Spivak: A comprehensive Introduction to Differential Geometry Vol 1

' -A Comprehensiue Introduction

to .

DIFFERENTIAL GEOMETRY

VOLUME ONE Third Edition

MICHAEL SPIVAK

PUBLISH OR PERISH, INC.

Houston, Texas 1999

ACKNOWLEDGEMENTS

1 am greatly indebted to

without his encouragement these volumes would have remained a short set of mimeographed notes

and

without his TE/X program they would never have become typeset books

PREFACE

T he preface to the first edition, reprinted on the succeeding pages, excused this book's deficiencies on grounds that can hardly be justified now that

these "notes" tmily have become a book. At one time I had optimisticaliy planned to completely revise ali this material

for the momentous occasion, but I soon realized the futility of such an under- taking. As I examined these five volumes, written so many years ago, I could scarcely believe that I had once had the energy to learn so much material, or even recall how I had unearthed some of it.

So I have contented myself with the correction of errors brought to my attention by diligent readers, together with a few expository ameliorations; among these is the inclusion of a translation of Gauss' paper in Volume 2.

Aside from that, this third and final edition diffiers from the previous ones only in being typeset, and with figures redrawn. I have merely endeavored to typeset these books in a manner befitting a subject of such importante and beauty

As a final note, it should be pointed out that since the first volumes of this series made their appearance in 1970, references in the text to "recent" results should be placed in context.

HOW THESE NOTES CAME TO BE

m d how they d id not come t o be a book

For many years 1 have wanted t o wr i te t h e Great American D i f f e r e n t i a l

Geometry book. Today a d i l e m a confronts any one i n t e n t on penetra t -

í ng t he mysteries of d i f f e r e n t i a l geometry. On t h e one hand, one can

consult numerous c l a s s i c a l t reatments of t h e subject i n an at tempt t o

f orm some idea how the concepts wi thin it developed. Unf o r tuna te ly ,

a modern mathemat i c a l education tends t o make c l a s s i c a l mathematical

works inaccess ib le , p a r t i c u l a r l y those i n d i f f e r en t i a lgeome t ry . Onthe

o ther hand, one can now f i n d t e x t s a s modern i n s p i r i t , and a s c lean i n

exposi t ion, a s Bourbaki's Algebra. But a thorough study of these books

usual ly leaves one unprepared t o consu l t c l a s s i c a l works, and e n t i r e l y

ignorant of t h e r e l a t i o n s h i p between e legant modern cons t ruc t ions and

t h e i r c l a s s i c a l c o u n t e r p a r t s . Most s tuden ts eventual ly f i n d t h a t t h i s

ignorance of the r o o t s of t h e subject has i ts p r i ce -- no one denies t h a t

modern d e f i n i t i o n s a r e c l e a r , e legan t , andp rec i s e ; i t ' s j u s t t h a t i t ' s

impossible t o comprehend how any one ever thought of them, And even a f t e r

one does master a modern treatment of d i f f e ren t i a l geometry , other modern

t reatments o f ten appear simply t o b e a b o u t t o t a l l y d i f f e r e n t sub j ec t s .

Of course, these remarks merely mean t h a t no matter how wel l some of t h e

present day t e x t s achieve t h e i r ob jec t ive , 1 never theless f e e l t h a t an

i n t r o d u c t i o n t o d i f f e r en t i a lgeome t ry ought t o have q u i t e d i f f e r e n t aims.

There a r e two main premises on which these notes a r e based. The f i r s t

premise i s t h a t it is a b s u r d l y i n e f f i c i e n t t o eschewthe modern language

of manifolds, bundles, forms, e t c . , whichwas developedprecise ly i n

order t o r i g o r i z e t h e concepts of classicaldifferentialgeometry.

Rephrasing every t h ing i n more element ary t e m s involves incred ib le

contor t ions which are not only unnecessary , but misleading. The work

of Gauss , f o r example, which uses in f in i t es imals throughout , i s most

na tura l ly rephrased i n terms of d i f f e r e n t i a l s , even i f it is possible

t o rewri te it i n terms of der iva t ives . For t h i s reason, the e n t i r e

f i r s t volume of these notes is devoted t o the theory of d i f f e r en t i ab l e

manifolds, the basic language of modern d i f f e r e n t i a l geometry. This

language is compared whenever possible with the c l a s s i c a l l a n p a g e , so

t h a t c l a s s i c a l works can then be read.

The second premise f o r these notes i s t h a t i n order f o r an in t roduc t ion

t o d i f f e r e n t i a l geometry t o expose t h e geometric aspect of the sub jec t ,

an h i s t o r i c a l approach is necessary; there is no point i n introducing

t h e curvature tensor without explaining how it was invented and what it

has t o do with curvature. 1 personally f e l t t ha t 1 could never acquire

a s a t i s f ac tory understanding of d i f f e ren t iab le geometry u n t i l 1 read

the o r i g i n a l works. The second volume of these notes gives a de t a i l ed

exposit ion of the fundamental papers of Gauss and Riemann. Gauss' work

is now avai lable i nEng l i sh (General Invest igat ions of Curved Surfaces;

Raven Press ) . There a re a l s o two Eng l i sh t r ans l a t i ons of Riemann's work,

but 1 have provided a (very f ree) t r ans l a t i on i n the second volume .

Of course, 1 do not th ink t ha t one should f ollow a l 1 the i n t r i c a c i e s of

t he h i s t o r i c a l process, with its inev i tab le dupl icat ions and f a l c e leads

What i s intended, ra ther , is a presenta t ion of the subject along the

l i n e s which i t s development might have followed; a s Bernard Morin s a i d

t o me, the re is no reason, i n mathematics any more than i n biology, why

ontogeny must recap i tu la te phylogeny. When modern terminology f i n a l l y

i s introduced, i t should be a s an outgrowth of t h i s (mythical) h i s t o r i c a l

development, And a l 1 the major approaches have t o be presented, f o r they

were a l 1 r e l a t ed t o each other , and a l 1 s t i l l play an important r o l e .

A t t h i s point 1 am reminded of a paper described i n Littlewood' S

Mathematician' S M i sce l lany . The paper began "The aim of t h i s paper i s

t o prove . . ." and it t ransp i red only much l a t e r t h a t t h i s aim was not

achieved (the author hadn ' t claimed t h a t it was) . What 1 have outl ined

above i s the content of a book the r ea l i z a t i on of whose basic plan and t h e

incorporation of whose d e t a i l s would perhaps be impossible ; what 1 have

wr i t t en is a second or t h i r d d r a f t of a preliminary version of t h i s book,

1 have had t o r e s t r i c t myself t o what 1 could wr i te and l ea rn about within

the present academic year, and a l l r e v i s i o n s and correct ions have h a d t o

be made within t h i s same period of time. Although 1 may some day be able

t o devote t o i t s completionthe time which suchanunder taking deserves,

a t prssent 1 have no plans f or t h i s. Consequent l y , 1 would l i k e t o make

these notes ava i lab le now, desp i te t h e i r de f ic ienc ies , andwith a l 1 the

compromises 1 learned t o make i n t h e ea r ly hours of the morning.

These notes were wr i t t en while 1 was teaching a year course i n d i f -

ferent ia lgeometry a t Brandeis Universi ty, during the academic year

1969-70. The course was taken by s i x juniors and sen iors , and audited by

a few graduate s tudents . Most of them were fami l ia r with t h e mater ia l i n

Calculus - on Manifolds, which i s e s sen t i a l l y regarded a s a p re requ is i t e .

More p rec i se ly , t he complete p re requ is i t es a r e advanced calculus using

l i nea r algebra and a bas ic knowledge of metric spaces. An acquaintance

with topological spaces i s even b e t t e r , s ince it allows one t o avoid the

t e chn i ca l t r oub l e s which a r e sometimes r e l e g a t e d t o t h e Problems, but 1

t r i e d hard t o make everything work without i t .

The mater ia l in thepresentvolume was covered i n t h e f i r s t t e r m , except

f o r Chapter 10, which occupied the f irst couple of weeks of the second

term, and Chapter 11, which was not covered i n c l a s s a t a l l . We f ound it

necessary t o t ake r e s t cures of near ly a week af t e r completing Caapters 2 ,

3, and 7 . The same mate r ia l could e a s i l y be expanded t o a f u11 year course

i n manifold theory with a pace t h a t few would describe as excessively

le i sure ly . 1 am gratef u1 t o the c lass f ú r keeping up w i t h my accelerated

pace, f o r otherwise the second half of these notes would not have been

wri t ten. 1 am a lso extremely gra te fu l t o Bichard Pa la i s , whose expert

knowledge saved me innumerable hours of labor.

TABLE OF CONTENTS

Although the chapters are not divided into sections. the listing for each chapter @ves some indication

which topics are treated. and on what pages .

CHAPTER 1 . MANIFOLDS

Elementary properties of manifolds . . . . . . . . . . . . . . . . 1 Exarnples of manifolds . . . . . . . . . . . . . . . . . . . . . 6

. . . . . . . . . . . . . . . . . . . . . . . . . . Problems 20

CHAPTER 2 . DIFFERENTI AL STRUCTURES

Cm structures . . . . . . . . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . . . . . . Cm functions 31

. . . . . . . . . . . . . . . . . . . . . . Partial derivatives 35 Critica1 points . . . . . . . . . . . . . . . . . . . . . . . . 40 Immersion theorems . . . . . . . . . . . . . . . . . . . . . 42 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . 50 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER 3 . THE TANGENT BUNDLE

The tangent space of IRn . . . . . . . . . Tlle tangent space of an imbedded manifold . Vector bundles . . . . . . . . . . . . . .

. . . . . The tangent bundle of a manifold Equivalence classes of curves. and derivations Vector fields . . . . . . . . . . . . . . . Orieil tation . . . . . . . . . . . . . . . Addendum . Equivalence of Tangent Bundles Problems . . . . . . . . . . . . . . . .

xiu

CHAPTER 4 . TENSORS

. . . . . . . . . . . . . . . . . . . . . . . The dual bundle 107 . . . . . . . . . . . . . . . . . The difkrential of a function 109

. . . . . . . . . . . . . . Classical versus modern terminology 1 1 1 . . . . . . . . . . . . . . . . . . . . . Mul tilinear functions 115

Covariant and contravariant tensors . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . . Mixed tensors. and contraction 121

. . . . . . . . . . . . . . . . . . . . . . . . . . Problems 127

CHAPTER 5 . VECTOR FIELDS AND DIFFERENTIAL EQUATIONS

. . . . . . . . . . . . . . . . . . . . . . . . Integral curves 135 . . . . . . . . . . . . . . . Existence and uniqueness theorems 139

. . . . . . . . . . . . . . . . . . . . . . . . The local flow 143 . . . . . . . . . . . One-parameter groups of difkomorphisms 148

. . . . . . . . . . . . . . . . . . . . . . . . Lie derivatives 150 . . . . . . . . . . . . . . . . . . . . . . . . . . . Brackets 153

. . . . . . . . . . . . . . Addendum 1 . Dserential Equations 164 Addendum 2 . Parameter Curves in Two Dimensions . . . . . . . 167

. . . . . . . . . . . . . . . . . . . . . . . . . . Problems 169

CHAPTER 6 . INTEGRAL MANIFOLDS

Prologue; classical integrability theorems . . . . . . . . . . . . 179 . . . . . . . . . . Local Theory; Frobenius integrability theorem 190

Global Theory . . . . . . . . . . . . . . . . . . . . . . . . 194 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 198

CHAPTER 7 . DIFFERENTIAL FORMS

. . . . . . . . . . . . . . . . . . . . . Alternating func tions 201 The wedge product . . . . . . . . . . . . . . . . . . . . . . 203

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms 207 DSerential of a form . . . . . . . . . . . . . . . . . . . . . 210

. . . . . . . . . Frobenius integrability theorem (secondversion) 215 . . . . . . . . . . . . . . . . . . . . Closed and exact forms 218

. . . . . . . . . . . . . . . . . . . . . Tlle Poincaré Lemma 225 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 227

CHAPTER 8 . INTEGRATION

Classical line and surface integrals Integrals over singular k-cubes . . The boundary of a chain . . . .

. . . . . . . . Stokes' Theorem Integrals over manifolds . . . . . Volume elements . . . . . . . .

. . . . . . . . Stokes' Theorem de Rliarn cohomology . . . . .

. . . . . . . . . . . Problems

CHAPTER 9 . RIEMANNIAN METRICS

. . . . . . . . . . . . . . . . . . . . . . . . Inner products 301 . . . . . . . . . . . . . . . . . . . . . Riemannian metrics 308

. . . . . . . . . . . . . . . . . . . . . . . LRngth of curves 312 The calculus of variations . . . . . . . . . . . . . . . . . . . 316 The First Variation Formula and geodesics . . . . . . . . . . . 323

. . . . . . . . . . . . . . . . . . . . . The exponential map 334 . . . . . . . . . . . . . . . . . . . . Geodesic com ple teness 341

Addendum . Tubular Neighborhoods . . . . . . . . . . . . . . 344 . . . . . . . . . . . . . . . . . . . . . . . . . . Problems 348

CHAPTER 1 O . LIE GROUPS

Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 371 L f t invariant vector fields . . . . . . . . . . . . . . . . . . . 374

. . . . . . . . . . . . . . . . . . . . . . . . . Lie algebras 376 Subgroups and su balgebras . . . . . . . . . . . . . . . . . . 379

. . . . . . . . . . . . . . . . . . . . . . . Homomorphisms 380 . . . . . . . . . . . . . . . . . . . One-parame ter subgroups 382

. . . . . . . . . . . . . . . . . . . . . The exponential map 384 Closed subgroups . . . . . . . . . . . . . . . . . . . . . . 391

. . . . . . . . . . . . . . . . . . . . . LRft invariant forms 394 . . . . . . . . . . . . . . . . . . . . . . Bi-invariant metrics 400

. . . . . . . . . . . . . . . . . . . The equations of structure 402 . . . . . . . . . . . . . . . . . . . . . . . . . Probleins 4 0 6

xuz Conlents

CHAPTER 1 1 . EXCURSION IN T H E REALM OF ALGEBRAIC TOPOLOGY

. . . . . . . . . . . . . . . . Complexes and exact sequences 419 . . . . . . . . . . . . . . . . . The Mayer-Vietoris sequence 424

. . . . . . . . . . . . . . . . . . . . . . . . Triangulations 426 . . . . . . . . . . . . . . . . . . . . The Euler characteristic 428

. . . . . . . . . . Mayer-Vietoris sequence for compact supports 430 . . . . . . . . . . . . . . . . . The exact sequence of a pair 432

. . . . . . . . . . . . . . . . . . . . . . . Poincaré Duality 439

. . . . . . . . . . . . . . . . . . . . . . . The Thom class 442 . . . . . . . . . . . . . . . . . . . . . Index oí' a \lector field 446

. . . . . . . . . . . . . . . . . . . Poincaré-Hopf Theorem 450 . . . . . . . . . . . . . . . . . . . . . . . . . . Problems 453

APPENDIX A

. . . . . . . . . . . . . . . . . . . . . . . . To Chapter I 459 . . . . . . . . . . . . . . . . . . . . . . . . . . Problems 467

. . . . . . . . . . . . . . . . . . . . . . . . To Chapter 2 471 . . . . . . . . . . . . . . . . . . . . . . . . . Problems - 4 7 2

. . . . . . . . . . . . . . . . . . . . . . . . To Chapter 6 473 . . . . . . . . . . . . . . . . . . . . . To Chapters 7.9. 10 474

. . . . . . . . . . . . . . . . . . . . . . . . . . Problem - 4 7 5

. . . . . . . . . . . . . . . . . . . . . . . NOTATlON INDEX 477

. . . . . . . . . . . . . . . . . . . . . . . . . . INDEX 481

A Comprehemive Introduction

t o

DIFFERENTIAL GEOMETRY

VOLUME ONE

CHAPTER 1

MANIFOLDS

he nices t example of a metric space is Euclidean n-space R", consisting of T al1 n-tuples x = ( x l , . . . ,Y) with each x' E IR, where R is the set of real numbers. Whenever we speak of R" as a metric space, we s h d assume that it has the "usual metric"

unless anotlier metric is explicitly suggested. For n = O we will interpret R0 as the single point O E R.

A manifold is supposed to be "locally" like one of these exemplary metric spaces R". To be precise, a manifold is a metric space M with the following proper ty:

If x E M, then there is some neighborhood U of x and some integer n 2 O such that U is homeomorphic to R".

The simplest example of a manifold is, of course, just Rn itself; for each x E R" we can take U to be ali of R". Clearly, R" supplied with an equivalent metric (one which makes it homeomorphic to R" with the usual metric), is also a manifold. Indeed, a hasty recollection of the definition shows that anything l-iomeomorphic to a manifold is also a manifold-the specific metric with which M is endowed plays almost no role, and we s h d alrnost never mention it.

[If you know anything about topological spaces, you can replace "metric space7' by "topological space" in our definition; this new definition allows some pathological creatures which are not metrizable and which fad to have other properties one might carelessly assume must be possessed by spaces which are locally so nice. Appendix A contains remarks, supplementing various chapters, which should be consulted if one allows a manifold to be non-metrizable.]

The second simplest example of a manifold is an open bali in R"; in this case we can take U to be the entire open ball since an open ball in R" is homeomorphic to R". This example immediately suggests the next: any open

subset V of R" is a manifold-for each x E V we can choose U to be some open ball with x E U c V. Exercising a mathematician's penchant for generalization,

we immediately announce a proposition whose proof is left to the reader: An open subset of a manifold is also a manifold (calIed, quite naturally, an open submanifold of the original manifold).

The open subsets of R" already provide many dXerent examples of manifolds Cjusi 11ow maiiy is the subject of Problem 24), though by no means all. Before proceeding to examine other examples, which constitute most of this chapter, some preliminary remarks need to be made.

If x is a point of a manifold M, and U is a neighborhood of x (U contains some open set V with x E V) which is homeomorphic to RR by a homeomorphism #: U + R", then #(V) c R" is an open set containing #(x). Conse-

quently, there is an open ball W with #(x) E W c $(V). Thus x E #-'(w) c V c U. Since # : V + R" is continuous, the set #-'(W) is open in V, and thus open in M; it is, of course, homeomorphic to W, and thus to R". This complicated little argumentjust shows tliat we can always choose the neighborhood U in our definition to be an open neighborhood.

Witli a little tliought, it begns to appear that, ii-i fact, U mtwf be open. But to prove this, we need the following theorem, staied I-iere without proof.*

1. THEOREM. If U c Rn is open and f : U + Rn is one-one and continuous, then f (U) c IR" is open. (It foUows that f (V) is open for any open V c U, so f -' is continuous, and f is a homeomorphism.)

Theorem 1 is called "Invariance of Domain", for it implies that the property of being a "domain" (a connected open set) is invariant under one-one continuous maps into R". The poof that the neighborhood U in our definition must be open is a simple deduction from Invariance of Domain, left to the reader as ail easy exercise (it is also easy to see that if Theorem 1 were false, then there would be an example where the U in our definition was not open).

We next turn our attention to the integer n appearing in our definition. Notice that n may depend on tlie point x. For example, if M c IR3 is

ihen we can clioose n = 2 for points in MI and ri = 1 for points in M2. Tliis

example, by the way, is an unnecessarily complicated device for ~roducing one manifold from two. In general, given Mi and Ms, with metrics di and d2, we can firsi replace each di witli an equivalent metric di such that d i ( x , y ) .E 1 for al1 X, y E Mi; for example, we can define

* Al1 proofs require some amount of macliiner)' The quickest routes arc probably provided by Vick, Hont o1o.g TJleory and Massey, Singular Hont ology Theo y. An old-fas hioned , but pleasanily geometric, treátment may be found in N e w a n , %pologp ofPlenr Se&.

Then we can define a metric d on M = Mi U M2 by

di(x, y) if therc is some i such that x , y E Mi d(x,v) =

o therwise

(we assume that Mi and M2 are disjoint; if not, they can be replaced by new sets which are). In tlie new space M, both Mi and M2 are open sets. If MI and M2 are manifolds, M is clearly a manifold also. This coiistruction can be applied to any number of spaces-even uncountably many; the resulting metric space is called the disjoint union of the metric spaces Mi. A disjoint union of manifolds is a manifold. In particular, since a space with one point is a manifold, so is any discrete space M, defined by the metric

Although ddfeient 11's may be required at ddferent points of a manifold M, it would seem tliat only one n can work at a giveii point x E M. For the proof of this ii-ituitively obvious assertion we llave recourse once again to Invariance of Domain. As a first step, we note that R" is not liomeomorphic to Rm wlien 11 # 111, for if 11 > 111, t1-iei-i iliei-e is a one-oiie continuous map from Rm into a iloii-open subset of R". The furtlier deduction, tliai the n of our definition is uilique at each s E M, is lefi to the reader. This unique n is called the dimension of M at x. A manifold has dimension 11 01- is n-dimensional or is an a-manifold if it lias dimension i~ at each point. It is convenient to refer to tlie inaiiifold M as M" wlieii we want to indicaie tliai M has dimensioii n.

Consider once more a discrete space, whicli is a O-dimensional maiiifold. The oiily compact subsets of such a space are finite subsets. Consequently, an ui-icountable discrete space is not o-compact (it cannot be written as a countable uiiioi~ of com11act sul~sets). Tlie same plieiiomenon occurs with higher-dime~i- sional manifolds, as we see by taking a disjoint uilion of uncountably many manifolds liomeomorpl~ic to R". In tliese examples, however, h e manifold is not coi~i-iected. Mie \vil1 often ileed to know that tliis is the only way in wliicl-i o-compactness can fail to hold.

2. THEOREM. If X is a connecied, Iocally coinpaci metric space, tl-ien X is o -compact.

PR001;. Foi- each 1- E X cconsider tl-iose numbers r > O such that the closed ball

(y E X : d(x-,y) 5 i ' )

is a compact set (there is at least one such r > O, since X is locally compact). The set of al1 such I. > O is an interval. If, for soine A-, tliis set includes all r > 0 , then X is o-compact, since

If not, then for each x E X define r ( x ) to be one-half the least upper bound of al1 su& r.

The triangle inequality implies that

so that

which implies that

Jnterchanging xi and x2 gives

so tlle function r : X + IR is continuous. This has tlle following important consequence. Suppose A c X is compact. Let A' be the union of all closed balls of radius r (y) aiid cei-iter y, for all 11 E A. Then A' is also compact. The poof is as follows.

Let z1,22,23,. . . be a sequence in A'. For eacli i there is a yi E A such that zi is in the ball of radius r(yi) with center yi. Since A is compact, some subsequence of the y i , which we might as well assume is the sequence itself, converges to some point y E A . Now the closed ball B of radius and

center y is compact. Since yi :i y and since the function r is continuous, eventually the closed b d s

are contained in B. So the sequence ri is eventually in the compact set B, and consequently some subsequence converges. Moreover, the limit point is actually in the closed ball of radius r(y) and center y (Problem 10). Thus A' is compact.

Now let AO E X and consider the compact sets

Their union A is clearly open. It is also closed. To see this, suppose that x is a point in the dosure of A . Tlicn there is sorne y E A with d(x, y) c i r ( x ) .

This shows tliat jf y E A,, tlien x E A,', so x E A . Siiice X is coi-ii-iected, and A # 0 is operi a ~ i d closed, it must be tliat X = A ,

wliich is o-compact. 6

After this liassle with point-se t topology? we present the long-promised exarnples of manifolds. Tlie oiily connected i-manifolds are tlie line IR and the circle, or I -dimensional sphere, S', defined by

The function f : (0,2ir) + S' defined by f (O) = (cos O, sin O) is a homeomorphism; it is even continuous, though not one-one, on [0,2n]. We will orten denote the ~ o i n t (cos O, sin O) E S' simply by O E [O, 2x1. (Of course, it is always necessary to check that use of this notation is valid.) The function g : (-s, s ) + S', defined by the same formula, is also a homeom~r~hism; together with f it shows that S' is indeed a manifold.

There is another way to prove this, better suited to generalization. The projection P from the point (O, 1) onto the line IR x (-1) c R x R, illustrated in

the above diagram, is a homeomorphism of S' - ((0, 1) ) onto R x (- 1): this is poved most simply by calculating P : S' - ((O,])) + R x (- 1 ) explicitly. The point (O, 1) may be taken care of similarly, by projecting onto R x (I), or it suffices to note that S' is "homogeneous"-there is a homeomorphism taking any point into any other (namely, an appropriate rotation of IR2). Considerations similar to these now show that the n-sphere

is an n-maiiifold. The 2-spliere S2, commonly known as "the sphere", is our first example of a compact 2-manifold or surface.

