gives rise, by means of integration, to a continuouslinear functional, denoted f -+ f f dµ or simply f - µ(f ), on the space CCof continuous functions with compact support. Here continuity is intendedin the sense that if { f} is a sequence of functions with support in a fixedcompact set K and tends to zero uniformly, then {µ(f,,,) } tends to zero. Themeasure u is determined by this functional, and all continuous functionalsare obtained in this way.

308 Appendix

A measure v is said to be absolutely continuous with respect to anothermeasure E.e if v(E) = 0 for every measurable set E for which j j (E) = 0. Inthis case there exists a measurable function f such that v = f p.

If E is any Borel set of positive measure in R7z, then there exists an openset U containing 0 such that U C E - E, where E - E is the set consistingofallx-y,x,yEE.

3.2. Hilbert spaces.A Hilbert space H is a vector space provided with a positive definite

Hermitian inner product, usually denoted (x, y) H (x, y), such that it iscomplete with respect to the metric space structure given by d(x, y) _IIx -Y 112 = (x - y, x - y). The space of measurable functions on a measurespace which are square summable (identifying functions which are equalalmost everywhere) is a Hilbert space with respect to the inner product(f, g) = f fgd1 .

Any Hilbert space admits an orthonormal basis, namely a set (ei) with(ei, ej) = biz such that any v E H is a countable linear combination of ei.If the space admits a countable orthonormal basis it is said to be separable.We only deal with separable Hilbert spaces.

3.3. Equicontinuity.A set S of functions on a metric space is said to be equicontinuous at

a point x if for every e > 0, there exists S > 0 such that I f (x) - f (y) I < ewhenever d(x, y) < S for all f E S. The point here is that 6 is independentof the function. The set is said to be equicontinuous if it is so, at all points.

3.4. Implicit and inverse function theorems.Let U be a domain in IIBk x Ill containing 0, and f : U - R1 a differ-

entiable function taking 0 to 0. If the matrix (ay) is invertible at 0, thenthere exist an open neighbourhood V of 0 in R' and a differentiable func-tion cp : V -+ lR such that V x So(V) is contained in U and f (x, cp(x)) = 0for all x E V. Any two such functions coincide in a neighbourhood of 0.This statement is also true with parameters, that is to say, if in additionf depends differentiably on a parameter in Ilg, then cp exists dependingdifferentiably on the parameter.

If f : U --> R7z is a differentiable map taking 0 to 0 and such that thematrix () is invertible, then there exists U' C U such that f maps U'bijectively onto an open neighbourhood of 0 and the inverse is differentiable.

3. Analysis 309

3.5. Existence theorem for ordinary differential equations.If U is a domain in Rk containing 0, and f is a differentiable function

I X U -* Rk, where I is an open interval containing 0, then there existsa differentiable function cp : I' -* U, where I' is a neighbourhood of 0contained in I, such that

dw(x)=

dxf(x,(p(x))

for all x E V. The solution co(t) is uniquely determined in a neighbourhoodof 0 by its initial value co(0). The same is true again with differentiabledependence on some parameter in IISI.

Bibliography

[1] M. F. Atiyah, R. Bott, and V. Patodi, On the heat equation and the indextheorem, Inv. Math. 19 (1973), 279-330.

[2] Jacques Chazarain and Alain Piriou, Introduction to the theory of partial dif-ferential equations, Studies in Mathematics and Applications, North-HollandPublishing Company, Amsterdam-New York.

[3] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie al-gebras, Trans. Amer. Math. Soc. 63 (1948), 85-124.

[4] W. Fulton, Intersection theory, Ergebnisse der Mathematik and ihrer Gren-zgebiete, Springer, 1984.

[5] Peter B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singerindex theorem, Math. Lecture series, Publish or Perish Inc., Delaware.

[6] P. R. Halmos, Measure theory, D. Van Nostrand Company, New York.