From these few manifolds we can already construct many others by noting tl-iat if Mi are manifolds of dimension ir j (i = 1,2), then MI x M2 is an (ni + n2)- manifold. In particular

n times

is called the n-torus, while S' x S' is commonly called "the torus". 1 t is obviously liomeomorphic to a subset of IR4, and it is also liomeomorphic to a certain subse t of IR3 which is what most people have in mind when they speak of

"the torus": This subset may be obtained by revolving the circle

{(O, y,a) E IR3 : (JJ - 1)2 + z2 = 1/41

around the z-axis. The same construction may be applied to any 1 -manifold

coiitaiiied ili {(O, jp, r) E IR3 : 1): > O). Tlie resul tiiig surface, called a surface of revolution, lias coinpoiients lioineomorpl~ic either to tlie toi-us or to the cyliiidei- Si x R, tlie latter of which is also homeomorphic to the annulus, the region of tlie plai-ie contail-ied be tweeii two concentric circles.

l'hc i-iest simplest compact 2-maiiifold is tlie 2-lioled torus. To provide a more

explicit description of tlie 2-lioled torus, it is easiest to begin witli a "l-iandle", a space liomeomorphic to a torus witli a hole cut out; more precisely, we tlirow

away d the points on one side of a certain circle, wbich remains in our llandle, and which will be referred to as the boundary of the handle. The 2-holed

torus may be obtaiiied by piecing two of tliese together; it is also described as

the disjoint uiiion of two handles witli corresponding points on the boundaries "identified".

Tlie >I-holed torus may be obtained by re~eated applications of this procedure. It is homeomorpliic to iIie space obtained by starting with the disjoint

union of i 7 liandlcs and a spliere with n holes, and then identif9ng points on the boundary of the ith Iiandle with corresponding points on the ith boundary piece of the sphere.

There is one 2-manifold of which most budding mathematicians make the acquaintance wheli tliey still know more aboui paper and paste than about

metric spaces-ihe famous Mobim S@, which you "make" by giving a strip of paper a half twist before pasting its ends together. This can be described

analytically as the image in R3 of the function f : [O, 2x1 x (- 1,l) + R3 defined

by e e f (B, i )= (2cosB+icos~cosB, 2sinB+icosIsinB,

If we define: f on [O, 2x1 x [- I,1] instead, we obtain the Mobius strip with a boundary; as iiivestigation of the paper model will show, this boundary is homeomorphic to a circle, not to two disjoint circles. With our recently introduced terminology, tlie Mobius strip caii also be described as [O, I] x (- 1,l) with (O, i) and ( 1, -1) "identified".

We have not yet liad to make precise this notion of ccidentification'y, but our next example will force tlie issue. We xvisli to identify eacli point x E s2 witli

its aiitipodal poiiit -x E S2. The space whicli results, the projecúve plane, IP2, is a lot harder io visualize than previous examples; indeed, there is no subset of IR3 which represents it adequately.

The precise definition of IP2 uses the same trick that mathematicians always use when they want two things wliich are not equal to be equal. The p0in.h- of P 2 are defilied to be the sets ( p , - p ) for p E s2. We wilI denote this set by [ p ] E IP2, so tliat [ - p ] = [ p ] . We thus have a map f : S2 + IP2 @ven by f ( p ) = [ p ] , for whicl-i f ( p ) = f ( q ) implies p = f q . We will postpone for a while the problem of defining the metric giving the distance between two points [ p ] and [q ] , but we can easily say wliat the open sets will turn out to be (and this is all you need to know iil order to check that P 2 is a surface). A subset U c p2 will be open if aiid only if f -' ( U ) c S2 is open. This just means that the open sets of P 2 are of tlie form / ( V ) where V c S2 is an open set with the addiiional important property that if it contains p it also contains - p .

In exacily ihe same wa): we could have defined ihe poinis of the Mohius sirip M io be

al1 points (s ,f) E (O, 1) x (-1,I) iogeiher wiih

allseis {(O,r),(l,-111, denotedby[(O,f)Ior[(I,-111.

(S, f ) if s # O, 1

[(s,f)] i f s = O or 1,

and U c M is open if and only if f -' (U) c [O, 11 x (- 1, l ) is open, so tliat tl-ie open seis of M are of the form f ( V ) ~rliere V is open and coniains (S, -1)

wlienevcr it coniains (S, f ) for s = O or 1.

To gei an idea of wliat p2 looks Iike, we can make tliings easier for ounelves by first ilirowiiig away all poiiiis of s2 helow tlie (x, y)-plane, siiice iliey are ideniified with poinis ahmre ihe (x, y)-plane anyway. This leaves the upper I-icniisplier-e (including tlie bounding ci~-cl c), wliicli is Iiomeoinor~~li ic io ilie disc

aiid wc must idcntify eacli p E Si wiili - p E S'. Squaring tliiiigs o K a bit, tliis is ilie same as identifying points on ilie sides of a square according to tlie sclienie shown below (poinis on sides wiil-i ihe same lahel are ideniified in sucli a way iliat ille heads of tlie arrows are ideniified wiih each otlier). Tlie dotied liiies iii ihis piciui-e are h e key io undersiaiiding IP2. If we disiori ilie

region beiween tl-iem a bit we see tliat the froni pari of B followed by t l ~ e I~ack part of A, ai the upper left, is to be ideniified wiih the carne thing ai the 10wer righi, in reverse direction; in other wods, we obtain a Mobius sirip with

a boundary (narnely, ihe doited line, which is a single circle). If this h4obius strip is removed, \ve are lefi wiih two pieces \vI~ich can be rearranged to form something homeomorphic to a disc. The projectiw plane is thus oh<ained from

tl-ie disjoini union of a disc and a Mobius sirip with a boundary, by identifying points on the boundary and poinis on the boundary of the disc, both of which are circles. Thus io make a model of iP2 we jusi have to sew a circular piece of cloth and a clotlr Mobius strip iogci1ie1- along ifieir cdges. Unforiunately a litile experimentaiion will convince you ilrai ihis cannot be done (without having tlle iwo pieces of doth pass through each other}.

The subsei of IR3 obiained as the union of ilie Mobius sirip and a disc, alihough noi hoiiicoiiioipliic to iP2, can siill be described mathematically in terms

of IP2. There is clearly a coiitinuous funciion /: iP2 + IR' wliose image is this subsei; moreovei; althougli f is not one-one, ii is locally one-one, that is, every point p E IP2 has a neigliborhood U on whicli f is one-one. Sucli a function f

is called a topological immersion (the single word "immersion" has a more v e - cialized meaiiiiig, explained in Chapter 2). IVe can thus say thai P2 can be topologicaIIy immersed in W ', al though noi topologically imbedded (ihere is no

liomeomoi-phism f fiom PZ LO a subset of W3), In W4, liowever, wiih an extra dimension to play around ivith, the disc can be added so as noi io interseci the Mobius strip.

Another topological immersioii of P2 in W3 can be obtained by first immersing ihe Mobius sirip so iflat its boundary cirde lies iri a plane; ibis can be done in the follo~ving way. The figures below show thai the Mobius sirip may be ob- tairled from an annulus by identifying opposite points of the inner circle. (This is also obvious from ihe fact thai ihe A4obius sirip is the projective plane wiih a disc removed.} TIlis inner circle can l ~ e replaced by a quadrilateral. When ihe

resuliing figure is di.a\vn up irito 3-space and the appropriatc ideniifications are niade we ol~tairi ibe "cross-cap". The cross-cap iogether wiih the disc ai tlie bo~tom is a iopologically immersed P2.

'llic onc gap in ihe precedii-ig discussion is ilre definiiion of a metric for P2. 'l'lir niissiiig nleir-ic can he supplied by an appeal to Problem 3-1, whicl~ will I;itri. l ~ e used quite often, and which the reader should peruse sometime hefoi-e 1-i.;icliiig Cliapter 3. Rouglily speaking, ii sliows ihai iliiiigs llke PZ, whidi ougli r t i ) be inanifolds, are. (Those who know ahout iopological spaces will recognize ii 21s a disguised case of ille Urysohn Metrizaiion Theorem.) For the preseiii, Iio~*rver, we will obtain our metric by a irick thai simultaneously provides an iiiibidding of P2 in JR4. Coiisider ihe fuiiciion f : s2 + JR4 defined by

f (x, y, 5 ) = (yz, xz, xy, x2 + 2y2 + 32) .

( :lr.asly f (p) = f (-p). We mainiain ihai f (p) = f (q) implies ihat p = f y. 'li) p~-ovetl~is~ suppose ihai f (xi y , ~ ) = f (ai b, c). We have, firsi of ail

I 1' a , b , c # 0, ihis leads to

r4

Now

xy = ab.

lience

(3)

Using (2), ihis gives

so s = f a . Similarly, Mre obiain y = f b , z = f b , wiih ihe same sign (which comes from (3)) holding for dl three equations. In this case we have proved our conteniion ~riihout even using the fourth coordinate of f. Now suppose a = 0. If x # 0; then (1) would immediately give y = z = O, so that

(x, y, z ) = (f l , O , O).

But y = ; = O implies (by (1) again) that bc = O, so b = O or c = O and

These equations clearly con tradict

Thus x = O aiso, and Mre have

But (6) implies ihat

2 2). + 3z2 = zY2 + 3(1 - y2) 2 = 3 - y ,

and similarly for b and c, so (5) gives

Now (4) $ves

(tlris holds eveil if y = b = 0, since then 1, c = f 1). Clearly, (4) also shows thai thc same sign holds in (7) arid (8). which completes the proof.

Sinre f (p) = f (q) precisely wlieii p = f q , we can define j : P2 + R4 b!,

This nlap is ont-oi~e and wc can LISC it to define the meti-ic in IP2:

Then one can check that the open sets are indeed the ones described above. By ihe way, the map g : P' + W3 defined by tlie firsi 3 componenis of j;

is a topologicai immersion of P' in W 3 . The image in W3 is Steiner's "Romaii surface".

Wiih the new surface P2 at our disposal, we can create other surfaces in t he same way as the n-holed torus. For example, to a handle we can attacli a projective space 14th a hole cut out, or, what amounts to the same thing, a h4obius strip. The closesi we can come io picturing ihis is by drawing a cross- cap sticking on a iorus. We can dso join together a pair of projective planes with holes cut out, whicll amounts to sewing two Mobius strips together along tlieir boundary. Although this can be pictured as two cross-caps joined together, it has a nicer, and famous, representation. Consider the surface obtained from ilre square with identifications indicated below; it may aiso be obtained from tlie cylinder [O, l] x S' by identifying (O, x) E [O, l] x S' with (1, x' ) , where x' is ihe reflection of x through a fixed diameter of ihe circle. Notice that ihe

identifications on the square force PI, P2, P3, and f i to be identified, so that the set {Pl , P2, P 3 , P4) is a single point of our new space. The doited Iines

beloy dividing the sides into thirds, form a single circle, which scparates the surface into two parts, one of which is shaded.

Rearrangemeni of ihe iwo parts shows thai ihis surface is precisely iwo Mobius strips witb corresponding points o11 iheir boundary ideniified. The description iti terms of [O, 11 x S' immediaiely suggests an immersion of the surface. Turriing one e i ~ d of the cylinder around and pushing ii through itself orients ihe left-hand boundary so that (O, x) is direcily opposite (1, x') , io which it can then be joined, forming the "Klein bottle".

1;xamples of higher-dimensional manifolds ~ ~ i l l noi be treated in nearly such ifvtail, I~ui! in addition to the famiiy of 11-manifolds S", we wiIl mention ihe i.i.lated fa mil^ of "projeciive spaces". Projective 11-space P" is defined as ihe i.olleciion of al1 sets (p, -p) for p E S". The description of the open sets in P" is ~x.ecisely analogous to tlie descripiion for P2. Alihough these spaces seem to li)i-in a family as regular as the family S", we ~cill see Iater that the spaces P" liii. cven n dXer in a very important way from the same spaces for odd n.

One further definition is needed io complete this introduction to manifolds. \%. have dready discussed some spaces which are noi manifolds only because tliey have a "boundarf"' for example, the h4obius strip and the disc. Points r i i i these "boundaries" do not have neighborhoods homeomorphic to IRn, but t l rey do llave neighhorhoods homeomorphic io an important subse t of IR". Tlie (closed) hdf-space W" is defined by

W" = {(xt , ..., x") E IRn : x" 2 O}.

!'I manifold-with-boundary is a metric space M with the following property:

If x E M , ihen there is some neighborhood U of x and some integer rr > O such ihai U is homeomorphic to either R" or W".

A poini in a manifold-with-boundary canriot have a neighborhood liomeo- mor-phic to both IR" and W" (Invariance of Domain again); we can therefore distirlguish ihose poinis x E M having a neighborhood homeomorphic to Hn. .]'he sei of al1 such x is cdled ihe boundary of M arid is denoted by aM. If M is ;ictually a manifold, tlien 8M = 0. Notice that if M is a subset of IR", then aM is noi necessarily the same as the boundary of M in the old sense (defined for iiny subset of IRn); indeed, if M is a manifold-wiih-boundary of dimension < n, tlien ali poinis of M wiil be boundary points of M.

If manifolds-~lith-boundary are studied as frequently as manifolds, it becomes I~oibersome io use tl-iis long designaiion. Ofien, ihe word "manifold" is used ior "manifold-~~ith-11oundary"- A manifold in our sense is then cdled "non- I~ounded"; a non-l~ounded compact mailifold is cailed a "closed manifold". We \vil1 stick to the other termiiiology, hui ~ l i l l sometimes use "bounded manifold" insiead of "manifold-with-boundary".

PRQBLEMS

1. Slio\v that if d is a metric on X, then both d = d/(l + d ) and d = min(1,d) are dso metrics and ihai tliey are equivdent to d (Le., the identiv map 1 : (X, d ) + (X, d) is a homeomorphism).

2. Jf (Xi, di) are metric spaces, for i E 1, with meirics di < 1, and Xi n Xj = 0 for i # j , tl~en (X, d ) is a meiric space, where X = Ui Xj, and d(x, y) = di(x, 1.) if A+, y E Xi for some i, while d(x, y) = 1 otherwise. Each Xi is an open subsei of X, and Y is homeomorphic to X if and only if Y = Ui Yi where the Yi are disjoint open sets and Yi is homeomorphic to Xi for each i. T l ~ e space (X, d ) (or any space homeomorphic io it) is cdled the disjoint union of ihe spaces Xi.

3. (a) Exre177 manifold is locally compaci. (b) Every manifold is locdly pathwise connecied, and a connected manifold is pathwise connected. (c) A connecied manifold is arcwise connected. (A path is a continuous image of [O, 11: but an arc is a une-ui~e continuous image. A difficult theorem states that every path contains an arc between iis end points, but a direct proof of arcwise-conneciedness can be @ven for manifol ds.)

4. A space X is cdled locdl)~ connected if for eacll x E X ii is ihe case that every neighborhood of x contains a connected neighborhood.

(a) Conncctedness does iiot imply local conneciedness. 01) An open subset of a locally coi~nected space is locally connected. (c) X is locally connected if and only if components of open sets are open, so evcry iieighboi-liood of a poini in a Iocally connected space coniains an open connected neigh borhood. (d) A locally connecied space is homcomorphic to the disjoint union of its com- ponen ts. (e) Every manifold is locally connected, and consequently homeornorpliic io the disjoin t unioii of its componeii ts, wliich are open submanifolds.

5. (a) Tlie neigliborhood U in oui. definition of a manifold is always open. (b) The integer n in our definition is unique for each x.

6. (a) A subset of aii n-manifold is an n-manifold if and only if it is open. (h) lf M is connected, ihen ihe dimension of M at x is the same for al1 x E M.

7. (a) lf U c IR is an intenlal and f : U + IR is contiriuous and one-one, ihen .f is eiiher increasing or decreasing.

(11) 'l'lie image f ( U ) is open. (f.) 'l'he inap f is a homeomorpl~ism.

8. Ikr t his problem, assume

(1) (Tlie Generalized Jordan Cunle Theorem) If A c Rn is homeomorphic to S"-', then IR" - A has 2 components, and A is the houndary of eacli.

(2) If B c IR" is homeomorphic to D" = (x E R" : d(x,O) 5 l), then R" - B is connecied.

(a) One componeni of R" - A (the "outside of A") is unhounded, and the other (ihc "inside of A") is bounded. (h) If U c W" is open, A c U is homeomorphic to SR-' and f : U + Ri" is oi~e-one aiid coniinuous (so ihai f is a honieomorphism on A), thei~ -/'(iiiside of A ) = inside of f (A). (First prove c .) (f.) Pi-ove Invariance of Domain.

9. (a) Give an elenientary proof tliat IRi is not liomeomorphic to R" for n > 1 .

(h) l'rove directly from the Generalized Jordan Cunre Theorem thai IRm is no1 Iiorneomorphic to IR" for nr # n .

10. In ihe proof of TI~eoirin 2, sliow that the limii of a convergent subsequence oi'the zi is aciudly i i i the closed ball of radius r(y) and cenier y.

11. Every connecied maiiifold (which is a metr-ic space) has a couniahle 11asc li~i- its iopology, and a countahle dense subset.

P 12. (a) Compute tlie composiiion f = S' - ((0, 1)) + Ri x (- 1 ) + Ri i:spIiciily for ihe map P on page 7, and show that it is a homeomorphism. (h) Do the same for f : S"-' - ((O,. . .,O, 1)) + IRn-'.

13. (a) Tlie text describes tlie open subsets of iP2 as seis of ihe form f(V); \\*liei-e V c S' is open aiid coiitaiiis - p wlienever it contains p. Show tliat this last condition is aciually unnecessary. (II) The andogous condiUon Zr necessary for the Mobius sirip, which is discussed iinmediately afterwards. Explain how ihe two cases differ.

14. (a) Check tliat tlie inetric defined for IP2 gises tlie open sets described in ihe texi. (II) Check that P* is a surface.

15, (a) S h o ~ l that P' is homeomorphic to S ' . (b) Since we can consider S"-' c S" , and since aniipodal points in S"-' are siill antipodal when considered as points in S", we can consider P"-' C P" in an obvious way. Show that P" - iPn-' is Iiomeomorphic to interior D" = (x E

R" : d(x,O) -= 1).

16. A classical tlieorem of topology states that every compact surface otl~er ~liaii S 2 is obiained by gluiiig iogether a certain number of ton and projective spaces, and that all compact surfaces-~4th-boundary are ohtained from these by cutting out a finite number of discs. To whích of these "standard" surfaces are the following homeomorphic?

a hole in a hole in a hole

17. Lei C c R c IR2 be the Cantor sei. Show thai W2 - C is homeomorpliir to tlie surface shown at the top of the next page.

illis uii.clc is tiol in tlic sur fa c.^

these circles are

18. A locdly compact (but non-compact) space X "has one end" if for every voinpact C c X there is a compact K such that C c K c X and X - K is c-onnected.

(a) R" Iias one end if n > 1, but not if n = 1. (13) Rn - (O) does not "have one end" so Rn - ( O ) is not homeomorphic to Rm.

l 9. This proaem is a seque1 to the previous one; it will be used in Problem 24. Aii end of X is a function E which assigns io each compaci subset C c X a iion-empty component E(C) of X - C, in sudi a way that Cl c C2 hplies ~ ( C Z ) C E(C1).

(a) If C c R is compact, ihen R - C has exactly 2 unbounded components, the "left" component coiltaining dl iiumbers < some N, the "right" one containing al1 ilumbers > some N. If E is an end of R, show thai E(C) is either dways tlie "lefi" componeni of IR - C, or dways the "rhiglli" one. Thus R has 2 ends. (11) Show that R" has only one end E for n > 1. More generdly, X has exactly oiie end E if and o~lly if X "has one end" in tlie sense of Problem 18. (c) This part requires some knowledge of topological spaces. Let E(X ) be the set of d ends of a coi-ii-iected, locally connected, locdly compact Hausdofl sl~ace X. Define a topology on X U E(X) by choosing as neighborhoods Nc (EO) of an end EO die sets

for- dl compact C. Sl-iow ihat X U G(X) is a compact HausdorK space. Whai is R U E(R), and R" U E(Rn) for n > l ?

20. Coilsider ihe folloiving tliree surfaces. -

(A) The infiniie-lioled torus:

(B) The doubly infiniie-holed torus:

(C) The infinite jail cell window:

_ k l l ~ : ~ ~ l l i ~ - - Ji!+ J\t - e * *

- - - - - - e * .

(a) Surfaces (A) and (C) liavc oiie end, wllile surface (B) does not. (b) Surfaces (A) and (C) are homeomorphic! Hi?zt: Tl-ie region cut out by ihe lines in tlie piciure below is a cylinder, wliicl-i occurs ai tlle left of (A). Now draw in two more liiies enclosirlg more holes, aild consider ihe region between the iwo pairs.

21. (a) The three open subsets of RZ shown below are Iiomeomorphic.

(b) The points kside the three surfaces of Problem 20 are homeomorphic.

22, (a) Every open subset of ES is homeomorphic to the disjoint union ofinter- vals. (b) Ther-e are only countably many non-homeomorphic open subsets of R.

23. For the purposes of this problem we will use a consequence of the Urysohn Metrization Theorem, that for any connected manifold M, there is a homeomorphism f from M to a subset of the countable product R x R x - .

(a) If M is a connected non-compact manifold, then there is a continuous function f: M + R such that f "goes to ca at m", i-e., if (x,) is a sequence which is eventually in the complement of every compact set, then f (x , ) + m. (Compare with Problem 2-30.) (b) Given a homeomorphism f : M + R x R x - - - and a g : M + R which goes to ca at m, define f : M + R x ( R x R x - - - ) by f ( x ) = ( g ( x ) , f ( x ) ) . Show that f ( ~ ) is closed. (c) Thei-e are ai most c non-liomeomorphic coniiected manifolds (where c = ZN" is the cardinality of ES).

24. (a) It is posible for IRZ - A and IR2 - B to be homeomorphic even though A and B are non-liomeomorphic closed subse ts. (b) If A c IRz is closed arid totally disconnected (the only components of A are points), then E(RZ - A) is liomeomorphic to A. Hence IR2 - A and IRZ - B are non- homeomorph ic if A and B are non-homeomorphic closed to tally disconnected sets. (c) The derived set A' of A is the set of al1 non-isolated points. We define A(") inductively by A ( ' ) = A' and A ( ~ + ' ) = (A("))'. For each n there is a subset A, of R such that A,(") consists of one poini.

*(d) There are c non-homeomorphic closed totally disconnected subsets of IRZ. Hint: Let C be the Cantor set, and ci C: cz C: c3 C: . -. a sequence of points in C. For each sequence nl r m r - - - , one can add a set A,, such that its n jth derived se t is (ci). (e) Tliere are c non-homeomorphic connected open subsets of IRZ.

25. (a) A manifold-with-boundary could be defined as a metric space M with the property tliat for each x E M there is a neighborhood U of x and an inieger n 2 O such thai U is homeomorphic to an open subset of W". (h) If M is a manifold-with-boundary, then aM is a closed subset of M and aM and M - aM are manifolds. (c) If Cj, i E 2 are the components of aM, and Ir c I , then M - UiEl , Ci is a manifold-with-boundary.

26. If M c Rn is a closed sei and an 11-dimensional manifold-with-boundary, ilien the iopological boundary of M , as a subsei of IR", is aM. This is not iiecessarily true if M is not a closed subset.

27. (a) Every poini (a, b, c ) oii Steiner's surface satisfies bZc2 + ozcZ + 02b2 = abc. @) If (a, b,c) saiisfies tliis equation and O # D = JbZcZ + oZcZ + oZbZ, then (a, b, c) is on Sieiner's suríiace. Hint: LRt x = bclD, etc. (c) The sei {(a, b, e) E IR3 : bZcZ + ozcz + azbz = abc) is the union of tlie Steiner surface and of the portions (-m, - 1 12) and (1 12, m) of each axis.

CHAPTER 2

DIFFERENTIABLE STRUCTURES

e are now ready to apply analysis to the study of manifolds. The necessary tools of "adva~lced calculus", which the reader should bring along

fieshly sharpened, are contained in Chapters 2 and 3 of Cizku1u.r on A4an8Ms. We will use freely the notation and results of these chapters, including some 11roblems, notably 2-9, 2-15, 2-25, 2-26, 2-29, 3-32, and 3-35; however, we wül denote the identity map from W" to Wn by 1, rather than by n (which will be used oRen enough iii other contexts), so that l i (x) = xi.

Ori a general manifold M the notion of a continuous function f : M + W inakes sense, but tlie notiori of a differentiable function f: M + W does not. This is the case despite the fact that M is locally like IR", where differentiability of funclions can be defined. If U c M is ari open set and we choose a homeomorpliism #: U + IRn, it would seem reasonable to define f to be differentiable on U if f o #-' : R" + W is differeiitiable. Unfortunately, if q : V + W" is anotl~er homeomorphism, and U n V # 0, then it is not necessarily true that f o SI-' : W" + W is also dfirentiable. Indeed, since

we can expect f o VI-' to be differentiable for al1 f wliich make f o#-' diñer- eiitiable only if # o q-' : Wn + W" is differeiitiable. This is certainly not always

tlie case; for example, one need merely choose # to be h o@, where h : R" + R" is a liomeomorphism that is not differentiable.

If we insist on defining differentiable functions on any manifold, there is no way out of this impasse. It is necessary to adorn our manifolds with a little additional structure, the precise nature of which is suggested by the previous discussion.

Among all possible homeomorphisms from U c M onto Rn, we wish to select a certain collection witli tlie properry that #o@-' is differentiable whenever #, @ are in the collection. This is precisely whai we shall do, but a few refinements will be introduced along the way.

First of all, we M i l 1 be interested almost exclusively in functions f : R" + R" wliich are Cm (that is, each component function f i possesses continuous partial derivatives of all orders); sometimes we will use the words "dflerentiable" or c'smoothy' to mean Cm.

Moreover, insiead of considering homeom~r~hisms from open subsets U of M orlto R" , it will sufice to consider Iiomeomorphisms x : U + x ( U ) c R" onto open subsets of R".

The use of tlie letters x, 11, etc., for tliese liomeoniorphisms, henceforth ad- Iiered to almost religiously is meant to encourage the casual confusion of a point p E M with x ( p ) E Rn: wliich has "coordinates" x' ( p ) , . . . , x n ( p ) . The only time tliis notation will be confusing (and it will be) is when we are referring to ihe maiiifold R", whcre ii is hard not to lapse back into the practice of denoting points by x and y. IVe will often mention tlie pair (x, U), instead of x aloile, just to provide a coii\rei~ieni name for the domain of x .

If U and V are opeii siibsets of M, two homeomorphisms x : U + x ( U ) c R" arid 11: V + j c ( V ) c R" are called Cm-related if the maps

are Cm. This make seiise, since x ( U n V ) and y(U n V ) are open subsets of R". Also, it makes sense, aild is automatically true, if U n V = 0.

A family of mutually Cm-related homeomorpliisms whose domains cwer M is called an atlas for M. A particular member ( x , U ) of an atlas 36 is called a chart (for tlie atlas 36):. or a coordinate system on U, for the obvious reason iliai it provides a way of assigiiing "coordinaies" to points on U , namely, tlic coordinates x' ( p ) , . . . , x n ( p ) to the point p E U.

I4Je can even imagine a mesh of coordinate lines on U , by considerirlg the

l)2jintz/ia ble Structure~

iiiverse images uiider x of lines in Rn parallel to one of the axes.

. . . . . . . . .......... ......... . . . . . . . . . ..m.. L....... ......s...... . . . . . . . . . . . . . . . . . . ............ - ...... -. .. . . . . . . . . . . . . . . . . . . ............. , ...... , . . . . . . . . . . . .................... d... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... , .......... . , - , . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ .- ...... -. .. . . . . . . . .

The simplest example of a manifold together ~r i th an atlas consists of Rn with ari atlas A of only one map, the identity 1 : R" + Rn. We can easily make the atlas bigger; if U a ~ i d V are llomeomorphic operi subsets of R", we can adjoin any homeomorphism s: U + V with the property that x and x-l. are Cm. Indeed, we can adjoiii as many such x's as we like-it is easy to check that they are al1 Cm-relaied io each other. Tlie advantage of this bigger atlas U is ihai the single word "cliart", wlieri applied to this atlas, denotes something which must be described in cumbersome lariguage if one can refer only to 36. Aside from this, U differ-s only superficially from 36; one can easily construct U from 36 (and one would be foolish not to do so once and for all). What has just been said for ihe atlas (1) applies to any atlas:

1. LEMMA. If 36 is an atlas of Cm-related cliarts on M, then 36 is contained in a unique maximal atlas 36' for M.

PR001;. Let A' be il-ic set of al1 cliarts y wliicll are Cm-related to al1 charts x E 36. It is easy to check that al1 charts in 36' are Cm-related, so 36' is an atlas, and it is clearly tlie uriique maximal atlas coiitaining 36. *:*

Mle now dcfine a Cm manifold (or differentiable manifold, or smooth manifold) to be a pair (M, A): wliere A is a niam'rna/ atlas for M. Thus, about the simplest example of a Cm manifold is (R", U), wliere U (the "usual Cm-structure for Rn") is tlie maxi~nal adas containing (1). Arioiher example is (R, V) where V contains the l~~rneornor~li ism x H x3, wliosc inverse is no/ Cm, together with al1 charts Cm-relaied io it. Although (R, U) and (R, V) are not tlie same, tliere is a one-one onto function f : R + R such thai

A- E U ifand only if x o f E V,

namely, the obvious map f ( x ) = x Thus (R, U) aiid (8, V) are the sort of structures one would want to cal1 "isomorphic". Tlie term actually used is

"diffe~mor~hic": two C* manifolds (M, A) and (N, a) are diffeomorphic if there is a one-one onto function f: M + N such that

x E J? if and only if A- o f E A.

Tl-ie map f is called a diffeomorphism, and f -' is clearly a difeomorphism also. If we had not required our atlases to be maximal, the definition of diffeomor.pliism would have had to be more complicated.

Normally, of course, we will suppress mention of tlie atlas for a dfireritiable mariifold, and speak elliptically of "the difirentiable manifold M"; the atlas for M is sometimes referred io as the dzjierentiabk structure for M. Ii wdl always be undersiood that IR" refers to the pair (IR", U).

Ii is easy to see that a difkomor-phism must be coritinuous. Consequently, its inverse musi also be coniinuous, so ihat a difeomorpliism is automatically a homeomorpl~ism. This r-aises the natural question whetlier, conversely, two liomeomorpliic maiiifolds are necessarily diffeomor-phic. Later (Pi.oblem 9-24} we will be able to prove easily that IR with any ailas is difkomorphic to (R, U). A proof of tlie corresporiding assertion for is much harder, the proof for IR3 would ccrtainly be ioo difficult for- iriclusion llere, aiid the proof of the essential uniqueness of C* structures on IR" for n 2 5 requires very difficuli techniques from topology,

In ihe case of spl-ieres, tlie projections Pl and P2 from the points (0,. . . ,O, 1) and (0, . . . ,O, - 1) of S"-' are easily seen to be Cm-related. They iherefore clctei-mine an ailz5-tl-ie "usual C* structure for S"-'". T1-iis atlas may also bc described in terms of the 2n homeomorpliisms

defined by J(s) = gi(x) = (x r , . . . , s i - ' ,x i+ ' , . . . ,xn), wbicli are Cm-related to P1 and P2. Tlrei-e are, up io diffeoniorpl-iism, unique differentiable struciures on S" for 17 5 6. But tl-iere are 28 difeomorpl-iism classes of differeniiable structures on S7. aiid over- 16 million on S3'. However, we shall not come close to proving iliesc asser-tioi~s, whicli are pari of tlie field called "differential topology", rather then diferential geometry. (hrhaps most astonishing of al1 is ilie quite receiit discovery that IR4 lias a difeirritiable structure that is not cliffromorphic io tlie usual diffei-entiable siructure!)

Otl-ier examples of differentiable manifolds will be @ven soon, but we cari already describe a diffcrentiable structure A' 011 any open submarlifold N of

a differentiable manifold ( M , A); tlie atlas A' consists of all (x, U) in 36 witli U c N.

Just as dfiomor-phisms are analogues for C* manifolds of homeomorphisms, there are analogues of continuous maps. A function f : M + N is called differentiable if for every coordinate system (x, U) for M and (y, V) for N, the map y o f o x-' : IRn + IRm is differentiable. More particularly, f

is called differentiable at p E M if y o f o x-' is differentiable at x(p) for coordinate systems (x, U) and (y, V) with p E U and f (p) E V. If this is true for one pair of coordinate systems, it is easily seen to be true for any otlier pair. We can thus define differentiability of f on any open subset M' C M ; as one would suspect, tliis coincides with differentiabilit~ of the restricted map f 1 M': M' + N. Clearly, a differentiable map is continuous.

A differeniiable function f : M + IR refers, of course, to the usual d f i r en - tiable structure oii IR, and lience .f is differentiable if and only if f o x-' is diflerentiable foi- cach chart x. It is easy to see that

(1) a function f : IR" + IR" is diflerentiable as a map between C* manifolds if and only if it is difler-entiable in tlie usual sense;

(2) a fuiiction f : M + IRm is differentiable ifand only if each f i : M + W m is diferen tiable;

(3) a coordinate system (x, U) is a diffeomorphism from U to x(U);

(4) a fuiiction f : M + N is diffei-eiitiable if and only if each y' o f is difirentiable for each coordinate system y of N;

(5) a difler-eritiable function f : M + N is a dfleomorphism if and only if f is one-o~ie onto and f U 1 : N + M is differentiable.

Tlie differe~itiable siructui-es on inaiiy manifolds are designed to make ccrtain fuilciions differ-eniiable. Consider first the product MI x M2 of two dfleren-

iial~le manifolds Mi, arid the two "projections" ni : MI x M2 + Mi defined by ni(pl, p2) == pi. It is easy to define a differentiable structure on MI x M2 which makes eac1-i ni difirentiable. For each pair (xi, Ui) of coordinate systems on Mi, we constmct the homeomorphism

defined by

Tlien we cxiend tliis atlas to a maximal one. Similarly, ihere is a difirentiable structure on Pn which makes the map

. f : Sn + Pn (defiried by f (p) = [p] = (p,-p)) difirentiable. Consider any coordinate system (x, U) for S", where U does not contain - p if it con- tair~s p, so tllat f 1 U is o~i,e-one. Tlie map x o (f 1 U)-" is a homeomorp1~ism on J'(U) c Pn, and any two such are Cm-related. The collectiori, of tliese lioineoi~iorpliisms can then be extended to a maximal atlas.

To obtain differentiable structures on other surfaces, we first note that a Cm manifold-witli-bour~dary can be defined in an obvious way. It is only necessary to know wlien a map f : W n + Rn is to be considered differentiable; we cal1 f differentiable d e n it can be extended to a dflerentiable function on an open i-ieiglibor-hood of W". A "Iiandle" is then a Cm manifold-with-boundary.

A diffei.eriiiable structure on tlie 2-holed torus can be obtained by "matcliiiig" tRe dflereiitiable structure on two l-iandles. Tl-ie details involved in tliis process are reserved for Pro blem 1 4.

To deal witl-i Cm fuilctions effectively, one needs to know that there are lots of ilicm. Tlie esistcrice of COQ furictions oii a manifold depends oii tlie existence of C m funciions on R" ~I i i c l i are O ouiside of a compact set. We briefly recall liere ilie necessat-y facis aboui such Cm fuiictions (c.f. Cakulus on MaliifoMs, pg. 29).

D@ben/ia bL Slructu res

(1) The function h : R + R defined by

is Cm, and h(")(0) = O for al1 n.

(2) The function j: IR + R defined by

e-(x-r)-2 e-(x+~)-2 x E (-1, 1) = { o x (-41) - 1 I I

is Cm. Similarly, there is a Cm function k : W + R which is positive on (0,S) and O

elsewhere.

I S

(3) The fuiiction I : R + R defined by

is Cm; ii is O for x 5 O, increasing on (O, S), and I for x 2 S.

(4) Tlie function g : R" + R defined by

is Cm; ii is positive on (-E, E) x . - - x (-E, E) and O elsewliere.

O n a Cm manifold M we can now produce many non-constant Cm functions. The closure (x : f(m) # O) is called tlie support of f, and denoted simply by support f (or sometimes supp f ).

2. LEMMA. Let C c U c M witli C conipact aild U open. Then there is a Cm function f : M + [O, 11 such tliat f = 1 on C and support f c U. (Compare h e 2 of the proof of Theorem 15,)

PROOF. For each p E C, choose a coordinate system (x, V) with c U and x(p) = O. Then x(V) 3 (-E,&) x - - - x (-E,&) for some E 3 O. The function g o x (wliere g is defined in (4)) is CDg on V. Clearly it remains Cm if we extend

it to be O outside of V. IRt Sp be the extended function. The function fp can be coiisir~ucted for each p, aiid is positive on a neighborhood of p whose closure is coniained iii U. Since C is compact, finitely many such neighborhoods cover C, aiid the sum, fp, + - - + - fp,, of the correcponding functions has support c U. 011 C it is positive, so on C it is 2 S for some 8 3 O. Let f = Io ( fp, + - . -+ fpt,), where I is defined in (3). *:e

By tlie way, we could have defined C r rnanifolds for each 1. 1, not just for 6 c

i- = m". (A function f : IRn + IR is Cr if it Iias continuous partid derivatives up to order i-). A "CO function" is just a continuous function, so a CO manifold is jusi a maiiifold in tlie sense of Chapter l. We caii also define analytic manifolds (a function f : W n + W is analytic at u E IRn if f can be expressed as a power series in the (x i - í i i ) which converges in some neighborhood of U ) . Tlie symbol CW staiids for analytic, and it is conveiiient to agree that r .: w c: o for each iiiteger i . > O. If a < B, tlien tlie cliarts of a maximal CB atlas are al1 Ca-related, bui ihis atlas can always be extended to a btggu atlas of Ca-related cliaris, as in Lemma 1. Tlius, a CB structure on M can always be extended io a C" structure i i ~ a uiiique way; tlie smaller siructui-e is tlie "stronger" one, tlie CO structure (coiisistiiig of al1 homeomoi~pliisms x: U + W " ) beiiig tlie lüigest. The coiiverse of this trivial remark is a hard tlieorem: For a 2 1 , ewry C" structurc roiitains a C B structurc for eacli /3 3 a; it is not unique, of c,ourse, but it is ui~ique up to diffeomorpliism. Tliis will not be proved liere.* In fact, C" maiiifolds for a # w will hardly ever be mentioned again. One irinark is iii ordei- 130~~; tlie proof of h m m a 2 produces an appropriate C" fiinciion f oii a C" inaiiifold, for O 5 a 5 w. Of course, for a = o the proof

* hi. a p1-00f see h4un krcs, EIentefztay Ditpereniial %/~olog)l.

fails completely (aiid ihe result is fahe-an analytic function which is O on an open set is O everywhere).

With difirentiable functions now at our disposal, it is fitting that we begin differentiating them. What we S hall define are the partid derivatives of a differentiable function f : M + R, with respeci to a coordinate system (x, U). At this point classical notation for partid derivatives is systematically introduced, so it is wortli recalling a logicai notation for the partial derivatives of a function f : R" + R. We denote by Di f (u) the number

f ( u ' y * . * y u i + h , - . - , u " ) - f(u) lim h-O I?

The Chaiii Rule states that if g : Rm + R" and f : R" + R, then n

Dj (f 0 g) (u) = Di f (g (u)) - ~j gi (u). i=21

Now, for a function f : M + R and a coordinate system (x, U) we define

a f (or simply - = Di (f o x-') o r, as an equation behveen functions). If we axl

define tlie curve ci : (-E, E) + M by

then this partial derivative is just

so it measures the rate change of f along tlie curve c;; in fact it is just (f oci)'(0). Notice tliat

ax3 1 i f i = j - (p) = 6; - a A- J

1f x liappens to be the identity map of Rn, then Dt f (p) = 8 f/axi(p), whidi is the classical synibol for this partid derivative.

Anot her classical instance of this notation, often not completely clarified, is thc use of the syrnbols a/ar and a/a9 in connection with "polar coordinates". On the subset A of JRZ defined by

we can i i i troduce a "coordinate system" P : A + IW' by

where r ( x , y) = m and 9(x, y) is the unique numbcr in (O, 2rr) with

Tliis really is a coordinate system on A in our sense, with its image being the set (r : r > O) x (0, k). (Of course, the polar coordinate system is often

not restricted to tlie set A . One can delete any ray other than L if 9(x, 1)) is restricted to lie in the appropriate interval (60, 90 + 2 ~ ) ; many results are essentially independent of which line is deleted, and this sometimes justifies the slopl~iness involved in the definition of the polar coordinate system.)

IVe liavc 1-eally defined P as an inverse function, whose inverse P-' is defined simply by

P-'(r,9) = (r cos9,r sino).

From this formula we can compute f i a r explicitly:

(f o P-')(r, O) = f (r cos O, i- sin O),

a f -(x? Y) ar = Di (f 0 p-'1 (P(x9 Y))

= DI f (P-' ( ~ ( x , Y)) Di [P-'11 (P(x, Y))

+ ~ z f (p- '(p(x, Y)) - Di [P- ' I~ (P(X~ Y))

by the Chain Rule

= DI f (x, y) tos O(x, y) + Dzf (x, Y) sino(x, Y).

Tliis formula just gives tlie value of the directional derivative of f at (x, JI), along a uiiit \lector v = (cos O (x, y), sin O (x, y)) poin ting outwards from the origin to (x, y). Tliis is to be expected, because cl, the inverse image under P

sin B(x,y)

of a culve along t l ~ e "1.-axis", is just a line in tliis direction. A similar computation gives

The lector w = (- siii O(x, y), cos 0(x, y)) is to v , and thus tlle direction, at tlie point (A-, y), of the curve cz ivliicli is tlie inverse image under P of a cunfe along tlle "O-axis". Tlie factoi- i. (x, y) appears because this cuive

goes around a circle of that radius as 8 goes from O to 2n, so it is going i-(x, y) times as fast as it should go in order to be used to compute the directional derivative of f in the direction w , Note that af/aO is independent of which line is deleted from the plai~e in order to define the function 8 unarnbiguously.

Using the notation a f/ax for Di f, etc., and suppressing the argument (x, 11) evenutliere (thus writing an equation about functions), we can write the above equations as

a f a f -c=-

a f cos 8 + - sin 8 a y ax ay

8.f ?f a.f - = - ( - )e sin 8) + -r cos 8. ae ax ay

111 particular, these formulas also te11 us what ax/ai- etc., are, where (x, y) rlrnotes ilie identit~ coordinate system of RZ. We have ax/ar = cose, etc., so oui- formulas can be put in the form

af afax - = -- +-- al- a ~ - a l - a y a ~

In classical notation, the Cl~ain Rule would always be written in this way. It is a pleasui-e to repoi-t thai Iienceforih this may always be done:

3 . PROPOSITION. If ( S , U) and (y, V) are coordinate systems on M, and f : M + R is differentiable, then on U n V we have

PROOF. It's tlie Chaii~ Rule, of course, if you just keep your cool:

At this point we could introduce the "Einstein summation convention". No- tice that the summation in this formula occurs for the index j , which appears both "above" fin axj /ayi ) and "below" (in af /ax j ) . There are scads of formulas in which this happens, often with hoards of indices being summed over, and the convention is to omit the C sign completely-double indices (wliicli by luck, tlie nature of t hings, and felicitous choice of notation, almost always occur above and below) being summed over. 1 won't use this notation because wheiiever 1 do, 1 sooii forget I'm supposed to be summing, and because by doing things "riglit", one can avoid what Élie Cartan lias called the "debauch of indices".

We will often write formula (1) in the form

here alay' is considered as aii operator taking the function f to af /ay i . The operator taking f to a f / a y i ( p ) is denoted by

For later use we record a property of f = a/axil,: it is a "point-derivation".

4. PROPOSITION. For any diflerentiable f , g : M + IR, and any coordinate system (A-, U ) with p E U, the operator l = a/axilP satisfies

K f g ) = f ( p ) f ( g ) + f ( f ) g ( p ) .

PROOF. Lft to the reader.

If (A-, U) and (x', U ' ) are two coordinate systems on M, the n x n matrix

is just the Jacobian matrix of x1 o A--' at x(p). It is non-singular; in fact, its inverse is clearly

Now if f : M" + Nm is Cm and (y, V ) is a coordinate system around f(p), t he rank of the m x 11 matrix

clearly does not depeiid on the coordinate system (x, U) or (y, V ) . It is called tlie rank of f at p. The point p is called a critical point of f if the rank of f at p is 4 i?7 (the dimension of the image N); if p is not a critical point of f, it is called a regular point of f. If p is a critical point of f, the value f (p) is called a cñtical value of f. Other points in N are regular values; thus y E N is a regular value if and only if p is a regular point of f for ewry p E f -' (y). Tliis is true, in particular, if q $ f (M)-a non-value of f is still a "regular value".

If f : W + W, tlien x is a critica] point of f if and only if f t (x) = O. It is ~ossible for al1 points of the intenral [u, b] to be critical points, although this can happen only if f is constant on [u, b] . If f : WZ + IR has al1 points as critical values, tlien Dl f = Dz f = O everywhere, so f is again constant. On tlie other hand, a function f : WZ + IRZ may have al1 points as critical points without being constant, for example, f ( x , y) = A. In this case, Iiowever, the iinage f (WZ) = W x { O ) c WZ is still a "small" subset of IRZ. The most importan1 theorem about critical points generalizes this fact. To state it, we will need some terminology.

Recall tliat a set A c W" has "measure zero" if for every E > 0 tliere is a sequeilce Bi, Bz , B 3 , . . . of (closed or open) rectangles with

and

wliere u(&) is the volume of B,l. We want to define the same concept for a subset of a manifold. To do this we need a lemma, wliich in turn depends on a Icmma from Caku/us on Adanfolds, which we merely state.

5. LEMMA. Let A C IR" be a rectangle and let f : A + Rn be a function such that ~ ~ ~ f ' l 5 K on A for i, j = 1 ,..., n. Then

for al1 x , y E A.

6. LEh4MA. If f : IR" + IR" is C' and A c IR" has measure O, then f ( A ) has measure O .

PROOF. We can assume that A is contained in a compact set C (since IR" is a countable union of compact sets}. Lemma 5 implies that there is some K such t hat

I f (x) - f (r)l 5 n2Klx - yl for al1 x , y E C. Tllus f takes rectangles of diameter d into sets of diameter

>I' Kd. This clearly implies that f ( A ) has measure O if A does. +3 -

A subset A of a Cm 71-manifold M Iias measure zero if there is a sequence of charts (xi, U i ) , with A c Ui Ui, S U C ~ that each set x i (A n Ui) c IR" has measure O. Using L m m a 6, it is easy to see tliat if A c M has measure O, then x(A n U ) c IR" Iias measure O for any coordinate system ( x , U). Conversely, if this condition is satisfied and M is connected, or has only countably many compoiieiits, then it follows easily from Theorem 1-2 that A has measure O. (But if M is the disjoint union of uncountably many copies of IR, and A consists of one poirit from each compoiient, then A does not have measure O). Lemma 6 thus implies another result:

7. COROLLARY. If f : M + N is a C' fu~iction between two n-manifolds and A c M has measure O, then f ( A ) c N has measure O.

PR001;. Tliere is a sequelice of charts (xi , Ui ) with A c Ui U; and each set xi(A n Ui) of measure O. If ( y , V ) is a cliart on N, then f ( A ) n V = Ui f (A n Ui) n V. Each set

Iias measure O, by Lemma 6. Thus y( f ( A ) n V ) has nieasure O. Since f (Ui Ui) is contained in tlie uiiion of at most countably many components of N, it follows that f ( A ) has measure O. e+

8. THEOREM (SARD'S THEOREM). If f : M + N is a C i map between n-manifolds, and M has at most countably many components, then the critical values of f form a set of measure O in N.

PROOI;. It clearly suffices to consider the case where M and N are Rn. But this case is just Theorem 3.14 of Calculw on M a n f o b .

Tlie stronger version of Sard's Theorem, which we will never use (except once, in Problem 8-24), states* tliat the critical values of a C' map f : M" + N"' are a set of measure O if k 2 1 + max(n - in,O). Theorem 8 is the easy case, and tlie case nz > n is the trivial case (Problem 20). Although Theorem 8 will be very important later on, for the present we are more interested in knowing wliat tlie image of f : M + N looks like locally, in terms of the rank k of f at p E M . More exact information can be @ven wlien f actually has rank k i n a neighborhood of p. It should be noted that f must have rank 2 k in some iieiglihor-hood of p, because some k x k submatrix of (a(yi o f )/axi) has non-zero determinant at p , and hence in a neighborhood of p.