[7] Noel J. Hicks, Notes on differential geometry, D. Van Nostrand Company, NewYork.

[8] F. Hirzebruch, Topological methods in algebraic geometry, Springer, 1966.

[9] Ji rgen Jost, Riemannian geometry and geometric analysis, Universitext,Springer.

[10] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geome-try, Vols. I and II, Interscience Publishers, New York.

[11] J: L. Koszul, Lectures on fibre bundles and differential geometry, T.I.F.R.Lectures on Mathematics, 20, Bombay.

[12] H. Blaine Lawson and Marie-Louise Michelsohn, Spin geometry, PrincetonMathematical Series, 38, Princeton University Press.

[13] Raghavan Narasimhan, Lectures on topics in analysis, T.I.F.R. Lectures onMathematics, 34, Bombay.

[14] M. V. Nori, The Hirzebruch-Riemann-Roch theorem, Michigan Math. J. 48(2000), 473-482.

311

312 Bibliography

[15] Richard Palais, Seminar on the Atiyah-Singer index theorem, Annals of Math-ematics Studies, 57, Princeton University Press, Princeton.

[16] L. Schwartz, Lectures on complex manifolds, T.I.F.R. Lectures on Mathemat-ics, Bombay.

[17] R. W. Sharpe, Differential geometry, Graduate Texts in Mathematics, 166,Springer.

[18] I. M. Singer, Differential geometry, Lectures at M.I.T., Cambridge, Mas-sachusetts.

[19] A. Weil, Introduction a 1'etude des varietes Kahleriennes, Publications del'Institut de Mathematique de l'Universite de Nancago, 1267, Hermann, Paris.

[20] R. 0. Wells, Differential analysis on complex manifolds, Graduate Texts inMathematics, 65, Springer, New York.

[21] T. J. Willmore, Riemannian geometry, Oxford Science Publications.

Index

SL(2)-modules, 282C-module, 130etale space of a presheaf, 5

a priori inequality, 251absolute derivative, 127adjoint of a differential operator,

86adjoint of a vector field, 86affine map, 132affine space, 131almost complex structure, 190almost Hermitian structure, 193Ambrose-Singer theorem, 167approximate identity, 237Ascoli's theorem, 254associated graded algebra, 64associated vector bundle, 137Atiyah-Singer index formula, 267

balanced map, 301barycentric subdivision, 110Bianchi identity, 178, 179blow-up, 297Bochner's vanishing theorem, 279Borel measure, 73, 231, 307

Cech cohomology, 115

canonical lift, 172Cartan connection, 182central simple algebra, 304change of variables formula, 82Chern character, 268Chern class, 161Chern connection, 288Chern form, 160Clifford algebra, 224Clifford group, 225Clifford structure, 226closed submanifold, 21cohomology exact sequence, 104cohomology of an elliptic

complex, 265complete vector. field, 41complexes, 57, 99composite of symbols, 70composition formula, 273conformal structure, 221connection, 126connection algebra, 64connection form, 144constant curvature, 216convex neighbourhoods, 176convolution of densities, 235covariant derivative, 127

313

314

current, 243curvature form, 129curvature space, 212

de Rham complex, 58de Rham's theorem, 117degree of a line bundle, 269density, 81, 232derivation of odd type, 55diffeomorphism, 18differentiable measure, 75differential manifold, 12differential of a map, 37differential operators of higher

order, 61differential operators on vector

bundles, 68Dirac current, 243Dirac operator, 226direct image of a sheaf, 11distribution, 243distribution section, 259distributional derivative, 243Dolbeault complex, 199Dolbeault resolution, 201

elliptic complex, 261elliptic operator, 251, 257enveloping algebra, 62equicontinuous, 308Euler class, 163exceptional divisor, 297exponential map, 174exponential map in Lie groups, 45extension of the structure group,

139exterior derivative, 54

filtered algebra, 64finiteness theorem, 260, 264first order operator, 29flabby sheaf, 94flat connection, 151