9. THEOREM. (1) If f : M" + Nm has rank k at p, then tliere is some coordinate system (x, U) around p and some coordinate system (y, V) around f (p) with y o f ox-' in the form

Moreover, given any coordinate system J?, the appropriate coordinar e syst em ori N can be obtained merely by pei-muting the component functions of y.

(2) If f has rank k in a rieiglil~orliood of p, theii tliere are coordinate systems (A-, U) and (y, V ) such that

Ren~ark: Tlie special case M = Rn, N = Rm is equivalent to the ge~ieral tlieorem, wliicli @\res only local results. If y is tlie identity of R", part (1) says thai ly first pei-formiiig a difleoniorpliism on Rn, arid tlieri permuting the coordi- iiates in Rm, we can irisure that f keeps tlie first k components of a poiiit fixed. Tliesc difTcornoiphisms on Rn and Rm are clearly necessary, since f may not everi be oiie-oiic on Rk x { O ) c RW aad itr image could, for example, contain oiily ~ o i n t s with first coordinate O.

* Foi- a proof, scc hqiliroi; To/)ologil h t n tlic Ditp~~'nl&inblc I4cuy)oi)ir or Sternberg, LEcllrres ott

Dflirenliol Geonzelv.

In part (2) we must clearly allow more leeway in the choice of y , since f ( I R n ) rnay not be contained in any k-dimensional subspace of I R n .

I'ROOF. (1) Choose some coordinate system u around p. By a permutation of ilie coordinate functions u' and y' we can arrange that

Define

Condition (1) implies that

This shows thai A- = (x o u- ' ) o u is a coordinate system in some neighbor- I~ood of p, since (2) and the Inverse Function Theorem show that x o u-' is a dfleomorphism in a ncighborhood of u ( p ) . Now

1 q = x- ' (a ' , . . . , an ) means x(q) = (a , . . . , an ) , i hence x ' ( ~ ) = a ,

SO

Y ~ f o x - l ( a l , . . . , a J 1 ) = y o f ( q ) for q = x-l (a 1 , . . . , a n ) 1 k = ( a ,..., a , ).

(2) Choose coordinate systems x and v so that v o f o x-' has the form in (1). Since rank f = k in a neigliborhood of p, the lower square in the matrix

must vanish in a neighborhood of p. Thus we can wnte

Define ya = va

k y '= vr - $ r o ( ~ ' , . . . , V ).

Since

has non-zero determinant, so y is a coordinate system in a neighborhood of f (p). Moreover,

Theorem 9 acquires a special form when the rank of f is n or nz:

10. THEOREM. (1) If m 5 ,I and f : M" -t Nm lias rank m at p , then for any coordinate sysiem (y, V ) around f (p), there is some coordinate system (x, U) around p with

y o f ox-'(a ' , . . . ,an) = (a ' , . . . ,am).

(2) If 11 5 m and $: Mn + N"' has rank ir ai 11, tlien for any coordinate system (x, U) around p , there is a coordinate system (y, V ) around $ ( p ) with

i 1 n y o f ~ x - ~ ( a , . . . , an ) = (a ,..., a , O ,..., 0).

PROOF. (1) This is practicdly a specid case of (1) in Theorem 9; it is only rlecessary to observe that when k = rn, it is clearly unnecessary, in the proof of this case, to permute ihe yi in order to arrange that

det ( a ( y ~ u ~ f ) ( p ) ) f 0 a,p = 1,. . .,m;

only the u' need be permuted. (2) Since the rank of f at any point must be 5 n, the rank of $ equals n in some neighborhood of p. It is convenient to think of the case M - R" and N = RJn and produce the coordinate system y for Rm when we are Siven ihe identity coordinate system for R". Par-t (2) of Theorem 9 yields coordinate systems # for Rn and S, for Rm such that

1 1 @ o f o#-'(a , ..., a") = (a ,..-, an,O ,..., O),

Even if we clo rioi pei-form #-'. firsi, tl-ie map f still takes EXn into the subset

(R") wliich S, takes to R" x (O) c Rm-the points of Rn just get moved to ihe wrong place in Rn x (O). This can be corr-ected by another map on R". Define h Ily

h(bi , . . . , bJn) = (4-' (bl , . . . , b"), b"+', . . . , bm).

Then

hoS,o$(ai , ..., a n ) = h o S , o $ o # - ' ( b 1 ,..., b")

for (b', . . . ,bn) = #(a)

= h(bl,. . . ,ón,0,. . .,O)

= ($-'(bl, . . ., b"),O,. . .,O) i R = (a , . . . , a ,O, . , O),

so h o S, is tl-ie desired ) J . If we are Siven a coordinate system x on R" other than the identify, we just define

h(bl, . . . , b"') = ( ~ ( 4 - ' ( b i , - . . , b')), b"+I,. . . , bm);

it is easily cliecked tliat y = h o S, is rlow the desired y. +3

Aliliough p is a regular point of f in case (1) of Theorem 10 and a critica1 point in case (2) (if n < m), it is case (2) \vhich most interests us. A dflerentiable funciion f: M n + N m is called an immersion if the rank of f is n, the dimension of the domain M, at al1 points of M. Of course, it is necessary that In > n, and i i is clear from Theorem 10(2) that an immersion is locally one-one (so it is a iopological immersion, as defiiied in Chapter 1). On the other hand, a differentiable map f need not be an immersion ewn if it is globally one-one. The simplest example is ihe function f: IR + IR defined by f ( x ) = x3, with f (O) = O. Another example is

A more illun~inating example is the function h : IR + IR2 defined by

alihough its image is tlie graph of a non-dflerentiable function, the curve itself manages to be differenkiable by slowing down to velocity O at the point (0,O). One can easily define a similar curve whose image looks like the picture below.

Tliree immersions of IR in IR2 are shown below. Althougll the second and iliii-d immei-sions Bi and B2 are one-one, iheir iinages are not homeomorphic

to R. Of course, even if tl-ie one-one immersion f : P + M is not a homeo- niorpliism onto iis image, tliere is certainly some metric and some dflerei~tiable sti-ucture on f ( P ) which makes the inclusion map i : f ( P ) + M an inlmersion. In general, a subset MI c M, with a dflerentiable structure (iiot nec- cssarily compatible with the metric Mi inherits as a subset of M), is called an immersed submanifoId of M if the inclusion map i : MI + M is an immersion. l'he follo\ving picture, indicating the image of an immersion B p : R + S' x S] ,

rS3

first time around second time around /

sl-iows that Mi may even be a dense subset of M. Despite these complications, if Mi is a k-dimensional immersed submanifold

of M n and Ui is a neighborhood in M' of a point p E M I , tllen there is a coordinate sysiem (y, V) of M around p , such that

tliis is an immediaie consequerice of Tlieorcm 10(2), with f = i. Tlius, if g ; Mi + N is Cm (co~isidered as a function oii tlie manifold Mi) in a neigliborhood of a point p E MI, ihen ihere is a Cm furiction g on a rieigliboi-liood V c M of p sucli that g = g O i on V n Mi -we can define

~ ' ~ ( q ' ) = ~'"(4) a = 1, ..., k S (4) = g (qi), where r = k + l , . s . , i z .

On the oilier hand, even if g is Cm on al1 of MI we may not be able to define g on M. For example, this cannot be done if g is one of the functions B;' j j ( M ) + R.

One other complicatioi-i arises with immersed submanifolds. If M I c M is an immersed submailifold, aild f : P + M is a Cm function with f ( P ) c Mi, i i is not necessarily true that $ is Cm when considered as a map into MI, with its Cm structure. The following figure shows that f might not even be continuous

as a map jnto M I . Actually, tliis is the only thing that can go wrong:

11. PROPOSITION. If MI c M is an immersed manifold, f : P + M is a Cm function witli f (P) c MI , and f is coniinuous considered as a map irlto MI , then f is also Cm coilsidered as a map into MI.

PROOF. Let i : MI + M be tlie inclusioil n-iap. We want to show that iP1 o j' is Cm if i t is continuous. Given p E P, choose a coordiilate system (y, V) for M around f (p) such that

is a iieigliboi-hood of f (p) in MI and (yi 1 U], . . . , yk 1 U]) is a coordiliate system of MI on U].

By assumption, i-' o f is continuous, so

f -' o i(open set) is an open set.

Since Ui is open in MI , this means that f -' (Ul) c P is open. Thus f takes some neighborhood of p E P into U,. Since al1 y j o f are Cm, and y ' , . . . , y k are a coordinate system on U1, the function f is Cm considered as a map into MI. 4 3

Most of these difficulties disappear wlien we consider one-one immersions f : P + M which are liomeomorphisms onto their image. Such an immersion is called an imbedding ("embedding" for the English). An immersed submanifold Mi c M is called siinply a (Cm) submanifold of M if the inclusion map i ; MI + M is an imbeddirlg; it is called a closed submanifold of M if MI is also a closed subset of M.

Tliere is one way of getiiilg submanifolds wl-iich is very important, aild @ves the sphei-e S"-' C W" - {O) c JR", defined as (x : 1x12 = 11, as a special case.

12. PROPOSITION. If f : M" + N lias constani railk k on a neighborhood of f -' (J!), then f -' (y) is a closed submanifold of M of dimension n - k (or is empty). Iii particular, if y is a regular value of f : M" + Nm, then f -' (y) is an (11 - 111)-dimensional submanifold of M (or is empty).

PROOF. Left to the reader. 43

It is to be hoped that however abstract the noiion of Cm manifolds may appear, sul~manifolds of J R N will scem like faii-ly concrete objects. Now it turns out tliat e u q r (connected) Cm manifold can be imbedded in some J R N , so that manifolds caii be pictured as subsets of Euclidean space (though this picture is iiot always die nlost uscful one). We will prove tliis fact oilly for compaci mailifolds, but we first develop some of the macl~iileiy which would be used in

the general case, since we xrtil1 need it later on anyway. Unfortunately, there are many definitions and theorems involved.

If O is a cover of a space M, a cover O' of M is a refinement of O (or "refines O ") if for every U in O' there is some V in O with U c V (the sets of O' are "smaller" than those of O)-a subcover is a very special case of a refining cover. A cover O is called locally finite if every p E M has a neighborhood W which intersects only finitely many sets in O.

13. THEOREM. If O is an open cover of a manifold M, then there is an open cover O' of M whicli is locally finite and which refines O. Moreover, we can choose al1 members of O' to be open sets dfleomorphic to R".

PROOF. \Ve can obviously assume that M is connected. By Theorem 1.2, there are compact seis Cl,C2, C3,. . - with M = CI U C2 U C3 U - - - - Clearly Ci has an open neighborliood U] with compact closure. Then U Ci has an open neighborhood U2 ~ 4 t h compact closure. Continuing in this way, we obtain open sets Ui, witli compact and c Ui+i, whose union contains aH C;, and lience is M. Let U-] = U. = 0.

- Now M is ilic union for i > 1 of the "aniiular" regons Ai = Ui - Ui-]. Since

eacli A i is compact, we can obviously cover Ai by a finite number of open sets, eacli coiiiaiiied in some membcr of O, and eacli coiitained in Vi = Ui+, - Ui-2. \Ve can also clioose these open sets to be diffeomoi.phic to RR. In this way we obiain a cover O' wliich refines O aild wliicli is locally finite, since a point in Ui is noi in 6 for j 3 2 + i. /f,

Notice that if O is an open locally finite cover of a space M and C C M is compact, then C intersects o~ily finitely many members of O. This shows that aii open IocaHy finite cover of a connected manifold must be countable (like the cover constructed in the proof of Theorem 13).

14. THEOREM (THE SHRINKING LEMMA). Let O be an open locally firiiie co\rer of a manifold M. Then it is possible to choose, for each U in O, an open set U' wiui c U in such a way that the collection of al1 U' is also an open cover of M.

PROOF. 1442 can clearly assume that M is connected. Let O = (Ui, U2, U3,. . .). Then

c1 = u, - (U2UU3 U . - - )

is a closed set co~itained in U], and M = Ci U U2 U U3 U. . . L t U{ be an open set witli Ci c U; c c U1 Now

is a closed set contained in U2, and M = U; U C2 U U3 U - . Let U; be an open set wiili C2 c U; c c U2. Continue in this way.

For aliy p E M tliere is a largest n witli p E U,, because O is locally finite. Now

p E u; UU; U - S U u; U (u.+, U UN+2 U . - . ) ;

it follows that

P E U / U U ; U a . * ,

since replacing U,+í by Un+i cannot possibly eliminate p. 34

15. THEOREL4. Let O be an open IocaHy firiite cover of a manifold M. Then ihere is a collection of Cm functioiis #u: M + [O, 11, one for each U in O, such tllat

(1) support $U C U for each U,

(2) $u(/)) = 1 for al1 p E M (this sum is really a finite sum in some U iieigliborliood of p, by (1)).

PROOF. Cme 1. Each U in O Rar ~omjia~t closure. Choose the U' as in Theorem 14. Apply Lemma 2 to c U c M to obtain a Cm function Jru : M + [O, 11 wliicli is 1 on aiid has support c U. Since the U' cover M , clearly

Define

Cme 2. Gerie~al c a e . Tliis case can be proved in tlie same way, provided diat Lemma 2 is irue for C c U c M with C closed (but not necessarily compact) and U open. But this is a consequence of Case 1:

For eacli y E C choose an open set Up c U with compact closure. Cwer M - C with open sets V. haviiig compact closure and contained in M - C. The open cover (Up, V..) lias an open locally fiilite refinement O to which Case I applies. Let

Tliis sum js Cm, silice it is a fi~iite sum iii a iieigliborliood of each point. Since xu #u ( p ) = 1 for al1 p , arid #u ( p ) = O when U c V., clearly J ( p ) = 1 fol. al1 p E C. Usirig tlic faci tliat O is locally finite, it is easy to see tliat support f c U. +:+

16. COROLLARY. If O is aiiy open cover of a n~aiiifold M , then ihere is a collection of Cm functions #i : M + [O, I ] such that

(1) tlie collection of sets ( p : #i ( p ) # O) js locally finite,

(2) ti #i ( p ) = I for al1 p E M ,

(3) for eacli i tliere is a U E O such that support #i C U.

(A collccti~ri (# i : M + [O, 11: satisfying (1) and (2) is called a partition of unity; if i t satisfics (3), ii is called subordinate to O.)

It is riow fairly easy to prove tlie last theorem of tliis cliapter.

17. THE0REh.I. If M" is a conipact C m mailifold, tlien tliere is aii iml~ed- cliiig /: M + IRN for some N.

1'1200F. TIiere are a £nite number of coordinaie systems (xl , Ul), . . . , (xk, Uk) witl-i M = Ul U - - -U Uk. Choose U; as in Tlieorem 14, and functions @i : M + [O, 11 whicli are 1 on and have support c Vi. Define f : M + RN, where N = n k + k , b y

Tl-iis is ai-i inlmersion, because aiiy poiiit p is in U; for some i, and on U,', where qfj = 1, ihe N x it Jacobian matrix

It is also one-one. For suppose that f ( p ) = f(q). There is some i such that p E U;. T1-iei-i @i(p) = 1, SO also @i(q) = 1- This shows that we musi llave y E Ui. Moreover,

@i ' xi (P) = @i - X i (Y),

Problem 3-33 sl-iows that, in fact, we can always choose N = 2rz + 1.

1, (a) Sliow tliat beir-ig Cm-r-elated is no.! a1.i equivalente relation. (b) 111 the pl-oof of Leninia 1, show ihat al1 charis in A' are Cm-related, as claimed.

2. (a) If M is a 1-ileti.i~ space togeiber wiil-i a collection of homeomorphisms x: U + Rn whose domains cover M and which are Cm-related, show that tl-ie tt at each yoiiit is ui-iique rtrillrouf using Irivariai-ice of Domain. (b) SI-iow sin-iilarly thai aM is well-defined for a Cm manifold-with-boundary M.

3. (a) Al1 Cm fui-ictions are continuous, ai-id il-ie coinposition of Cm functions is Cm. (b) A funciion f : M + N is Cm if al-id only if g o f is Cm for every Cm func tion g : N + R.

4. How inaiiy clistii-ict Cm stl-uctures are thel-e 011 R? (Tlleie is only one up to diffeon-iorpliisin; il-iat is i-ioi the question beil-ig asked.)

5. (a) If N c M is open and A' consists of al1 ( x , U) in A with U c N, show thai A' is maximal for N if A is maximal for M. (b) Show that A' can also be described as ihe set of al1 (xlV í l N, V í l N) for (x, V) in A. (c) SIlow ihai ihe inclusion i : N + M is Cm, and that A' is the unique atlas with this property.

6. Clieck thai tlie iwo projections Pl and P2 on S"-] are Cm related to the 2n l~omeomor~hisms _I;: and gj.

7. (a) If M is a connected Cm marlifold and p,q E M, tl-ien ihere is a Cm c u m c : [O, I ] + M with c(0) = p and c(I) = q. (b) It is even possible to clioose c to be one-one.

8. (a) Sliow ihat (MI x M2) x M3 is dfleomorphic io MI x (M2 x M3) and iI-iai MI x M2 is dfleomorphic to M2 x MI . (b) The dflcrentiable structure on MI x M2 makes the "slice" maps

of MI , M2 + M1 x M2 diffel-entiable for a11 pl E Mi , p2 E M2. (c) Pl4ore gerierally, a map f : N + MI x M2 is Cm if and only if the compo- siiions rrl o f : N + Mi and rr2 o f : N + M2 are Cm. Moreover, ihe Cm structure \ve llave defined for Mi x M2 is the only one wiih this property. (d) If fj : N + Mi are Cm (i = 1,2), can one determine ihe rank of ($1, $2) : N + Mi x M2 (71 p in terms of the ranks of _f;: at p? For fj : Ni + Mi, show thal Si x f2: Ni x N2 + Mi x M2, defined by ji x f i(pi , p2) = (Si(pi1, Sz(p2)), is Cm and determine its rank in terms of the ranks of ji.

9. Let g : S" + P" be tlie map p H [p]. SIiow that f : P" + M is Cm if and only if j' o g : S" + M is Cm. Compare the rank of f and the rank of f o g .

10. (a) If U c R" is open and f : U + R is locally Cm (every point has a iieiglil~orliood ori wllich f is Cm), tllen f is Cm. (Obvious.) (b) If f : W" + R is locally Cm, then f is Cm, i-e-, f can be exieilded to a Cm function on a ileighborhood of M". (Not so obvious.)

1 1 . If f : W" + R Ilas hvo extensions g, h to Cm functions in a neighborhood of M", illeli Qg and Qh are ihe same at points of R"-' x (O) (so we can speak of Di f at these points).

12. If M is a Cm rnanifold-wiil-i-boundary, tlien there is a unique Cm struciure ori aM sucll iliai il-ie inclusion map i : aM + M is an imbeddiilg.

13. (a) Let U C M" be an open set sucb ihat boundary U is an (n - 1)-dimensional (dflereiitiable) submanifold. Show tliat U is aii 11-dimensional manifold- with-boundary (It is well to bear in mind the following example: if U = {x E EX' : d(x , O) < 1 or 1 < d(x , O) < 21, then Ü is a manifold-with-boundary, but aÜ # boundary U.) (b) Consider the figure shown below. This figure may be extended by putting

srnaller copies of tlie two parts of S' into the regions indicated by arrows, and ihen repeatirig tliis coristi-uction iudefinitely. The closure S of the final resulting figure is known as Alexnnder's Houied SSplnr. Sliow tliat S is liomeomorphic to s2. (Hinl: Tlie additional poinis in tbe closure are bomeomorphic to the Cantor sei.) If U is tlie uiibourided component of W3 - S, ihen S = boundary U, but Ü is not a 2-dimensional manifold-with-boundary, so part (a) is true only for differen tiable su bmanifolds.

14. (a) Therc is a map f : EX2 + EX2 SUC~I tliat

(1) f (x, O) = (x, O) for al1 x,

(2) f(x, y) c EI2 for y > O,

(3) f ( ~ , y ) c W 2 - IN2 for y < O,

(4) f restricted to the upper half-plane or the lower half-plane is Cm, but f itself is not Cm.

(b) Suppose M and N are Cm manifolds-with-bounda~y and f : aM + aN is a difiomorphism. Let P = M U 1 N be obtained from the disjoint union of M and N by identifying x E aM with f (x) E aN. If (x, U) is a coordinate system around p E aM and ( y , V) a coordinate system around f (p ) , with f ( U n a M ) = V n a N , and ( y o f)IU n a M = xlU n a M , we can define a homeomorphism from U U V c P to IRn by sending U to Wn by x and V to tlle Iower half-plane by the reflection of y. Show that this procedure does not

define a CbO structure on P. (c) Now suppose t11at there is a neighborhood U of aM in M and a dfleomorphism a : U + 8M x [O, l ) , such that a ( p ) = (p,O) for al1 p E aM, and a similar diffeomorphism p : V + aN x [O, 1). (M'e will be able to prove later that such diffeomorpllisms always exist). Show that tllere is a unique CbO structure

on P sucl-i tl-iat t l~e ii~clusions of M and N are CbO and such that the map from U U V to aM x (- 1,l) induced by a and B is a dfleomorphism. (d) By usiilg two difTei-ent pairs (a, p), define two dflerent Cm structures on EX2, collsidcred as thc union of two copies of W 2 witli corresponding points on a W 2 iclei~tified. SIlow that the resulting CbO manifolds are dfleomorphic, but that the difKeomorphism ca?~nol be chosen arbitrai-iIy close io the identity map.

15. (a) Find a Cm structure on W' x M' which makes the úidusion into EX2 a Cm map. Can the inclusioi-i be an imbedding? Are the projections on each factor Cm maps? (b) If M and N are manifolds-with-boundary, construct a Cm structure on M x N such that ali the "slice maps" (defined in Problem 8) are Cm.

16. Show that the fui-iction f : IR -+ IR defined by

is Cm (the formula e-'ix2 is used just to get a function which is > O for x < 0, and e-'/lXl could be used just as weil).

17. Lemma 2 (as addended by the proof of Theorem 15) shows that if Cl and C2 are disjoint closed subsets of M, then there is a Cm function f : M + [O, 11 such that Ci c f -'(O) and C2 c f -'(I). Actually, we can even find f with Cl = /-'(O) and C2 = f- ' (1). The proof turns out to be quite easy, once you know the trick.

(a) It suffices to find, for any closed C c M, a Cm function f with C = /-'(O). (b) Let ( U i ) be a countable cover of M - C, where each Ui is of the form

for some coordinate system x taking an open subset of M - C onto EXn. Let Si : M + [O, 11 be a Cm function with _I;: > O on Ui and j;. = O on M - Ui. Functions like

- a x i ' a x í a x k 5 "'

will be called mixed partials of A, of order 1,2,. . . . Let

ai = sup of all mixed partials of Si,. . . , f;. of al1 orders 5 i.

Show that " Si f=C- i=I ai2'

is Cm, and C = f -l (O).

18. Consider the coordinate system (y1, y2) for IR2 defined by

y' (a , b) = a

y2(a,b) = a + b.

(a) Compute f/ayr(a, b) from the definition. (b) Also compute it from Proposition 3 (to find aI i /ayj , write each 1' in terms of y' and y2).

Notice that a f /ayl # 8 / / a l r even though = I r ; the operator alay' depends on y and i, not just on y'.

19. Compute the "Laplacian"

in terms of polar coordinates. (First compute a/ax in terms of alar and a/a9; r a a a r a then compute a2/ax2 from this). Amwer: ; a;) + ae (,=)J.

20. If : M" + Nm is C1 and m > n, then /(M) has measure O (provided that M has only countably many components).

21. The following pictures show, for 11 = 1,2, and 3, a subdivision of [O, 11 x [O, 11 into 22n squares, An,i,. . . , An,2b; square An,k is labeled simply k. The numbering is determined by the following conditions:

(a) The lower left square is A,,].

(b) The upper left square is An1221f.

(c) Squares A,,k and AR,k+l have a common side.

(d) Squares An,41+1 , An,41+2, An,41+3i An,41+4 are contained in A,-, ,[+l.

Define f : [O, 11 + [O, 11 x [O, 11 by the condition

k - 1 f( t) E An,k for al1 - k

22" < t s - . - 22"

Show that f is continuous, onto [O, 11 x [O, 11, and not one-one.

22. For p/2" E [O, 11, define f (p/2") E lR2 as shown below.

(a) Show that f is uniformly continuous, so that it has a continuous extension g: [O, 11 + lR2. Show that g is one-one, and that its image wiil not have measure O if tlie shaded triangles are chosen correctiy. (b) Consider tlie homeomorphic image of S' obtained by adding, below the image of g, a semi-circle with diameter tlie line segment AB, JVhat does the inside of this curve look like?

23. Let c : [O, 11 + Rn be continuous. For eacli partition P = {to,. . . , fk ) of [O, 11, define

k

The curve c is rectifiable if {l(c, P)) is bounded above (with length equal to sup{l(c, P))). Show tliat the image of a rectifiable curve has measure O.

24. (a) If M is a Cm manifold, a set M i c M can be made into a k-dirnen- sional submanifold of M if and only if al-ound each point in M I there is a coordinate system (x , U) on M such tliat M i n U = {p : xk+'(p) = = x" (p) = O). (b) Tlie subset Mi can be made into a closed submanifold if and only if such coordinate systems exist around every point of M .

25. Tlie sei {(x, 1x1); x E R) is noi tlie image of aiy immersion of R into IR2.

26. (a) If U c Rk is open and f: U -t IRn-' is Cm, then the graph of f = {(p, f(p)) E R": p E U) is a submanifold of Rn. (b) Every submaliifold of R" is locally of tiiis form, after renumbering coordinates. (Neitlier Theorem 9 nor 10 is quite strong enough. You wiil need

the implicit function theorem (Gilculw un Manzjbolds, pg. 44)). Theorem 10 is essentially Theorem 2-1 3 of Celculw on ManifoHs; comparison with the implicit function theorem will show liow some information has been allowed to escape.)

27. (a) An immersion from one n-manifold to another is an open map (the irnage of an open set is open). (b) If M and N are n-manifolds with M compact and N connected, and f : M + N is an immersion, then f is onto.

28. kove Proposition 12: If f : M" + N has constant rank k on a neighborhood of f - ' ( y ) , then f -' ( y ) is a (closed) submanifold of M of dimension n - k (or is empty).

29. Let f : P' + W 3 be the map

defined in Chapter 1, whose image is the Steiner surface. Show that g fails to be an immersion at 6 points (tlie image points are the points at distance &1/2 on each axis). There js a way of immersing P2 in IR3, known as Boy's Surface. See Hilbert and Col1 n-Vossen, Geomeiy and !he Imagination, pp. 3 17-32 1.

30. A continuous f~inction f : X + Y is proper if f - ' ( C ) is compact for evcry compact C c Y. The limit set L ( f ) of f is the set of al1 y E Y such ihat y = lim f(x,) for some sequence xl,x2,x3,. . . E X with no convergent subsequence.

(a) L( f ) = 0 if and only if f is proper. (b) f ( X ) c Y is closed if and only if L ( f ) c f ( X ) . (c) Tliere is a continuous f : W + I W Z with f(W) closed, but L ( f ) # 0. (d) A onc-one continuous function f : X + Y is a homeomorphism (onto its image) if and only if L ( f ) n f ( Y ) = 0. (e) A submanifold Mi c M is a closed submanifold if and only if the inclusion map i: MI + M is proper. ( f ) If M is a manifold, there is a proper map f : M + I W ; the function f can be made Cm if M is a Cm manifold.

31. (a) Fiiid a cover of [O, 11 which is not locally finite but wliich is "point- finite": every point of [O, 11 is in only finitely many members of the cover. (b) Prove the Shrinking Lemma when the cover O is point-finite and countable (notice that local-finiteness is not really used). (c) Prove the Shrinking Lemma when O is a (not necessarily countable) point- finite cover of any space. (You will need Zorn's Lemma; consider collections C

of ~ a i r s (U, U') where U E O, U' c U, and t he union of all U' for (U, U') E C, togetlier with all other U E O covers the space.)

32. (a) If MI c M is a closed submanifold, U 2 Mi is any neighborhood, aiid f : Mi + R is Cm, then there is a Cm function f: M + W with / = f on Mi, and with support J c U. (b) This is false if M = W and MI = (0,l). (c) This is false if R is replaced by a disconnected manifold N.

Remark: It is also false X M = RZ, MI = N = S', and f = identity; in fact, in tliis case, f has no contiiiuous extension to a map from R2 to S', but the proof requires some topology. However, f can always be extended to a CbO function in a neighborhood of MI (extend locally, and use partitions of unity).

33. (a) The set of aH non-siiigular n x n matrices with real entries is called GL(i1, R), tlie general linear group. It is a CbO manifold, since it is an open

2 subset of W" . The special linear group SL(n, R), or unimodular group, is the subgroup of al1 matrices witli det = 1 . Usirig tlic formula for D(det) in CahuEw on A4aa@ds, pg. 24, show tliat SL(n, R) is a closed submanifold of GL(n, R) of dimension n2 - 1. (b) The symmetric n x 11 matrices may be thought of as R"("+')/~. Define $: GL(n, W) + (symmetric matrices) by $(A) = A . A', wliere A' is tlie traiispose of A . Tke subgroup $-'(I) of GL(J~,W) is called the orthogonal group O(ri). Sliow tliat A E O(n) if arid only if tlie rows [or columns] of A are ort honormal. (c) Show that O(n) is compact, (d) For any A E GL(JI, R), define RA : GL(!r, R) + GL(n, R) by RA(B) = BA. Show tliat RA is a dfleomorphism, aiid ihat $ o RA = @ for ali A E O(n). By applying tlie chain rule, show that for A E O(n) the matrix

a @ij lias the same rank as (-(1)) .

axkl

(Here xk' are the coordinate functions in WnL, and @'j the n(n+ 1)/2 component functions of @.) Conclude from Proposition 12 that O(n) is a submanifold of GL(n, R). (e) Using the formula

62

show that

\ O otherwise.

Show that the rank of this matrix is n(n + 1)/2 at I (and hence at A for al1 A E O(n).) Conclude that O(n) has dimension n (n - 1)/2. (fj Show that det A - & 1 for al1 A E O@), The group O(n) nSL(n, R) is called the special orthogonal group SO@), or the rotation p u p R(n).

34. Let M(nr, 11) denote the set of al1 n.i x n matrices, and M(n2,n; k) the set of al1 n? x n matrices of rank k.

(a) For every Xo E M(m,n; k) there are permutation matrices P and Q such that

PXoQ = B o ) , where A. is k x k and non-singular. c o Do

(b) There is some E > O S U C ~ that A is non-singular whenever al1 entries of A - A. are cl E .

(4 If A B

P X Q = ( C D)

wbere the entries of A - A. are < E , then X has rank k if and only if D = CA-'B. Hinl: If Ik denotes the k x k identity matrix, then

(d) M (112, n; k) c M (m, n) is a submanifold of dimension k (m + n - k) for a11 k 5 in,n.

CHAPTER 3

THE TANGENT BUNDLE

A point v E R" is frequentiy pictured as an arrow from O to V. But there are many situations where we would like to picture this same arrow as starting

at a dXerent poini p E Rn:

For example, suppose c : EX + R" is a differentiable curve. Then cr( t ) = (cl ' ( t) , . . -, cnr(t)) is just a point of R", but the line bemeen c ( t ) and c ( t ) + cr( t ) is tangent to the curve, aiid the "velocity vector" or "tangent vector" cr ( i ) of the curve c is customarily pictured as the arrow from c ( i ) to c ( t ) + cr(t) .

To give tliis picture mathematical substance, we simply describe the "arrow" from p to p + v by the pair (p, u). The set of al1 such pairs is just R" x R", wliich we will also denote by T R", the "tangent space of R" "; elements of T R" are called "tangent vectors" of R". We will often denote (p, u) E TR" by U,

("the vector v at p"); in conformjty with this notation, we will denote the set of al1 (p,v) for v E R" by Rnp. At times, it is more convenient to denote a member of TR" by a single letter, like v. To recover the first member of a pair 2i E TRn, we define tlie "projection" map rr : R" x IR" + R" by ~ ( a , b) = a . For any tangent vector u, the point X(V) is "where it's at".

The set rr-l ( p ) may be pictured as al1 arrows starting at p. Alternately,

ii can be pictur-ed more geometi-ically as a particular subset of R" x R", the oiie visualizable case occurring when 11 = l . This picture gives rise to some

tei.minology-we call rr-'(p) ihe fibre over p. Tliis fibre can be made into a

The Tazgeni Bundle

vector space in an obvious way: we define

(p, u) (p, w) = (PY u + w) a (p, u) = (p,a u)-

(The operations $ and * should really be thought of as defined on

( ) x ~ ( ~ ) , and R x T R", res~ectively. ,€E"

Usually we will just use ordinary + and . instead of $ and .)

If f : Rrl + Rm is a differentiable map, and p E Rn, then the linear transformation Df(p): Rn + Rm may be used to produce a linear map from Rnp + Rni (,, defined by

This map, whose appareiitly anomalous features will soon be justified, is denoted by S,,; tlle symbol denotes tlie map f* : TR" + TRm which is the union of al1 f*,. Sincc j;,(u) is defined to be a vector E Rmftpl, the followiiig diagram "commutes" (the two possible compositions from TR" to Rm are equal), f* TR" - TRm

J IR" - Rm

Tlius, $, has the map f , as well as al1 maps Df(p), built into it. This is noi the only reason for defining f* in this particular way, however.

Suppose that g : Rm + IRk is another dzerentiable function, so that, by the chain rule,

By our definition,

This looks horribly complicated, but, using (11, it can be written

thus we have

g*of* = (g of)*.

Thjs relation would clearly fall apart completely if f*(vp) were not in R T ( p i ; with our present definition of f*, it is merely an eiegant restatement of the chain rule.

Henceforth, we will state almost al1 concepts about Jacobian matrices, like rank or singularity, in terms of f,, rather than Df. The "tangent vector" of a curve c: EX + R" can be defined in terms of this concept, also. The tangent vector of c at t may be defined as

[If c happens to be of the form

then

tllis \lector lies along the tangent line to tlie graph of f at (1, f (l)).] Notice thai the tangeni vector of c at 1 is the same as

c*(l*) = [Dc(t)(UIc(t) = (cl/(t), . - , cns(l))c(*),

wliere l t = (1,I) is iIie "unit" tangent vector of R at 1.

If g : Rn + E X m is differentiable, then g o c is a curve in Rm. The tangent

T h 7angent Bundle

vector of g o c at 1 is

( g O c)*(lr) = g*(c*(lr)) = g,(tangent vector of c at t ) .

Consider now an n-dimensional manifold M and an imbedding i : M + W N . Suppose we take a coordinate system ( x , U ) around p. Then i o x-' is a map from EXn to W N with rank n. Consequently, (i o x- ' ) , (W\(,)) is an n-dimensional subspace of W N i ( , ) . This subspace doesn't depend on the coordinate

system x., for if y is another coordinate system, then

(i o y- ') , = ( i O x-' o x O y - 1 ) ,

= (i o x - ' ) , o ( X O y-1)*

and

(x 0 y-')*y(,) : + W n x ( P )

is an isomorphism (with inverse ( y o x - ' ) , ~ ( , ) ) .

There is another way to see this, which justifies the picture we have drawn. If c: ( - & , E ) + W" is a curve with c(0) = x(p), then a = i o x-' o c is a curve in EXN which lics in i(M), and every dxerentiable curve in i (M) is of this

form (Proof?). Now ar(lO) = (i O x-')* o lo),

so the tangent vector of every a is in (i ox-'), (EXn,(,)). Moreover, every vector in this subspace is the tangent vector of some a, since every vector in EXn,(,) is tlie tangent vector of some curve c. Thus, our n-dimensional subspace is just the set of al1 tangent vectors at i(p) to dxerentiable curves in i(M). We will denote this n-dimensional subspace by (M, i),.

We now want to look at the (disjointj union

We can define a "projection" map

As in the case of TWn, each "fibre" x- ' (p) has a vector space structure also. Beyond this we have to look a little more carefully at some specific examples.

Coiicider first the maiiifold M = S' and tlie inclusion i : S' + EX2. Tlie curve c(O) = (cos @,sin O) passes through every point of S', and

cr(0) = (- sin O, cos O) # O.

For eacli p = (cos O, sin O) E S', let up = (- sin O, cos O), (it clearly doesn't matter wliich of the infinitely many possible 0's we choose). Then (S', i), con-

The Ta?gel2t Bundle 69

sists of al1 multiples of the vector u,,. We can therefore define a homeomorphism

h : T(S1, i) + S' x IR1 by h (hu,,) = ( p , A), which makes the following diagram commu te.

If we define tlie "Abres" of n' to be the sets n'-' (p), then each fibre has a vector space structure in a natural way. Commutativity of the diagram means that $i takes fibres into fibres; clearly f] restricted to a fibre is a linear isomorphism onto the image.

Now consider the manifold M = s2 and tlie inclusion i : s2 c R3. In thjs case tliere is «o map f2 : T ( s ~ , i ) + s2 x R2 with the properties of the map h. If there were, then, for a fixed vector u # O in R2, the set of vectors

would be a collection of non-zero tangent vectors, one at each point of s 2 , which varied coiiiiriuously. It is a well-known (hard) theorem of topology that this is impossible (you can't comb the hair on a sphere).

There is another example where we can prove that no appropriate homeomorphism T ( M , i) -t M x EX2 exists, without appealing to a hard theorem of topology. The rnap i will just be the inclusion M +- where M is a Mobius strip, to be precise, the particular subset of IR3 defined in Chapter 1-M is the image of the map f : [O, 2x1 x (- 1 , l ) + R3 defined by

At each point p = (2 cos 8 , 2 sin 8 , O ) of M , the vector

u,, = (-2 sin 8,2 cos 0, O)p - - f*((l f0)(e,0))

is a tangent vector. The same is true for al1 multiples of f,((O, I ) ( ~ , ~ ) ) , shown as dashed arrows in the picture. Notice that

while

This means tliat we can never pick non-zero dashed vectors continuowEy on the set of al1 points (2 cos 8,2 sin @ , O ) ; If we could, then each vector would be

for some continuous function A: [0,2rr] + R. This function would have to be non-zero everywhere and also satisfy h(2rr) == -h(O), which it can't (by an easy theorem of topology). The impossibility of choosing non-zero dashed vectors contiiiuously clearly sliows that there is no way to map T ( M , i), fibre by fibre,

The 7 a z g e ~ Bundle 71.

homeomorphically onto M x IR2. We thus have another case where T (M, i) does not "look like" a product M x Rn.

For any imbedding i ; M + RN , however, the s tmc ture of T (M, i) is always simple locab: if ( x , U ) is a coordinate system on M , then n - ' ( U ) , the part of T ( M , i) over U, can always be mapped, fibre by fibre, homeomorphically onto U x Rn. In fact, for each p E U, the fibre

( M , i ) , equals (i o x- ' )*~( ,) (WnX(, ) ) = mp (Rnx(,)) ,

where the abbreviation 172, lias been introduced temporarily; we can therefore define

/: n-'(U) + U x R"

by / (m, (U,(,))) = ( p , u) -

In standard jargon, T(M, i) is "locally trivial". This additional feature qualifies T ( M , i) to be included among an extremely important class of structures:

An ~r-dimensional vector bundle (or n-plane bundle) is a five-tuple

where

(1) E and 3 are spaces (tlie "total space" and "base space" of 6, respec tively),

(2 ) n ; E + B is a continuous map onlo B,

(3) $ and O are maps

with $ ( n - ' ( p ) x n - ' ( p ) ) c n - ' ( p ) and @ ( W x n - ' ( p ) ) c n-' (p), which make each fibre ir-' ( p ) into an n-dimensional vector space over IR,

such that the following "local triviality" condition is satisfied:

Foi. each p E B, there is a neighborhood U of p and a homeomorphism t : n -' ( U ) + U x 8" which is a vec tor space isomorphism from each n - ' ( q ) onto q x Rn, for al1 q E U.

Because tliis local triviality condition really is a local condition, each bundle 6 = (E ,n , B, $,O) automatically gives rise to a bundle 6lA over any subset A c B; to be precise,

Notation as cumbersome as al1 tliis invites abuse, and we shall usually refer simply to a bundle rr ; E + B, or aren denote the bundle by E alone. For vectors u, w E n-' (p) and a E R, we wiil denote $(u, w) and @(a, v) by v + w, and a . v or av, respectively.

The simplest example of an n-plane bundle is just X x Rn with x : X x R" + X the projection on tlie first factor, arid tlie obvious vector space structure on each fibre. This is called the trivial H-plane bundle over X and will be denoted by E"(X). Tlie "tangent bundle" TR" is just E"(R").

The bundle T(S' , i) considered before is equivalent to E' (S'). Equivalente is here a tecl~iiical te1.m: Two vector bundles 61 = XI : El + B and 6 2 = n2 : E2 + B are equivalent (ti h., C2) if there is a homeomorphism h : El + E2 which takes each fibre ni- ' (p) isomorphically onto n2-' (p). The map h is called an equivalence. A bundle cquivalent to E" (B) is called trivial. (The local triviality condition for a bundle 6 just says that ,$[U is trivial for some neighborhood U of p.)

The bundles T ( s ~ , i) aiid T(M, i) are not trivial, but tliere is an even simpler example of a non-trivial buiidle, The Mobius strip itsey (not T(M, i)) can be i.,ir~sidered as a 1-dimeiisional vector bundle over S' , for M can be obtained 1iom [O, I ] x R by identifying (0,a) with ( 1 , -a), wliile S' can be obtained from

The Tangen.? Bundle 73

[O, 11 by identifying O with 1; the map rr is defined by x(1.a) = t for O cr r cr 1 aild ~ ( ( ( 0 , a), (1, -a))) = {O, 1). The diagram above illustrates local triviality iiear the point {O, 1) of S'. SupPose that S: S' + M is a continuous function with rr o S = identity of M (such a function is called a section). Such a map

corresponds to a continuous function S : [O, 11 + R with S(0) = -S(l). Since S must be O soinewliere, t he section s must be O somewliere (that is, S(@) E rr-l(8) must be the O vector for some 0 E S '1. This surely sliows that M is not a trivial bundle.

An equivalente is obviously the analogue of an isomorphism. The analogue of a homomorphism is the following.* A bundle map from c1 to c2 is a pair of continuous maps (7, f), with f : El + E2 and f : Bl + Bz, such that

(1) the following diagram commutes

(2) /: nl-'(p) + rrz-' (f (p)) is a linear map.

Tlie pair (f., f ) is a bundle map kom T R k to TR' for any dxerentiable f : IRk + IR'. If MW c Rk and N m c IR1 are submanifolds. i : .M + IRk and j : N + IR1 are tlie inclusions, and the rnap f satisfies f (M) C N, then f,

* Tliere are actually severa1 possjble choices, depending o11 whether one is considering al1 buridles at once, fixed bundles over various spaces, or a fwed base space with varyiiig buiidles. Thus f may be restricted to be an isomorphism on fibres and f to be tlie identity or a Iion~eomorphism. The relations between some of these cases are considered in the problems.

takes T ( M , i ) to T ( N , j ) ; to see this, just remember that v E T ( M , i ) p is the tangent vector of a curve c in M , so &(u) is the tangent vector of the c u m f o c in N, and consequently f+ (u) E T ( N , j ) . In this way we obtain a bundle map from T ( M , i) to T ( N , j ) . Actually, it would have sufficed to begin with a Cm function f : M + N, since f can be extended to IRk locally. In fact, this construction could be generalized much further, to the case where i and j are merely imbeddings of two abstract manifolds M and N, and f: M + N is Cm; we just consider the function j o f o i-' : i ( M ) + i ( N ) and extend it locally to 3. The case which we want to examine most carefully is the simplest: where M = N and f is the identity, while i and j are two imbeddings of M in IRk and dR', respectively. Elements of T ( M , i)p are of the form (i o x-' ). (tu)

for w E dnxnx(p), while elements of T ( M , j )p are of the form ( j Q x-'). ( w ) for w E IRnx(,). If we map

(i O x-'), ( w ) H (j o X-'), ( w )

we obtain a bundle map from T ( M , i)lU to T ( M , j)lU, which is obviously an equivalence. The map ( M , i)p + ( M , j ) p iiiduced on fibres is independent of the coordinate system x , for if ( y , V) is another coordinate system, then

We can therefore put al1 tliese maps together, and obtain an equivaEence from T ( M , í) to T ( M , j ) . In otlier words, the dependence of T ( M , i) on i is almost illusory; we could abbreviate T ( M , i) to T M , if we agreed that T M really denotes an equivalence class of bundles, rather than one bundle. That is the

The 7angenl Bundle 75

sort of thing an algebraist might do, and it is undoubtedly ugly. What we would like to do is to get a single bundle for each M, in some natural way, which has al1 the properties any one of these particular bundles T (M, i) has. Can we do ihis? Yes, we can. When we do, TRn will be dflerent from our old definition (namely, &"(IRn)), and so will f, for f : Rn + Rm, so in stating our result precisely we will write "old f," when necessary.

f . THEOREM. It is possible to assign to each n-manifold M an n-plane bundle TM over M, and to each CbO rnap f : M + N a bundle map (f,, f ) , such that:

(1) If 1 : M + M is the identity then 1,: TM + T M is the identity. If g : N + P, then (g o f) , = g, o f,.

(2) There are equivalentes t" : T Rn + E" (R") such that for every Cm function f : Rn + Rm the following commutes.

(3) If U C M is an open submanifold, then T U is equivalent to (TM) 1 U, and for f : M + N the map (f IU), : TU + T N is just the restriction of f,. More precisely, there is an equivalence T U 2 (TM)IU such that the following diagrams commute, where i : U + M is the inclusion.*

PR00F. T h e construction of TM is an iilgenious, tllough quite natural, sub- terfuge. We will obtain a single bundle for TM, but the e h e n t s of TM will eacli be large equivalence classes.

* M7hen usiiig the notation $*, it must be understood that the symbol 'y" realiy refers to a triple (f, M, N) wliere f : M + N. The identitymap 1 of U to itself and the inclusion map i : U + M have to be considered as different, since the maps 1, : TU + TU and i, : T U + TM are certainly dfleren t (they map T U into two different sets).

The construction is much easier to understand if we first imagine that we already had our bundles T M . Then if ( x , U ) is a coordinate system, we would have a map x , : TU + T ( x ( U ) ) , and this would be an equivalence (with inverse (x-').). Since T U should be essentially (TM)IU3 and T ( x (U)) should essentially be x ( U ) x Rn, a point e E n - ' ( p ) would be taken by x. to some ( x ( p ) , u) . Here v is just an element of Rn (and every v would occur, since x* maps a-' ( p ) isomorphically onto { p ) x R"). If y is another coordinate system, then y, (e) would be ( y ( p ) , w ) for some w E Rn. 'We can easily figure out what tlle relationship between v and w would be; since ( x ( p ) , u ) is taken to ( y @ ) , w ) by y, o x,-' = ( y o x - ' ) , , and ( y o x- ' ) , is supposed to be the old ( y o x-'). , we would have

This condition makes perfect sense without any mention of bundles. It is the clue wliicli enables us to now define T M .

If x and y are coordinate systems whose domains contain p, and u , w E R", we define

(x, U ) ; ( y , w ) if (a) is satisfied.

It is easy to check (using the cbain rule) that - is an equivalence relation; the P

equivalence class of ( x , u ) will be denoted by [ x , u], . These equivalence classes will be called tangent vectors at p3 and TM is defined to be the set of al1 tangent vectors at al1 points p E M ; the map n takes - equivalence classes to p . We

P define a vector space stmcture on n-' ( p ) by the formulas

tliis definition is independent of the particular coordinate system x or y , because D ( y o x - ' ) (x ( p ) ) is an isomorphism from RR to R".

Our definition of TM provides a one-one onto map

(b> tx : n-' ( U ) -t U x Y , namely [ x , vJq H (q , u) .

We want tliis to be a homeomorphism, so we want tx-' ( A ) to be open for every open A C U x RR3 and thus we want any union of such sets to be open. There is a metric witli exactly these sets as open sets, but it is a little ticklish to produce, so wc leave this one part of the proof to Problem 1.

We now have a bundle n : TM + M . We will denote the fibre n - ' ( p ) by Mp, iii conformity with the notation Rnp, tbough TMp might be better. If

The Znge~d Bundle 77

J': M + N, and (x, U) and (y, V ) are coordinate systems around p arid f (p), r-espectively, we define

Of course, it must be cbecked that this definition is independent of x and y (the chain rule again),

Condition (1) of our theorem is obvious. To prove (2), we define t n to be t l , where I is the identity map of R" and t ,

is defined in (b); it is trivial, though perhaps confusing to the novice, to prove commutativity of the diagram.

Condition (3) is pi4actically obvious also. In fact, the fibre of TU over p E U is almost exactly the same as the fibre of TM over p ; the only difirence is that each equivalence class for M contains some extra members, since in M there are more coordinate systems around p than there are in U c M. *3

Henceforth, the bundle rr : TM + M will be called the tangent bundle of M. If i : M + Rk is an imbedding, then TM is equivalent to T (M, i). In fact, if (x, U) is a coordinate system around p, and I is tbe identity coordinate system of IRk, then

the composition tnic is easily seen to be an equivalence. But T (M, i) will play no further rolc in this story-the abstract substitute TM will always be used instead.