Index

formula for Laplacian, 276Fourier inversion, 241Fourier transform of densities,

238Fourier transform of Schwartz

functions, 239frame of a vector bundle, 138free product of groups, 305Friedrichs lemma, 249Frobenius theorem, 47

gauge transformation, 132Gauss' lemma., 210Gauss-Bonnet theorem, 164geodesic, 172, 207geodesic vector field, 171geometric complex, 275

harmonic decomposition, 264harmonic sections, 263Hilbert space, 308Hirzebruch class, 268Hirzebruch-Riemann-Roch

theorem, 267Hodge decomposition, 279holonomy group, 165homotopic morphisms, 101homotopy between morphisms, 58horizontal lift, 165

immersed manifold, 21index of an elliptic complex, 267index of an elliptic operator, 267infinitesimal transformation, 40injective sheaf, 94inner product, 55integrability of an almost

complex structure, 198integral curve, 41integration by parts, 87interior regularity, 251inverse image of a sheaf, 10

Index

kernel function, 229Kodaira's imbedding theorem,

292Kodaira's vanishing theorem, 289Kahler class, 206Kahler manifold, 206

Laplacian, 224Laplacian of an elliptic complex,

263Levi-Civita connection, 204Lichnerowicz' vanishing theorem,

291Lie algebra, 29Lie algebra of a Lie group, 42Lie derivative, 51, 75Lie group, 23lift of adjoint symbol, 179lift of higher order symbols, 271lifting symbols, 128local system, 32locally free sheaf, 31

morphism of complexes, 58, 99

normal bundle, 37normal coordinates, 175

orientation, 79, 185outer gauge group, 202outer linear group, 201

partition of unity, 19Pfaffian, 162Plancherel theorem, 242Poincare duality, 266polynomial differential operators,

234Pontrjagin class, 162presheaf, 2principal bundle, 135product manifold, 20product of measures, 74

315

regularisation, 229regularity theorem, 260Rellich's theorem, 254resolution, 99restriction of the structure group,

140Ricci curvature, 220Ricci endomorphism, 220Riemannian connection, 204Riemannian curvature, 211Riemannian density, 222Riemannian structure, 186

Schur's theorem, 216Schwartz functions, 234Schwartz space of functions, 237sectional curvature, 213Serre duality, 266sheaf, 2sheaf cohomology, 102short exact sequence, 304shrinking, 306simply connected space, 307singular cohomology, 58, 107singular complex, 58singular simplex, cochain, 3Sobolev chain, 250Sobolev space of sections, 259Sobolev's theorem, 233, 249soft sheaf, 94space form, 218Spin-group, 226Spin-manifold, 227stalk, 4star operator, 223Stokes theorem, 86, 118symbol of an operator, 67

tempered current, 244tempered distribution, 244theorema egregium, 216Todd class, 268torsion of a linear connection, 177

316 Index

torsion of an almost complex vanishing of elliptic solutions, 276structure, 197 vector bundle, 33

torsion of structure on manifold,196 Weingarten map, 215

twisted integers, 80 Weitzenbock formula, 279Weyl curvature, 221

The power that analysis, topology and algebra bring to geometry has revolu-tionised the way geometers and physicists look at conceptual problems. Some ofthe key ingredients in this interplay are sheaves, cohomology, Lie groups, connec-tions and differential operators. In Global Calculus, the appropriate formalism forthese topics is laid out with numerous examples and applications by one of theexperts in differential and algebraic geometry.

Ramanan has chosen an uncommon but natural path through the subject. In thisalmost completely self-contained account, these topics are developed fromscratch.The basics of Fourier transforms, Sobolev theory and interior regularityare proved at the same time as symbol calculus, culminating in beautiful results inglobal analysis, real and complex. Many new perspectives on traditional andmodern questions of differential analysis and geometry are the hallmarks of thebook.The book is suitable for a first year graduate course on Global Analysis.

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