Having succeeded in producing a bundle over each M, which is equivalent to T(M, i), we next ask how fortuitous this was. Can one find other bundles with the same properties? Tlie answer is yes, and we proceed to define two different such bundles.

For tbe first example, we consider curves c: (-&,E) + M, each defined on some interval around 0, witb c(0) = p. If (x, U) is a coordinate system around p , we define

x ocl and x oc2, mapping IR to IR", cl 7 c2 if and only if:

have the same derivative at O.

The equivalence classes, for al1 p E M, will be the elements of our new bundle, T'M. For f: M + N tliere is a map S# taking the % equivalence class

P

of c to the Jz) equivalence class of f o c. Without bothering to check details,

we can already see that this example is "redly the same" as TM -

the 7 equivdence class of x-' o y, [x , v J p corresponds to:

where y is a c u m in Rn with y'(0) = u;

under this correspondence, $# corresponds to f,. In the second example, things are not so simple. We define a tangent vector

at p to be a linear operator which operates on al1 CbO functions f and which is a "derivation at p":

We have already seen that the operators = a/axilp have this property For these operators, clearly t ( f ) = l ( g ) if f = g in a neighborhood of p. This condition is actually tme for any derivation t. For, suppose that f = O in a neighborhood of p. There is a Cm function h : M + R with h ( p ) = 1 and support h c f - ' ( O ) . Then

Thus, if f = g in a neighborhood of O , then O = e ( f - g ) = e ( f ) - l ( g ) . If f is defined only in a neighborhood of p, we may use this trick to define e ( f ) : choose h to be 1 on a neigliborhood of p, with support h c f -l ( O ) , and define [ ( S ) as [ ( f h ) .

Tlie set of al1 sucli operators is a vector space, but it is not a jjriori clear what its dimension is. This comes out of the following.

2. LEMMA. Let f be a CbOfunction in a convex open neighborhood U of O in IRn, with f ( 0 ) = O. Then there are CbO functions gi : U + R with

(Tlie second condition actually follows from the first.)

PROOF. For x E U, let Ax ( t ) = f ( t x ) ; this is defined for O 5 t 5 1 , since U is convex. Then

Therefore we can let g ( x ) = J: D i f ( t x ) d t . 9

The Tangeni Bundle 79

3. THEOREM. The set of al1 linear derivations at p E M" is an n-dimensional vector space. In fact, if (x, U ) is a coordinate system around p, then

span tbis vector space, and any derivation l can be written

(so e is determined by the numbers l ( x i ) ) .

PROOE Notice that

so l ( 1 ) = O. Hence l ( c ) = c l(1) = O for any constant function c on U. Consider the case where M = R" and p = 0. Assume U is convex. Given f on U, choose gi as in Lemma 2, for the function f - f (O) . Then

n

l ( f ) = e ( f - f ( o ) ) = e (li denotes the ith coordinate func tion)

This sliows tliat a / a liio span the vector space; they are clearly linearly independent. It is a simple exercise to use the coordinate system x to transfer this result fmm R" to M. 4 3

Froni Tbeorcm 3 we caii see tbat, once again, a bundle constructed from al1 derivations at al1 points of M is "really tlie same" as TM. We can let

the formula a

P j = ]

derived in Chapter 2, shows that

n a n a ayj x a i - 1 = b i - 1 ifand only d bj = i= 1

a x i i = ~ ayi p i =l

ax i

and this is precisely the equation which says that (x ,a) -, (y,b) . It is easily checked that under this correspondence, the map which corresponds to f, can be defined as foilows:

E f * ( 0 1 ( g ) I= e ( g o f n

Notice that if x denotes the identity coordinate system on Rn, then a' - axl ,

corresponds to ap when we identi@ TRn with E" (Rn). i=l . a l Wc will usually make no distinction whatsoever between a tangent vector

v E MP and the linear derivation it corresponds to, that is, between Ex, a], and

consequently, we will not hesitate to write u ( f ) for a differentiable function f defined in a neighborhood of p. In fact, a tangent vector is often most easiiy described by tclliiig what derivation it corresponds to, and the map f, is often most easiiy analyzed from the relation

11 is customary io denote the identity coordinate system on IR' by r , and to write

ior 11 ; dt 2 , at 1,

this is a basis for Rz,,. If c : R + M is a differentiable curve, then

is called tlie tangent vector to c at lo. We will denote it by the suggestive symbol

The 7 a n g n l Bundle 81

This symbol will be subjected to the standard abuses one finds (unexplained) in calculus textbooks: the symbol

- dC will often stand for 1 , dt dt l

the subscript "t" now denoting a particular number t E R, as well as the identity coordinate system.

As you might well expect, it is no accident that our second and third exarnples turned out to be "really the same" as TM. There is a general theorem that a11 "reasonable" examples will have this property, but it is a littie delicate to state, and quite a mes5 to prove, so it has been quarantined in an Addendum to this chapter.

The tangent bundle TM of a Cm manifold has a little more stmcture than an arbitrary n-plane bundle. Since TM locally looks like U x Rn, clearly TM is itself a manifold; there is, moreover, a natural way to put a Cm stmcture on TM. If x : U + IRn is a chart on M, then every element u E (TM)IU is uniquely of the form

Let us denote a ' by X i (u). Then the map

U H (,Y1 (H (U)), . . . , xn (ir (U)), x1 (U), . . . , Xn (U)) E IR2"

is a homeomorphism from (TM)IU to x (U) x R". This map, ( x o x, X), is simply the map x, when we identify TU with U x Rn in the standard way If (y, V) is anotlier coordinate system, and

then, as we have already seen,

This shows that if (t,a) = (zl,. . . , t n , a l , . . . , an ) E IR2", then

Y* 0 (x*)-I (1, a)

= ( y o x - 1 (t), xFZl ai Di (y1 0 x-l) (t)> - - - , ai ~i ( Y" 0 x-l) (t))-

This expression sliows that y* o (x*)-' is Cm.

\/Ve thus have a collection of Cm-related charts on T M , which can be extended to a maximal atlas.

With this Cm structure, the local trivializations x, are Cm. In general, a vector bundle r r : E + B is called a Cm vector bundle if E and B are Cm manifolds and there are Cm local trivializations in a neighborhood of each point. It follows that rr : E + B is Cm.

Recall that a section of a bundle rr : E + B is a continuous function s : B + E such that rr OS = identity of B; for Cm \lector bundles we can also speak of Cm sections. A section of T M is called a vector field on M ; for submanifolds M of Rn, a vector field may be pictured as a continuous selection of arrows tangent to M. The theorem that you can't comb the hair on a sphere just states that

there is no vector field on s2 which is werywhere non-zero. We have shown that there do not exist two vector fields on the Mobius strip which are everywhere linearly independent.

Vector fields are customarily denoted by symbols like X, Y, or 2, and the \rector X(p) is often denoted by X, (sometimes X may be used to denote a single vector, i ~ i some M,). If we think of T M as the set of derivations, then for any coordinate system (x, U), we have

n

X(P) = C ~ ' ( P ) for al1 p E U. i=l

The functions a' are continuous or Cm if and only if X: U + T M is continuous or Cm.

If X and Y are two vector fields, we define a new vector field X + Y by

Similarly, if f : M + R, we define the vector field JX by

The 7angeni Bundle 83

Clearly X + Y and f X are Cm if X, Y, and f are Cm. On U we can write

the symbol a/axi now denoting the vector field

If f : M + R is a Cm function, and X is a \lector field, then we can define a new fwidion 2 ( f ) : M + B by letting X operate on f at each point:

It is not hard to check that if X is a Cm vector field, then f ( f ) is Cm for every Cm function f ; indeed, iflocally

then

which is a sum of products of Cm functions. Conversely, if y( f ) is Cm for e u q Cm function f , then X is a Cm vector field (since 2 ( x i ) = a').

Let F denote the set of al1 Cm functions on M. We have just seen that a Cm vector field X @ves rise to a func tion 9 : P + P. Clearly,

tlius 2 is a "derivation" of the ring F. Ofien, a Cm vector field X is identified witli the derivation f . Tlie reason for this is that if A : P + 7 is any derivation, then A = 2 for a unique Cm vector field X. In fact, we clearly rnust define

X , ( f > = A ( f ) ( P ) ,

and the operator XP thus defined is a derivation at p.

The tangent bundle is the tme beginning of the study of dflerentiable manifold~, and you should not read further until you grok it.* The next few chapters constitute a detailed study of this bundle. One basic theme in all these chapters is that any structure one can put on a vector space leads to a structure on any vector bundle, in particular on the tangent bundle of a manifold. For the present, we will discussjust one new concept about manifolds, which arises in this very way from the notion of "orientation" in a vector space.

The non-singular linear maps f : V + V from a finite dimensional vector space to itself fall into two groups, those with det f > 0, and those with det f < 0; linear transformations in t he first group are called orientation preserving and the others are called dentat ion reversing. A simple example of the latter is

1 the map f : Rn + R" defined by f (x) = (x , . . . ,xn-l, -xn) (reRection in the llyperplane xn = O). There is no way to pass continuously between these two groups: if we identify linear maps Rn + R" with n x n matrices, and thus with IRn2, then the orientation presewing and orien tation reversing maps are disjoint open subsets of the set of al1 non-singular maps (those with det # 0). The terniinology "orientation presewing" is a bit strange, since we have not yet defined anything called "orientation", which is being preserved. The problem becomes more acute if we want to define orientation preseMng isomorphisms between two dflerent (but isomorphic) vector spaces V and W; this clearly makes no sense unless we supply V and W with more structure.

To provide tliis extra structure, we note that two ordered bases (vi,. . . , u,) and (vf1,. . . , vfn) for V determine an isomorphism f : V + V with f(vi) = vfi; the matrix A = (aij) of f is given by the equations

We cal1 (vi , . . . ,u,) and (vfl,. . .,u',) equally oriented if det A > O (i.e., if f is orientation preserving) and oppositely oriented if det A < 0.

Tlle relation of being equally oriented is clearly an equivalence relation, dividing tlie collection of al1 ordered bases into just two equivalence classes. Either of iliese two equivalence classes is called an orientation for V. The class to which (vi,. . . ,un) belongs will be denoted by [vi,. . . ,vn], so that if p is an orientation of V, then (vi , . . . ,un) E ,u if and only if [vi,. . . ,un] = p. If ,u denotes one

* A cult word of the sixties, "grok" was coined, purportedly as a word from the Martian language, by Robert A. Heirilein in his pop ccience fiction novel Shnger in a Slrange Lnttd. lts sense is nicely coiiveyed by the defiriitjoii in The Anzen'can Hdhge Diclio?tag*: "To understand profoundly through intuition or empathy".

The 7angeni Bundle

Examples of equally oriented ordered bases in R, IR2, and IR3.

orientation of V, tbe other will be denoted by - p , and the orientation [ e ] , . , . , e,] for R" will be called the "standard orientation".

Now if (V, p) aild (W, u ) are two ri-dimensional vector spaces, together with orientations, an isomorphism f : V + W is called orientation preserving (with respect to p and u) if [ f (v i ) , - . -, f ( v , ) ] == v whenever [ V I , . . . , u,] = p; if this holds for any one ( v i , . . . , u,), it clearly holds for all.

For tlie trivial bundle P ( X ) = X x R" we can put the "standard orientation" [ ( x , el), . . . , ( x , e,)] on each fibre {x) x Rn. If f : e n ( X ) + e n ( X ) is an equivalente, aiid X is connected, tlien f is either orientation presewing or orientation reversing on eacli fibre, for if we define the functions aij : X + R by

then det (a i j ) : X + R is coniinuous and never O . If rr : E + B is a non- trivial 11-plane bundle, an orientation p of E is defined to be a collection of orientatioiis pp for rr-I ( p ) wllicll satisfy the following "~ompatibilit~ condition" for any open connected set U C B:

lf t : x-' ( U ) + U x IRn is an equivalence, and the fibres of U x IR" are given tlie standard orientation, tlien t is either orientation presewing or orientation reversing on all fibres.

Notice tliat if tliis condition is satisfied for a certain t, and t' : n-' ( U ) + U x Rn is another equivalence, then t' automatically satisfies the same condition, since

t 'o t -' : U x 1" + U x 1" is an equivalente. This shows that the orientations pp define an orientation of E if the compatibility condition holds for a collection of sets U which cover B.

If a bundle E has orientation p = {,up), it has another orientation -p = {-,up), but not every bundle has an orientation. For example, the Mobius strip, considered as a 1-dimensional bundle over S' , has no orientation. For, although the Mobius strip has no non-zero section, we can pick two vectors from each fibre so that the totality A looks like two sections. For example, we can let A be [O, i ] x {- 1, 1) with (O, a ) identified with (1, -a); then A just looks like the boundary of the Mobius strip obtained from [O, i] x [- 1 , 11. If

we had compatible orientations p,,, we could define a section S: S' -, M by choosing s (p ) to be the unique vector s (p ) E A nn-'(p) with [s(p)] = pp.

A bundle is called orientable if it has an orientation, and non-orientable otherwise; an oriented bundle is just a pair (6, p ) where p is an orientation for 6. Tliis definition can be applied, in particular, to the tangent bundle TM of a Cm manifold M. In this case, we cal1 M itself orientable or non-orientable depending on whether TM is orientable or non-orientable; an orientation of TM is also called an orientation of M, and an oriented manifold is a pair (M,p ) where p is an orientation for TM.

Tlle manifold Rn is orientable, since T W 2 E" (Rn), on which we have the standard orientation. The sphere S"-' c Rn is also orientable. To see this we

note that for each p E S"-' the vector w = pp E en(Rn) 2 TR" is nol in i. (Sn-',,) E T Rn,, (Roblem 2 l), so for VI , . . . , v.- 1 E Sn-lp we can define (vi , . . . , un- ' ) E ,up if and only if (w, i,(vi), . . . , i, is in the standard orientation of Rnp. The orientation p = {/L, : p E S"-' ) thus defined is called the "standard orientation" of S"-'.

The iorus S' x S' is aiiotlier example of an orientable manifold. This can be seen by noting that for any two manifolds Mi and M2 the fibre (Ml x M2)p of T (MI x M2) can be written as VI $ V2p where (ni)* : K,, + (Mi),, is an isomorphism and the subspaces Vi, vary continuously (Problem 26). Since Ts' is trivial, this shows that T (S' x S') is also trivial, and consequently orientable. Any n-holed torus is also orientable-the proof is presented in Problem 16, which also discusses the tangent bundle of a manifold-with-boundary.

The Mobius strip M is the simplest example of a non-orientable 2-manifold. For tlie imbedding of M considered previously we have aiready seen that on the

subset S = ((2 cos 8 ,2 sin 8, O)) c M there are continuously varying vectors u,,, but tliat it is impossiWe io clioose continuously from among the dashed vectors wp = f*((O, I ) ( ~ , ~ ) ) and their negatives. If we had orientations /l, for p E S, then we could simply choose wp if [up, wp] = pp and -wp otherwise.

The projective plane IP2 must be non-oiieniable also, since it contains the Mobius strip (for any orientable bundle 6 = rr : E + By the restriction 6IB' to aiiy subset B' c B is also orientable). Non-orientability of IP2 can be seen in aiiother way, by considering the "antipodal map" A : S2 -t S2 defined by A(p) = -p. This rnap is just the restriction of a linear map A: R3 + R3 defined by tlie same formula. The rnap A * : S2,, -t s ~ ~ ( ~ ) is just ( p , ~ ) H

A(v)), wlien S2,, is identified with a subspace of {p ) x R3. The map A is orientation reversing, so if vi = (p, ui) E SZP, the bases

(ui, u2, p) and (A(ui), A(u2), A(P))

are oppositely oriented. Tliis shows that if ,u is the standard orientation of S2 and [ul, vi] E ,up, then [A,v1, A,v2] E -/,LA(,,). Thus the map A : S2 + s2 is

"orientation reversing" (the notion of an orientation preserving or orientation reversing map f : M + N makes sense for any imbedding f of one oriented manifold into another oriented manifold of the same dimension). From this fact it follows easily that IP2 is not orientable: If IP2 had an orientation v = {v[,l) and g : s2 -t IP2 is the map p H [ p ] , then we could define an orientation {ji,) on S" by requiring g to be orientation preserving; the map A would then be orientation preserving with respect to f i , which is impossible, since k. = p or -p.

For projective 3-space IP3 the situation is just tlle opposite. In this case, the antipodal map A : S' + s3 ir orientation presening. If g : S 3 + IP3 is the map p H [P], we obviously can define orientations v, for IP3 by requiring g to be orientation presewing. In general, these same arguments show that IP" is orientable for n odd and non-orientable for n even.

There is a more "elementary" definition of orientability which does not use the tangent bundle of M at all. According to this definition, M is orientable if there is a subset A' of the atlas A for M such that

(1) tlle domains of al1 (x, U) E A' cover M,

(2) for al1 (x, U) and (y, V ) E A',

An orientation p of TM allows us to distinguish the subset A' as the collection of al1 (x, U) for which x, : TM lU + T(x(U)) i x(U) x IRn is orientation preserving (when x(U) x IR" is given the standard orientation). Condition (2) liolds, because it is just the condition that (y o x-'),: T(x(U)) + T(x(U)) is orientation preserving. Conversely, given A' we can orient the fibres of TM IU in such a way that x, is orientation preserving, and obtain an orientation of TM. Although our original definition is easier to picture geometrically, the determinant condition will be very important later on.

The 7angeni Bundle

ADDENDUM EQUIVALEMCE OF TANGENT BUNDLES

The fact that al1 reasonable candidates for the tangent bundle of M turn out to be essentially the same is stated precisely as foilows.

4. THEOREM*. If we have a bundle T'M over M for ea& M, and a bundle map (h, f ) for each Cm map f : M + N satisfying

(1) of Theorem 1,

(2) of Theorem 1, for certain equivalences t'",

(3) of Theorem 1, for certain equivalences T'U h: (TfM)IU,

then there are equivalences

eM : TM + T'M

such that the following diagram commutes for every Cm map f : M + N.

PROOF. The details of this proof are so horrible that you should probably skip it (and you should definitely quit when you get bogged down); the welcome symbol 4 4 occurs quite a ways on. Nevertheless, the idea behind the proof is simple eilough. If (x, U) is a chart on M, tllen both (TM)I U and (T'M) 1 U "look like" x(U) x Rn, so there ought to be a map taking the fibres of one to the fibres of the other. Wliat we have to hope is that our conditions on TM and T'M make them "look alike" in a sufficiently strong way for this idea to really work out. Those who have been through this sort of rigamarole before know (i-e., have faith) that it's going to work out; those for whom this sort of proof is a new expei-iencc should find it painful and instructive,

* Functorites will iiotice that Theorems 1 and 4 say that there is, up to natural equivaleiice, a uiiique functor from the category of Cm manifolds and Cm maps to the category of bundles and bundle maps which is naturally equi\~alent to (E", old f,) on Euclidean spaces, and to the restriction of the functor on open submanifolds.

Let ( x , U ) be a coordinate system on M . Then we have the following string of equivalences. Two of them, which are denoted by the same symbol 2 , are the equivalences mentioned in condition (3). Let a, denote the composition a, = ( f n l x ( u ) ) 0- 0 x* 0 (-)-l.

Similarly, using equivalence 2' for TI , we can define p,.

Then

Bx-' o ax : (TM)IU -t (T'M)IU

is an equivalence, so it takes tlie fibre of T M over p isomorphically to the fibre of T I M over p for ea& p E U . Our main task is to show that this isomorphism between the fibres over p is independent of the coordinate system ( x , U ) . This will be done in three stages.

(1) Suppose V c U is open and y = xlV. We will need to name all the inclusion maps

To compare a, and ay, consider the following diagram.

TIze 7angeni Bundle 91

Each of the four squares in this diagam commutes. To see this for square O, we enlarge it, as sllown below. The two triangles on the left commute by con- - dition (3) for TM, and the one on the right commutes because i o j = r .

Square @ commutes because k o y = x o j. Square @ commutes for the same reason as square @; tlie inclusions x ( U ) + Rn and y(V) + IRn come - into play. Square (4) ob~iously commutes. Chasing through diagram (1) now shows tllat the follo-ving commutes.

This means that for p E V, the isomorphism ay bemeen the fibres over p is the same as a,. Clearly the same is true for B, and By, since our proof used only properties (l), (21, and (3), not the explicit construction of TM. Thus By-' o (Y,, = B x - l O ax on the fibres over p, for every p E V.

(11) We now need a Lcmma which applics to both TM and TIM. Again, it will be proved for TM (where it is actually obvious), using only properties (l), (2), and (3), so that it is also true for T'M.

LEMMA. If A c IRn and B c IRm are open, and f : A + B is Cm, then the following diagram commutes.

PROOF. Case 1. %e ir a map /: Wn + Rm with 7 = f on A . Consider the following diagram, where i : A + IRn and j : B + IRm are the incIusion maps.

Everythirlg in tliis diagram obviously commutes. This implies tliat the two compositions

Ev tnlA C old /; TA 2 (T R")I A d &"(Rn)I A d E" ( I R n ) - .em(Rm)

and

are equal and this proves the Lemma in Case 1, since the maps "old T*'' and "old f," are equal on A.

&e 2. General' cae. For each p E A , we want to show that m o maps are the same on the fibre over p. NOW there is a map j : IRn + IRm with j' = f on an open set A', wl~ere p E A' C A. We tben have tlie following diagram, wliere every i comes from the fact tliat some set is an open submanifold of another,

The Gngeni Bundle

and i: A' + A is the inciusion map.

Boxes @, 0, and @ obviously commute, and commutes by &e l . To see that square @ (whidi has a triangle witliin it) commutes, we imbed it in a larger diagram, in which j: A + R" is the incIusion map, and other maps have also been named, for ease of reference.

T Rn

h/ ( u )

TA E

(TRn)IA

i*]

TA1 E

+ ( T P ) I A r

To prove that h o i, = p o K, it suffices to prove that

since v is one-one. Thus it suffices to prove j* o i* = v o p o o, which amounts to proving commutativi ty of the foilowing diagram.

Since j o i is just the inclusion of A' in Rn, this does commute.

Commutativity of diagram (2) shows that the composition

coincides, on the subsel (TA)I A', with the composition

and on A' we can replace "old /," by "old f,". In other words, the two compositions are equal in a neighborhood of any p E A, and are thus equal, which proves the h m m a .

(111) Now suppose ( x , U) and (y, V) are any two coordinate systems with p E U n V. To prove that By-' 0 a, and Bx-' o a, induce the same isomorphism un the fibre of TM at p , we can assume without loss of generality that U = V, because part (1) applies to x and xl U n V, as well as to y and y1 U n V.

Assuming U = V, we have the following diagram.

Tlle triangle obviously commutes, and tlle rectangle commutes by part (11). Diagram (3) thus shows tbat

ry

(3) (TM)lU 4 TU

Exactly the same result llolds for T':

(y 0 h.-1)* old (y o x-'),

The desired result B,-' o a, = B,-' o a, follows immediately. Now that we have a well-defined bundle map TM + T'M (the union of al1

Bx-' 0 a,), it is dearly an equilraience e M . The proof that eni o f, = ffi o e M is left as a masochisiic exercise for the reader. *:*

a . - O-R")ly(U)

n l e Zngeni Bundle

PROBLEMS

1. Let M be any set, and { ( x i , Ui)) a sequence ofone-one functions xi : Ui + Rn with Ui c M and x (Ui) open in Rn, su& that each

is continuous. It would seem that M ought to have a metric which makes each Ui open and each xi a homeomorphism. Actually, this is not quite true:

(a) Let M = R U {s), where s 6 R. Let U] = R and x l : Ul + R be the identity, and let U2 = R - {O) U {s) , with x2: U2 + R defined by

Show that there is no metric on M of the required sort, by showing that every neighborhood of O would have to intersect every neighborhood of s. Never- theless, we can find on M a pseudometric p (a function p : M x M + R with a11 properties for a metric except that p (p, q) may be O for p S; q) such that p is a metric on each Ui and each xj is a homeomorphism: (b) If A C IRn is open, then there is a sequence A l , A2, A j, . . . of open subsets of A such that every open subset of A is a union of certain Ai's. (c) There is a sequence of continuous functions 1;. : A + [O, l], with support J c A, which "separates points and closed sets": if C is closed and p E A - C, then there is some j;: with fi (p) 6 (A n C). Hint: First arrange in a sequence

all pairs (Ai, Aj) of part (b) with C A l . (d) Let Alj, j = 1,2,3,. . . be such a sequence for each open set xi (Ui)- Define gi,j : M + [o, 11 by

Arrarlge a11 gil] in a single sequence Gl, G2, G3,. . . , let d be a bounded metric on R, and define p on M by

Show that p is the required pseudometric. (e) Suppose that for every p,q E M there is a Ui and Uj wit.h p E Ui and q E Uj and open sets Bi c xi(Ui) and Bj c xj(Uj) SO that p E xi-'(Bi), q E xj-'(l?j), and xi-'(Bi) n xj-'(Bj) = 0. Sliow that p is actually a metric on M.

Tle Tangeni Bundle 97

9. (a) Show that the correspondence between TM and equivalence classes of curves under which [x , u], corresponds to the 7 equivalence clau of xd l o y, for y a curve in R" with y'(0) = u, makes f, correspond to $#. (b) Show that under the correspondence [x,a] , n Ci aia/axilp, the map f , can be defined by

Ef*re)l(g) = [ (g f ) .

10. If V is a finite dimensional vector space over R, define a Cm structure on V and a homeomorphism from V x V to T V which is independent of choice of bases. As in the case of R", for u, w E V we will denote by u, E Vw the vector corresponding to ( w , u).

1 1. If g ; R + R is Cm show that

for some Cm function h : R + R.

12, (a) Let F, be the set of al Cm functions f : M + R with f ( p ) = 0, and 1et e : F, + R be a linear operator with e( fg) = O for al1 f,g E F,. Show that l has a unique extension to a derivation. (b) Let W be the vector subspace of 3, generated by al1 products fg for f , g E

F,. Show that the \rector space of al1 derivations at p is isomorphic to the dual space (.7;p / W ) *. (c) Since (F,/ W)* has dimension n = dimension of M, the same must be true of F,/ W. If x is a coordinate system with x ( p ) = O, show that x1 + W , . . . , x" + W is a basis for F,/ W (use Lemma 2). The situation is quite different for C1 functions, as the next problem shows.

13, (a) Let V be the \lector space ofall C1 functions f : R + R with f (0) = 0, and let W be the subspace generated by ail products. Show that lim f ( x ) / x2

x-o exists for al1 f E W. (b) Foro < E < 1, jet

Show that al1 $, are in V, and that they rcpresent linearly independent elements of V / W. (c) Conclude that (V / W)* has dimension cC = 2'.

14, If f ; M + N aild f, is the O map on each fibre, then f is constant on each component of M.

15, (a) A map f : M + N is an immersion if and only if f* is one-one on each fibre of TM. More generally, the rank of $ at p E M is the rank of the linear transformation f , : Mp + N f ( p l .

@) If f o g = f , where g is a difkomorphism, then the rank of f o g at a equals the rank of f at g(a ) . (Compare with Problem 2-33(d).) 16. (a) If M is a manifold-with-boundary, the tangent bundle T M is defined exactly as for M ; elements of Mp are y equivalente classes of pain ( x , u). Although x takes a neighborhood of p E aM onto W n , rather than R", the ueciors v still run through IR", so Mp still has tangeilt vectors "pointing in ail directions". If p E aM and x : U + W" is a coordinate system around p, then

x,-' (Rn-',(,)) c M, is a subspace. Show that tliis subspace does not depend on the choice of x; in fact, it is i,(aM),,, where i : aM + M is the inclusion. (b) Let a E IRn-' x { O ) c W n . A tangent vector in W", is said to point "inward" if, under the identification of T W n with & " ( M n ) , the vector is (a , u) where un > O. A vector v E Mp which is not in i,(aM)p is said to point "inward" if

x* (u) E Wnqp1 points inward. Show that this definition does not depend on the coordinate system x. (c) Show tllat if M has an orientation p, then aM has a unique orientation a p such that [v i , . . - , = (ap),, if and only if [w , i,vi,. . . , i , ~ , - ~ ] = pp for every outward pointing w E Mp. (d) If p is the usual orientation of W", show that a,u is (- l ) n times the usual orientation of IRn-' = aWn. (The reason for this choice will become clear in Chapter 8.) (e) Suppose we are in the setup of Problem 2-14, Define g : 8M x [O, 1 ) + aN x [O, 1 ) by g ( p , t) = ($ (p ) , 1 ) . Show that T P is obtained from T M U T N

Tlze Z q m i Bundle

(f) If M and N have orientations p and u and f : (aM,ap) + (aN,au) is orientation-reuer~rg, show that P has an orientation which agrees with p and u on M c P and N c P. (g) Suppose M is s2 with two holes cut out, and N is [O, 11 x S ' . Let f be a dfiomorphism from M to N which is orientation preserving on one copy of S' aiid orientation reversing on the other. Wliat is the resulting manifold P?

17. Sliow that T P ~ is homeomorphic to the space obtained from T ( s ~ , ~ ) by identifying ( p , u) E (s2, i), with ( -p , -u ) E (s2, i)-,.

18. Although there is no everywhere non-zero \rector field on s2, there is one on s2 - { ( O , 0, l ) ) , which is dfiomorphic to IR2. Show that such a vector field can be picked so that near ( O , 0 , l ) the vector field looks like the following picture (a "magnetic dipole"):

19. Suppose we have a "multiplication" map (a , b) H a - b from IR" x R" to R" that inakes R" into a (non-associative) division algebra. That is,

(al + a2) b = al b + a2 b

a (bi +b2) = a .bl +a .b2

h(a - b ) = (ha) b = a (hb) for h E R

a ( l , O , . , . , O ) = a

and there are no zero divisors:

a , b # O ab#O.

(For example, for 12 = 1, we can use ordinary multiplication, and for n = 2 we can use "complex multiplication", (a, b) (c, d) = (ac - bd, a d + bc).) Let e l , . . . ,e, be the standard basis of Rn,

(a) Every point in S"-' is a - el for a unique a E Rn. @) If a # O, then a el, . . , , a en are linearl~ independent. (c) If p = a + el E S"-', then the projection of a - ez,. . . ,a . e, on (Sn-', i), are linearly independent. (d) Mu~tiplication by a is continuous, (e) TSn-' is trivial. (f) TP"-' is trivial.

The tangent bundles TS3 and TS' are both trivial. Multiplications with the required properties on R4 and R8 are provided by the "quaternions" and "Cayley numbers", respectively; the quaternions are not commutative and the Cayley numbers are not even associative, It is a classical theorem that the reals, complexes, and quaternions are the only associative examples. For a simple proof, see R. S, Palais, TLv Classi;cation oJReal DiuisiDn Akebrdi, Amer, Math. Monthly 75 (1968), 366-368. J. E Adams has proved, using methods of algebraic topology, that n = 1, 2, 4, or 8.

pncidentally, non-existence of zero divisors immediately implies that for a # O there is some b with ab = (1,0,. . . ,O) and b' with b'a = (1,0,. . . ,O). If the multiplication is associative i t follows easily that b = b', so that we always have multiplicative inverses. Conversely, tliis condition implies that there are no zero divisors if the multiplication is associative; otherwise it suffices to assume the existence of a unique b with a b = b a = (1, O , , . , , O).]

20. (a) Consider the space obtained from [O, 11 x Rfl by identifying (0, u) with (1 , Tu), where T : R" + Rn is a vector space isomorphism. Show that this can be made into the total space of a vector bundle over S' (a generalized Mobius strip), @) S11ow that the resulting bundle is orientable if and only if T is orientation preserving.

21. Show that for p E s 2 , the \rector pp E R3, is not in i.(s2,) by showing that the inner product (p, c'(0)) = O for al1 curves c with c(0) = p and Ic(l)l = 1 for al1 t . (Recall that

wliere denotes the traiispose; see C a h b on Man@Idr, pg. 23-1

22, Let M be a Cm manifold, Suppose that (TM)I A is trivial whenever A c M is homeomorphic to S'. Show that M is orientable. Hinl: An arc c from

The h g m t Bundle 101

po E M to p E M is contained in some cuch A so (TM)lc is trivial. Thus one can ('transportJi the orientation of Mpo to Mp. It must be checked that this is independent of the choice of c. First consider pairs c, c' which meet only at p0 and p, The general, possibly quite messy, case can be treated by breaking up c into s m d pieces contained in coordinate neighborhoods,

Remark: Using results from the Addendum to Chapter 9, together with Prob- lem 29, we can conclude that a neighborhood of some S' c M is non-orientable if M is non-orientable,

The next two problems deal with important constructions associated with vector bundles.

23. (a) Suppose 6 - x : E + X is a bundle and f : Y + X is a continuous map. Let E' c Y x E be the set of al1 ( y , e ) with f ( y ) = x (e ) , define n': E' + Y by x r ( y , e ) = y, and define f: E' + E by j ( y , e ) = e. A vector space structure can be defined on

by using the vector space structure on x-'( f (y ) ) . Show that x': E' + Y is a bundle, aiid ( j , f ) a bundle map which is an isomorphism on each fibre. This bundle is denoted by f * ( C ) , and is called the bundle induced (from 6 ) by f . (b) Suppose we have anotlier bundle 6" = n": E" + Y and a bundle map - (Y , f ) from 6" to 6 wliich is an isomorphism on each fibre. Show that 6" 2 - ' = f * ) Hid: Map e E E" to (+"'e), ?(e)) E E'. (c) If g : + y, then ( f og)*(6) 8*(f*(6)) . (d) If A C X and i : A + X is the inclusion map, then i* ( 6 ) 2 6 1 A. (e) If 6 is orientable, then f *(e) is also orientable. (f) Give an example where 6 is non-orientable, but f * (6 ) is orientable. (g) Let e = n : E + B be a vector bundle, Since n : E + 3 is a continuous map from a space to the base space B of 6 , the symbol n * ( ( ) makes sense, SIiow that if 6 is not orientable, then x*(6) is not orientable.

24, (a) Given an n-plane bundle 6 = x : E + B and an m-plane bundle 7 =

rr': E' + 3, 1et E" C E x E' be the set of al1 pairs (e,ef) with x ( e ) = x1(e'). Let x"(e,el) = x ( e ) = nl(e'). Show that n": E" + B is an (n + m)-plane I~undle. It js called the Whitncy sum 6 $ 7 of 6 and 7; the fibre of 6 $ 7 over p is the direct sum n-' ( p ) $ n/-' ( p ) . (b) If f : Y + 3 , show that f 4 ( 6 $ 7 ) 2r S*(.$) $ f4 (7) .

(c) Given bundles ti = ni: Ei + Bi, define X : El x E2 + B1 x B2 by rr(el,e2) = (rr1(ei),rr2(e2)). Show that this is a bundle x c2 over Bl x B2. (d) If A: B + B x B is the "diagonal map", A(x) = (x,x), show that e $ z ~ EZ

A*(6 ZJ). (e) If 6 and q are orientable, show that e $ z~ is orientable. (f) If 6 is orientable, and ZJ is non-orientable, show that $ ZJ is also non- orientable. (g) Define a "natural" orientation on V $ V for any vector space V, and use this to show that 6 $ 6 is always orientable. (11) If X is a "figure eight" (c.L Problem 5), find two non-orientable 1-plane bundles 6 and q over X such that 6 $ q js also non-orientable.

25, (a) If rr r E + M is a Cm vector bundle, then rr, has maximal rank at ixch point, and each fibre n-' (p) is a Cm submanifold of E. (b) The O-section of E is a submanifold, carried dfiomorphically onto B by rr.

26. (a) If M and N are Cm manifolds, and l i ~ [or rrN] : M x N + M [or N] is the projection 017 M [or N], then T(M x N) h. XM*(TM) $ rrhr*(TN). (b) If M and N are orientable, then M x N is orientable. (c) If M x N is orientable, then both M and N are orientable,

27. Show that the Jacobian matrix of y, o (x.)-' is of the form

Tliis shows that the manifold TM is alz~rqs ork~ttabk, i,e., the bundle T(TM) is orientable. (Here is a more coilceptual formulation: for v E TM, the orientation for (TM), can be defined as

thc form of y, o (x,)-' shows that this 01-ientation is iildependeni of the choice of A-.) A differeilt proof that TM is orieiltable is $ven in Problem 29.

28, (a) Let (x, U) be a coordinate system on M with x(p) = O and let v E Mp he Cy=l a' a/axi lp . Consider the curve c in TM defined by

Tlze 7ageni Bundle

Show that

(b) Find a curve whose tangent vector at O is a/a(xi o rr)I,.

29. This problem requires some familiarity with the notion of exact sequences - f 2

(c.f. Chapter 11). A sequence ofbundle maps El -+ E2 -+ E3 With $ = g = identity of B is exact if at each fibre it is exact as a sequence of vector space maps.

(a) If ,$ = rr : E + B is a CbO vector bundle, show that there is an exact sequence

O + rr*(.$) + TE + rr*(TB) + O.

f i t : (1) An element of the total space of ~ " ( 6 ) is a pair of points in the same fibre, which determines a tangent vector of the fibre. (2) Map X E (TE)e to (e, GX). (b) If O + El + Ez + E3 + O is exact, then each bundle Ei is orientable if the other two are. (c) T(TM) is always orientable, (d) If rr r E + M is not orientable, then the manifold E is not orientable. (This is why the proof that the Mobius strip is a non-oriei-itable manifold is so similar to the proof that the Mobius bundle over S' is not orientable.)

T h e ilext two Problems contain more information about the groups introduced in Problem 2-33. 111 addition to being used in Problem 32, this information wiH al1 be important in Chapter 10,

30. (a) Let po E S"-' be the point (0,. . . ,O, 1). For n 3 2 define / : SO(n) + S"-' by / ( A ) = A(po). Show that / is continuous and open. Show that /-' (po) is homeomorphic to SO(n - I), and then show that j-' (p) is homeomorphic to SO(n - 1) for ali p E S"-'. (b) SO(1) is a point, so it is connected. Using part (a), and induction on n, prove that SO(n) is connected for ali n > 1 . (c) Show that O(tz) has exactly two components.

31, (a) If T : IRn + IR" is a linear transformation, T*: IRn + Rn, the adjoint of T, is defined by (T*v, w) = (u, Tw) (for each u, the map w H (u, Tw) is linear, so it is w H (T*v, w) for a unique T*v). If A is the matrix of T with respect to the usual basis, show that the matrh of T* is the transpose A'.

(b) A linear transformation T : R" + Rn is self-adjoint if T = T*, so that (Tu, w ) = (u, Tw) for a l u, w E R". If A is the matrix of T d t h respect to the standard basis, then T is self-adjoint if and only if A is symmetric, A~ = A. It is a standard theorem that a symmetric A can be written as CDC-1 for some diagonal matiix D (for an analy& proof, see CaEculw on Man@ldr, pg. 122). Show that C can be chosen orthogonal, by showing that eigenvectors for distinc t eigenvalues are orthogonal. (c) A self-adjoint T (or the corresponding symmetric A) is called positive semi- ddn i te if (Tu, u) > O for al1 u E Rn, and positive definite if (Tu, u) > O for al1 v # O. Show that a positive definite A is non-singular. Hznt: Use the Schwarz inequality. (d) Show that A' - A is always positive semi-definite. (e) Sliow that a positive semi-definite A can be written as A = B2 for some B. (Remember that A is symmetrjc.) (f) Show tliat every A E GL(n, R) can be written uniquely as A = A l . A2 where A i E O(n) and A 2 is positive definite. Hht: Consider A' . A, and use part (e). (g) The matrices A l and A2 are continuous functions of A. Hin t : If A(") + A and A(") = A(")] ~ ( " 1 2 , then some subsequence of ( A ( " ) ~ ) converges. ) GL(i1, R) is homeomorpliic to O(n) x R"("("*')/~ and has exactly two components, ( A : det A > 0) and ( A : det A < O). (Notice that this also gives us anotller way of fiilding the dimension of O(n).)

32, Two continuous fuilctioils So, 5 : X + Y are called homotopic if there is a coiltjiluous function H: X x [O, i ] + Y such that

The functions H; : X + Y defined by Hl (x) = H(x, t ) may be thought of as a path of fuilctions fi-om Ho = fa to HI = J. The map H is called a homotopy between fo and J .

The i-iotation f : (X, A) + (Y, B), for A c X and B c Y, means that f : X + Y and f (A) c B. We cal1 f l ~ , J : (X, A ) + (Y, 3) homotopic (as maps from (X, A ) to (Y, B)) if there is an H as above such that each Hr : (X, A ) + (Y, B)-

(a) If A : [O, 11 + GL(n, R) is continuous aild H : R" x [O, 11 + R" is defiiled by H(x, 1 ) = A(r)(x), show that H is continuous, so tllat Ijg and Hl are homotopic as nlaps from (R" , R" - (O)) lo (Rn, R" - (O)). Conclude that a non-singular linear trailsformation T : (R", R" - (0)) + (R", R" - (O)) with det T > O is homotopic to ihe identity map. (b) Suppose f : R" + R" is Cm and f(0) = 0, while f (R" - (0)) c R" - (O). If Df(0) is non-singular, sl-iow that f : (R", Rn - (O)) + (R", R" - (O)) is

homotopic to Df (O) : (Rn,Rn - (0)) + (Rn,Rn - (O)). Hint; Define H(x , I ) = f (tx) for O < 1 5 1 and H(x, O) = D f (O) (x). To prove continuity at points (x, O), use Lemma 2. (c) Let U be a neighborhood of O E R" and f: U -4 Rn a homeomorphism with f (O) = O. Let B, C V be the open ball with center O and radius r , and let h : Rn + B, be the homeomorphism

then f 0 , z : (R",rw" - (O)) + (Rn,Rn - (O)).

MTe will say that f is orientation preserving at O if f o h is homotopic to the identity map 1 : (Rn, Rn - (0)) + (Rn, Rn - (O)). Check that this does not depend on the choice of B, c V. (d) For p E R", Jet Tp: R" + Rn be T'(q) = p + q . If f: U 4 V is a l-iomeomorphism, where U, V c Rn are open, we will say that f is orientation preserving at p if T-J(p) O f O T,, is orientation preserving at O. Show that if M is orientable, then there is a collection C of charts ~lhose domains cover M such that for every (x, U) and (y, V) in e, the map y ox-' is orientation presening at x ( ~ ) for a11 p E U n V. (e) Noiice that the condition on y o xd' in part (d) makes sense even if y ox-' is not djffereniiable. Thuus, if M is any (not necessarily dxerentiable) manifold, we can define M to be orientable if there is a collection C of homeomorphisms x: U + R" whose domains cover M, such that C satisfies the condition in part (d), To prove that this definition agrees with the old one we need a fact from algebraic iopology: If f : Rn + Rn is a homeomorphism with f (O) = O and T ; Rn S'> R" is ~ ( x ' , . . . ,x") = (x ' , . . . ,xn-l, -xn), then precisely one of f and T o f is orientation presen~ii-ig at O. Assuming this result, show that if M has such a collection C of l-iomeomorphisms, then for any Cm structure on M the tailgeiit bundle TM is orientable.

33. Let M" c RN be a Cm n-dimensional submanifold. By a chord of M we meai-i a point of IRN of the form p - q for p ,q E M.

(a) Prove tliat if N > 212 + 1, then there is a vector v E sN-l such that

(i) no chord of M is parallel to u,

(ii) no tangent plane M,, contains v .

Hint; Coilsider ceriain n-iaps from appropriate open subsets of M x M and TM to S"-'.

(b) Let IRN-l c IRN be the subspace perpendicular to U, and n: IRN + IRN-l

the correspoilding projection. Show that n l M is a one-one immersion. In particular, if M is compact, then rrl M is an imbedding. (c) Every compact Cm rl-dimensional manifold can be imbedded in IR2"+'.

Note: This is the easy case of M'hitney's classical theorem, which gives the same resul t even for non-compact manifolds (H. Mfhitney, fizermtzabk man@ldí, Ann. of Math. 37 (1935), 645-680). Proofs may be found in Auslander and MacKenzie, Intmduction ¿o Dzffmentiabk A4un@lds and Sternberg, Latures on fiy

fmential Geotne~t In Munkres, Eln~tentary Dflia~tial Zpology, there is a dflerent sort of argumerlt to prove that a not-necessarily-compact n-manifold M can be imbedded in some IRN (in fact, With N = (n + I ) ~ ) . T h a l we may show that M imbeds in IR2"+' using essentiaily the argument above, together with the existence of a proper map f : M + IR, given by Problem '2-30 (compare Guillemin ,:lid Pollack, fizmnlial Topology). A much harder result of Whitn y shows that M" can actually be imbedded in IR2" (H. Wllitney, nie selj-intersectioonr oja mooU1 u-t~ranfold in 2n-space, Ai111. of h4ath. 45 (1 944), 220-246).

CHAPTER 4

TENSORS

A I I the constructions on vector buildles carried out in this chapter have a common feature. In each case, we replace each fibre x - ' ( p ) by soirie

other vector space, and then fit al1 these new vector spaces together to form a new vector bundle over the same base space.

Tlie simplest case arises when we replace each fibre V by its dual space V*. Recal1 that V* deilotes the vector space of al1 linear functions A : V + R. If f r V + W is a linear transformation, then there is a linear transformation f * : W* + V* defined by

It is clear that if 1 v : V + V is the ideiitity, tlien 1 v* is the identity map of V* and if g : U + V, then (f o g)* = g* o f *. These simple remarks already sliow tliat f * is an isomorphism if f : V + W is, for (f -' o f )* = 1 v* and (f 0 f-l)* = 1W+.

Tlie dimensjon of V* is tlie same as that of V, fol. finite dimeilsional V. In fact, if vi,. . . , v, is a basis for V, theii the elements E V*, defined by

are easily cliecked to be a basis for V*. The linear fuiiction v*i depeilds on tlle eiltire set vi,. . .,v,, not just on vi alone, aiid tlle isomorphism from V to V* obtained by sending vi to is not indepeildeilt of the choice of basis (consider what happens if vi is replaced by 2vl),

On the othei. hand, if v E V, we caii define u** E V** = (V*)* unambiguously by

If v**(h) = O for every h E V*, then h(v) = O for al1 h E V*, which implies that

v = O. Thus the map v w v** is an isomorphism from V io V*'. It is called the natural isomorphism from V to V**.

(Problem 6 gives a precise meaning to the word "natural", formulated only after the term had long been in use. Once the meaning is made precise, we can prove that there is no natural isomorphism from V to V*.)

Now let 6 = n: E + B be any vector bundle. k t

and define the function x': E' + B to take each [n-'(p)]* to p- If U c B, lind I : ?r-' (U) + U x R" is a trivialization, then we can define a function

I' : nf-' (U) + U x (Rn)*

in the obvious way: sincc the map r restricted to a fibre,

tP : H-'(~) + {p) x R",

is an isomorphism, it gives us an jsomorphism

(g*)-' : [ir-'(p)]* + {p} x (R")'

We can makc n': E' + B into a vector bundle, t l ~e dud bundle 6' of 6, by requiring that al1 such i r be local trivializations. (We first pick an isomorphism from (Rn)* to R", once and for all.)

At fii-st jt n-iigllt appear tliat e* 2 6, since eacli n-'(p) is isomorphic to x ( p . However, tliis is true merely because t l~e two vector spaces havc tlie same dime~ision. Tl-ie lack of a natural isomorpl-iism from V to V* prevents us from constructing an equivalence between 6' and 6. Actually, we will see later that in "most" cases e* Zr equivalei-it to 6; for the present, readers may ponder tl-iis question for tliemselves. In contrast, the bundle e** = (t*)' is nlu~oys equivalcnt to 6. We consti-uct the cquivalence by mapping the fibre V of 6 over p to the fibre V** of t'* oler p by the natural isomorphism. If you

tl-iink about how e* is constructed, it will appear obvious that this map is indeed an equivalence.

Even if 6 can be pictured geometrically (e.g., if 6 is TM), there is seldom a geometric picture for C*. Rather, 6* operates on 6: If s is a section of 6 and o is a section of t*, then we can define a function from B to R by

s ( p ) E f i ( p ) P G ( P ) ( S ( P ) )

o ( p ) E x'-l ( p ) = ,-' (p )" .

This function wjll be denoted simply by o ( s ) .

When this construction is applied to the tangent bundle TM of M, the resulting bundle, denoted by T*M, is called the cotangent bundle of M; the fibre of T*M over p is (M,)". Like TM, the cotangent bundle T*M is actuaiiy a Cm vector bundle: since two trivializatjons X* and y, of TM are Cm-related, the same is clearly true for x.' and y,' (in fact, y.' o (x.')-' = y, o (x , ) - ' ) . 1% can thus define Cm, as wcll as continuous, sections of T*M. If w is a Cm section of T*M and X is a Cm lector field, then w ( X ) js the Cm function P l+ ~ ( P ) ( X ( P ) ) .

If f : M + R is a Cm function, then a Cm section df of T*M can be defined by

d f ( p ) ( X ) = X ( f ) for X E M,.

The sectio~i df is called tlie differentd of f . Suppose, in particular, that X is dcldt 11,,, where c(io) = p. Recall that

This means that

Adopting the elliptical notations

this equation takes the nice form

If (x, U) is a coordinate system, tlien tlie dxi are sections of T*M over U. Applyiiig the definition, we see tliat

Tlius dxl(p) , . . . , dxn(p) is just tlie basis of M,' dual to tlie basis a/axl l,, . . . , a/axnIP ~f M,.

Tliis means tl~at every section w can be expressed u~~iquely on U as

for certaiii functio~~s w; o11 U. Tlie section w is coiitiiluous or Coa if aiid only if the functions wi are.

We can also write

if wc define sums of sections and products of functions and scctioiis in the olwious way ("poii~rwise" additioii ai-id multiplication).

Tlie sectioii <f must liave some sucli expressioii. In fact, we obtain a classical formula:

1. THEOREM. If (x, U) is a coordiiiate system and f is a Cm function, then

tlien

Thus

C lassical differential geome ters (and classical aiialysts) did not liesi tate to talk about "infiiijtely small" clianges dxi of the coordinates x i , just as Leibnitz had. No one wanted to admit tliat tliis was nonsense, because true results were ob- taincd wlien tliesc iiifinitely small quantities wcre divided into each other (provided one did it in tlie right way).

Eventually it was rcalized that the closest one can come to describing an infinitely sinall cliange is to describe a directiori in wliicli tliis change is supposed to occur, Le., a tangent vector. Since df is supposed to be the infinitesimal changc of f uiidei- aii iiifiiiitesimal chaiige of tlic poirit, df must be a function of this chaiige, whicli means that df should be a function on tangent vectors. The dxi tliemselves tlien metamorpliosed into fuiictions, and it became clear tliat tliey must be distinguislied from tlie tangent vectors a/dxi.

Oiice this realization came, it was only a matter of making new definitions, which presenled tlie o k i notation, and waitiiig for everybody to catch up. In sliort, al1 classical iiorioiis involviiig iilfinitely small quantitics became functions on tangent vectors, likc df, eexcept for- quotieiits of infinitely smaü quantities, which became taiigent vectors, like dc/dr.

Looking back at tlic classical works fi-om our modern vantage point, one can usually see that, 110 niatter liow obscui*ely expressed, this point of view was in

some sense the one always taken by classical geometers. In fact, the differential d f was usually introduced in the following way:

CLASSICAL FORMULATION MODERN FORMULATION

LRt f be a function of the x', . . . , x", say f = f ( x l , . . . , x").

Let x i be functions of r , say x' = x i ( l ) . Then f becomes a function of 1 , f ( t ) = f ( x l ( r ) , . . . , -Y"( ¡ ) ) .

We now have

d f " a f -=EGd;. dr i= I

(Tlle classical notation, which suppresses tlie curve c , is still used by physicists, as we shall point out once again in Chapter 7.)

Multiplying by d r gives

" a.f d f = E - dx i . 1= 1

axl

(Tl-iis equation signifies that true results are obtained by dividing by dr again, 710 mal& what ~e functions x i ( t ) are. It is the closest approach in classical analysis to the realization of d f as a function on tangent vectors.)

k t j be a function on M, and x a coordinate system (so that f = f o x

for some function f on W", namely j = f 0.-'1.

Let c : W -. M be a curve. Then f o c : W + R, where

f O c(r ) = j ( x l o c ( I ) , . . . ,xn O e(¡)).

We now have

( f O c)I(r) n

= E D; j ( x ( ~ ( t ) ) ) m (xi o c ) ( ( I ) i = I

" a f = E ,(c(r)) - (xi 0 c)I(r) f =I

a~ or

d ( f ( c ( r ) ) ) " a f dx i (c (r ) ) dr =Ea;i(c(0). i= 1 di

Consequently,

i=l

Since every tangent vector at c(r) is of the form dcldr, we have

" a f d f = E - d x i . i= 1

a ~ i

In preparation for our reading of Gauss and Riemann, we will continually examine tlie classical way of expressilig al1 concepts which we introduce. After a whiie, the "translation" of classical terminology becomes only a little more difficult than tlie translation of the German in which it was written.

Recall that if f : M + N is Cm, then there is a map f,: TM + TN; for each p E M, we have a map f*p : Mp + Nf ( p ) . Since f Z p is a linear transformation between t~7o vector spaces, it gives rise to a map

Strict notational propriety would dictate that this map be denoted by (f,,)*, but everyone denotes it simply by

Notice tliat we cannot put al1 4 together to obtain a bundle map from T*N to T*-M; in fact, tlle same q E N may be f (pi) for more than one pi E M, and there is no reason why fWp, should equal f,p,. On the other hand, we can do sometl~ing with t l~e cotaiigent bundle that we could not do with the tangent bundle. Suppose w is a section of T*N. Then we can define a section rj of T*M as follows:

V(P) = w(f (p ) ) 0 f*p,

(The complex symbolism tends to hide the simple idea: to operate on a vector, we push it over to N by f*, and then operate on it by u . ) This section q is denoted, naturally enough, by $*m. There is no corresponding way of trans- ferring a vector field X on M wer to a vector field on N.

Despite these differences, we can say, roughly, that a map f : M + N produces a map f* going in tlie same direction on tlle tangent bundle and a map f * going in the opposite direction on tlle cotangent bundle. Nowadays such situations are always distinguished by calling the things which go in the same direction "covariant" and the tliings which go in the opposite direction "contravariant". Classical terminology used these same words, and it just happens to llave reversed tllis: a vector field is called a contravariant vector fieId, whiie a section of T*M is called a covariant vector fieId. And no one has had the gall or autllority to reverse terminology so sanctified by years of usage. So it's very easy to remember which kii-id of vector field is covariant, and which contravariant -itJs j ust t l ~e opposite of wliat it logically ought to be.

The rationale behind the classical terminology can be seen by considering coordinate systems x on R" which are linear transformations. In this case, if x(vi) = el, then

1 1 x ( a vi + . v . + a%,)= (a , . . . ,a") ,

so h e x coordinate system is just an "oblique Cartesian coordinate system".

If x' is aiiotlier such coordinate system, then x'j = =yzI aijxi for certain aij. Clearly aij = ax"/axi, so

ilib can be seeii directly from die fact that h e matrix (ax'J/axi) is the constant inatrix D(xt o -Y-' ) = xt o x-' . Comparing (*) with

fi-om Tlieorem 1, we see tliat tlie differeiitials dxi "cliange in tlie same way" as tlic coo~diiiatcs x t , lience tliey are "covai-iant". Coiisequently, any combination

is also called "covariant". Notice that if we also llave

o = oti dx",

t l~en \ve can express the wti in terms of the wt. Substituting

n axi

dx' = E dxtJ j==l

into tlle first expression for w and comparing coeffrcients with the second, we find that

1 W j =

t-1

On the otller hand, given two expressjons

for a vector field, tlie functions atj must satisfy

Tllese expressioiis can always be remembered by iioting that indices which are summed over always appear once "above" and once "below". (Coordinate functions x ' , . . . , x" used to be denoted by xl, . . . ,A-.. Tliis suggested subscripts wi for covariaiit vector fields and superscripis o' íor coiiti.avariant vector fields. M- ter tllis was fiimly established, the indices oli the x's were shifted upstairs again to inake tlle summation convention work out.)

Covariar-it and contravariant vector fields, i.e., sections of T*M and TM, respectively, are also called covariant ar-id coi1 travarian t tensors (or tensor fields) of order 1 , which is a warning that worse things are to come. We begin with some worse algebra.

If VI, . . . , Vm are vector spaces, a function

is limar for each choice of VI,. . . ,vk-l,vk+lr.. . ,um. The set of aii such T is clearly a vcctoi- space. If VI , . . . , Vm = V, this vector space will be denoted

by Tm(V). Notice that 7' (V) = V*. If f : V + W is a linear transformation, then there is a linear transformation $" : T m ( W) + Tm (V), defined completely analogously to the case m = 1:

For T E Tk (V), and S E 7' (v) we can define the "tensor product" T @S E

T'+'(v) by

Of course, T @ S is not S @ T. On the otl-ier lland, (S @ T) @ U = S @ ( T @ U), so we can define n-fold tensor products unambiguously; this tensor product operation is itself multilinear, (Sl + S2) 8 T = SI 8 T + S2 8 T, etc. In particular, if V I , . . . , vn is a basis for V and v * ~ , . . . , v*, is the dual basis for V* = T' (V), then the elemen ts

are easily seen to be a basis for T ~ ( V ) , which ihus has dimension nk. We can use this new algebraic construction to obtain a new bundle from any

\rector bundle 6 = ?r : E + B. We let

E' = u Tk(?r-l (p)), P@

and let n i : E' + 3 take T~ (s-' (P)) to p.

If U c B and r : n - I ( u ) + U x w n

is a ti+i\4alization7 then the isomorphisms

ip : n-I (p) + {p} x Rn

yield isomorpllisms

(rpL)-' : Tk (n-' (p)) + {p} x 7 " ~ " ) .

If we choose an isomorphism Tk(wn) + Rnk once and for all, these maps can bc put together to give a map

\Ve rnake kerr>: E' + B into a vector bundle T~ (6) by requiring that al1 such i' 1)e local trivializations. The bundle 6* is the special case k = 1 .

For the case of T M , h e bundle T ~ ( T M ) is called the bundle of covariant tensors of order k, and a section is called a cwariant tensor fidd of order k. If ( x , U) is a coordinate system, so that

is a basis for (Mp)* , tlien tlie k-fold tensor products

are a basis for T k (M, ) . Thus, on U every covariant tensor feld A of order k can be written

A ( P ) = Ai, ... i , ( p ) dx i ( ( p ) @ - . @ dxik ( p ) , i l ,..., ik

or simply

A = E Ai, dxi1 8 - - . @ d x f k ,

wllere dxil 8 - - @ dx-'k riow denotes a section of T ~ ( T M ) . If we also have

then

(tlie pi-oducts are just 01-diiiary products of fuilctions). To denve this equation, we just use equation (**) on page 114, and multilinearity of B. The section A is continuous or Coo if and only if the functions Ai,.. .;, are.

A covariant tensor feld A of order k can just be tliought of as an operation 2 on k vector fields XI , . . . , Xk which yields a function:

Notice that A is rnultilinear on h e set V of Cm vector fields:

Moreover, because A is defined 'pointwise", it is actually linear ouer h e Cm fuízdions F; ie. , if f is Coo, then

for we have

\Ve are finally ready for anotlier theorem, one that is used over and over.

is liiiear over Y, theii tliel-e is a unique tensor field A with A = A. PXOOF. Note first that if v E Mp is any tangent \lector, then there is a vector Geld X E V witll X(p) = v. In fact, if (x, U) is a coordinate system and

then we can define

( 0 outside U,

~ ~ l i e r e eacli af now denotes a constant function and f is a Cm function with f (p) = 1 and support f c U.

Now if v1, . . . , vk E Mp are extended to vector fields Xf , . . . , Xk E V we clearly must define

The problem is to prove tl-iat tl-iis is well-defined: If Xr(p) = Yi(p) for eacl-i i, we claim that

d(x1, - -, Xk)(p) = A(Y1 3 - ., Yk)(p)*

(The map A "lives at points", to use tlie in terminology.) For simplicity, take the case k = 1 (tlle general case is exactly ailalogous). The proof that A ( X ) ( p ) = A ( Y ) ( p ) when X ( p ) = Y ( p ) is in two steps.

(1) Suppose first that X = Y in a neighborhood U of p. Let f be a Cm function witli f ( p ) = I and support f c U. Then f X = f Y, so

evaluating at p gives ~ ( X ) ( P ) = & ( Y ) ( P ) .

(2) To prove tlle result, it obviously suffrces to show that A ( X ) ( p ) = O if X ( p ) = O. k t (x, U) be a coordinate system around p , so that on U we can write

n m

x = ~~~d where al1 b ' ( p ) = O. axi

f - l

If g is 1 i ~ i a iieigliboi-liood V of p, and support g c U, tlien

is a well-defined Cm \rector field on al1 of M wliich equals X on V, so that

Now

= O, since b f ( ~ ) = 0. +3

Because of Tlieoi-em 2, we will ilevei- distinguisli between tlie tensor field A and tlie ol~csation A, ilor will we use the symbol A any longer. Note that Tlieorem 2 applies, in particular, to tlie case k = 1, wliere 7 ' ( T M ) = T'M, tlie cotangent bui-idle: a fiiilctioii fi-om V + 9 wliicll is linear over 9 comes fi-om a covariaiit vectoi- field u. Just as witli covariant vector fields, a Cm map f : M + N @ves a map f* taking covariant tensor fields A of order k on N to covariaiit tei-isor fields f * A of 01-der k on M:

Moreover, if A and B are covariani tensor fields of orders k and I , respectively, then we can define a new covariant tensor field A @ 3 of order k + I :

( A @ B) (p) = A (p) @ B(p) (operating on Mp x . x Mp k + I times).

Although covariant tensor fields will be our main concern, if only for the sake of completeness we should define coiitravariant tensor fields. Recall that a contravariant vector field is a seciion X of TM. So each Xp E Mp. Now an element v of a vector space V can be thought of as a linear function v : V' + R; we just define v(h) to be h(v). A contravariant tensor field of order k is just a section A of the bundle T ~ ( T * M ) ; thus, each A(p) is a k-linear function on Mp*. We could also use the notation Tk(TM), if we use Tk (V) io denote al1 k-linear functioiis on V'. In local coordinates we can write

(remember that each a/axj 1, operates on Mp*), or simply

If we have another such expression,

tl-ien we easiiy compute that

A contravariant tensor field A of order k can be considered as an operator A iaking k covariant vector fields wl , . . . , wk into a function:

Naturally, tliere is aii analogue of Tlieorem 2, proved exactly the same way, that allows us to dispense with the notation A, and to identify contravariant tensor fields of order k with operators on k covariant vector fields that are linear over ,!he Coo funciioni F.

Finally, we are ready to introduce "mixed" tensor fields. To make the introduction less painful, we consider a special case first. If V is a vector space, let q' ( V ) denote al1 bilinear functions

A vector bundle e = a : E + B gives rise to a vector bundle q' ( e ) , obtained by replacing each fibre K-' ( p ) by q' (a -' ( p ) ) . In particular, sections of ( T M ) are called tensor fields, covariant of order 1 and contravariant of order 1.

There are al1 sorts of algebraic tricks one can play with 7,' ( V ) ; although they should be kept to a minimum, certain ones are quite important. k t End(V) denote the vector space of al1 linear transformations T: V + V ("endomor- phisms" of V ) , Notice that each S E End( V ) @ves rise to a bilinear S E II;' ( V ) ,

by the formula

Moreover, the correspondence S I+ S from End(V) to q' ( V ) is linear and one- one, for S = O implies that A ( S ( v ) ) = O for al1 A, which irnplies that S ( v ) = O, for al1 v . Since both End(V) and ( V ) have dimension n 2 , this map is an isomorphisrn. The inverse, however, is not so easy to describe. Given 3, for each v the vector S ( v ) E V is merely determined by describing the action of a h on it according to (*). It is not hard to check tl-iat this isomorphism of End(V) and T 1 ( V ) makes the identity map 1: V + V in End(V) correspond to the c c evaluation" map

e : V x V * + W in T'''(v) given by

e ( v , A) = A(v) .

Generally speaking, our isomorpliism can be used to transfer any operation from End(V) to ?''(V). In particular, given a bilinear

we can take the trace of the corresponding S : V + V ; this number is called the contraction of T . If v i , . . . , v , is a basis of V and

then we can find the matrix A = (aij) of S, defined by

in terms of the T,' ; in fact,

contraction of T = T:.

(The term "contraction" comes from the fact that the number of indices is contracted from 2 to O by setting tl-ie upper and lower indices equal and summing.)

Tl-iese identifications and operations can be carried out, fibre by fibre, in any fibre bundle T,' (5). Thus, a section A of 7'' (5) can just as well be considered as a section of tlie bundle End(h), obtained by replacing each fibre a-' (p) by ~nd (a - ' ( p ) ) . In tliis case, each A(p) is an endomorphism of a-'@). More- over, each section A gives rise to a function

(contraction of A) : B + R

defined by

p I+ contraction of A (p)

= trace A ( p) if we consider A (p) E ~ n d (n-' @))

In particular, given a tensor field A, covariant of order 1 and contravariant of order 1, wliich is a section of (TM), we can consider each A (p) as an endomorpllism of M,, and we obtain a function "contraction of A". If in a coordinate system

tl-ien

(contraction of A ) = y A:.

Tlle general notion of a mixed tensor field is a straightfonvard generalization. Define 7jk ( V) to be the set of al1 (k + !)-linear

k times l times

Every bundle 5 gives rise to a bundle ~ ' ( 5 ) . Sections of q ( T M ) are called tensor fields, covariant of order k and contravariant of order i, or simply of type (f), an abbreviation that also saves everybody embarrassment about the use of

the words "covariant" and "coníravarian t". Locally, a tensor field A of type (f ) can be expressed as

and if

then

Classical differen tia1 geometry books are filled with monstrosities like this equation. 111 fact, the classical definition of a tensor field is: an assignment of nk+* functions to every coordinate system so that (*) holds between the nk+' functions assigned to any two coordinate systems x and x'. (!) Or even, "a set of ,rk+' functions which changes according to (*)" . Consequently, in classical differential geonletry, al1 important tensors are actually defined by defining the

functions A;, in terms of the coordinate system x, and then checking that (*) holds.

Here is an important example. In every classical dxerential geometry book, one will find the following assertion: 'The Kronecker delta 6; is a tensor." In other words, it is asserted that if one cliooses the same n2 functions 6; for each coordinate system, then (*) holds, i.e.,

this is certainly true, for

From our point of view, what this equation shows is that

a A = C6; dx' @ -

i, j a x j

is a certain tensor field, independent of the choice of the coordinate system x To identify the mysterious map

we consider v E Mp and h E Mp* with the expressions

then

Thus A ( p ) is just tl-ie evaluation map M p x Mp* + R; considered as an ei-ido- morphism of M,, it is just the identity map.

Tl-ie conti.actjon of a tensor is defined, classically, in a similar manner. Given a tensos, Le., a collection of functions A!, one for each coordinate system, satisfying

we note that

n axi =EA/E-- i , j a-l

axiU a x j

so tliat tliis sum is a well-defined function. Tliis calculation tends to obscure the one part which is really necessary-verification of the fact that the trace of a linear transformation, defined as the sum of the diagonal entries of its matrix, is indcpendent of the basis with respect to which the matrix is written.

Incideiitally, a teiisor of type (f) can be contracted with respect to any pair of upper and lower indices. For example, the functions

a-l

"transform correctly" if the AaPY do. If we consider each A (p) E T.. ( M ~ ) , then we are taking B(p) E qZ(Mp) to be

~ ( p ) ( v ~ , vz, Al, Az) = contraction of: (v, A) H A(p)(v, vi, vz7 A l 7 hz, A).

Wlliie a corltravai-iant vector feld is classically a set of n functions which "transforms in a certain way", a vector at a single point p is classically just an assigiiment of n tiumbns u ' , . . . , a n to each coordinate system x , such that the numbers a", . . . ,a'" assigned to x' satisfy

Tllis is preciseb the defi~iition we adopted when we defined tangent vectors as equivalente classcs [A-, a],. The revolutioil iil tlle modern approach is tliat the set of al1 vectors is made into a bundle, so t l~at \lector fields can be defined as sections, ratller tlian as equivalente classes of sets offunctions, and that al1 other

types of tensors are constructed from this bundle. The tangent bundle itselfwas almost a victim of the excesses of revolutionary zeal. For a long time, the party line held tliat TM must be defined either as derivations, or as equivalence classes of curves; the return to the old definition was influenced by the "functorial" point of view of Theorems 3-1 and 3-4.

The modern revolt against the classical point of view has been so complete in certain quarters that some matliematicians will give a three page proof that avoids coordinates in preference to a three line proof that uses them. We won't go quite that far, but we will give an "invariant" definition (one that does not use a coordinate system) of any tensors that are defined. Unlike the "Kronecker delta" and contractions, such invariant definitions are usually not so easy to come by. As we shall see, invariant definitions of al1 the important tensors in differential geometry are made by means of Theorem 2. We seldom define A ( p ) directly; instead we define a function A on vector fields, which miraculously turns out to be linear over the Coo functions Y, and hence must come from some A . At t l~e appropriate time we will discuss wliether or not this is al1 a big cheat.

PROBLEMS

l. LRt f : M" + Nm, and suppose that (x, U) and (y, V ) are coordinate systems around p and f (p), respectively.

(a) If g : N + R, then

(Proposition 2-3 is tlie special case f = identity.) (b) Show that

and, more generally, express j;(C;,, aia/axi 1 P ) in terms of the a/ayjIp. (c) Show that

(d) Express

in terms of the dx'.

2. If f , g: M + N are Cm, sl~ow tllat

3. LRt f : M + R be Cm. For v E Mp, show tllat

4. (a) Sllow that if the ordered bases vi , . . . , v, aiid wl , . . . , W , for V are equaiiy oriented, tllen tlle same is true ofthe bases v*I, . . . , v*, and w * ~ , . . . , u*, for V*. (b) Show that a bundle 6 is orientable if and only if e* is orientable.

5. The following statements and problems are al1 taken from Eisenhart's classical work Riemalznian Geometgt. In each case, check them, using the classicai methods, and then translate the problem and solution into modern terms. An "invariant" is just a (well-defined) function. Remember that the summation convention is always used, so h i p i means x7=, h i p i . Hints and answers are given at the end, after (xiii).

(i) If the quantity hip i is an invariant and either hi or pi are the components of an arbitrary [covariant or coiitravariant] vector field, tlie otlier sets are components of a vector field. (ii) If haIi are the components of n vector fields [in an n-manifold] , where i for i = 1 , . . . , I Z indicates the component and a for a = 1 , . . . , n the vector, and tbese vectors are independent, that is, det(hUii) # O, then any vector-field h' is expressible in the form

h' = ~ ~ h , ~ ' ,

where the a's are invariants. (iii) If p; are the componeiits of a given vector-field, any vector-field hi satisfyiiig h i p i = O is expressible linearly in terms of ,z - 1 independent vector fields h,lr for a = 1 , . . . , n - 1 wliicli satisfy the equation. ( i ) If uij = aj i for the components of a tensor field in one coordinate system,

. . then U"J = arJ' for the coordinates in any other coordinate system. (1) If uij and bi; are compoiieii ts of a teiisor field, so are aij + b * ~ . If aiJ aiid bki are components of a tensor field, so are ai jbki . (vi) If uohihJ is an invariant for 1' an arbitiary vector, then uij + aji are the componeiits of a teiisor; in particular, if uUhihj = O, then aij + aji = O. (vii) If uijhihJ = O for al1 vectors hi sucli tliat hip i = O, wliere pi is a given covariant vector, ir v i is defined [c.f. (iii)] by a i j h U i i ~ j = 0, a = 1,. . . , n - 1 and pivi # 0, and by definition

1 tlieii ( ~ ' j - 7 p i o 1 ) e i e j = O is satisfied by every vector field P i , and consequently

(viii) If a,, are tlie componeiits of a tensor aiid b aiid c are invariants, sl~ow tliat if bu,, + ea,, = O, tlien either b = -c and u,, is symmetric, or b = c and u,, is skew-symme tric. (ix) By delinition tlie rnlzk of a tensor of the second order a;j is tlie rank of the matrix (u t j ) . Show tliat the rank is invariant under al1 transformations of coordinates.

(x) Show tliat the rank of the tensor of components aibj, where ai and bj are the components of two vectors, is one; show that for the syrnmetric tensor aibj + ajbi the rank is two. (xi) Show that the tensor equation aijhi = ahj , where a is an invariant, can be written in the form (aij - aGij)hi = O. Show also that aij = Gija, if the equation is to hold for an arbitrary vector hi. (xii) If aijh< = a h j holds for al1 vectors hi S U C ~ that pihi = O, where ,ui is a @ven vector, then

i a j = 0161 + oi.pi.

( O if j, = j~ for some a # or i, = ip for some a # fi . . ~JI...JI> - or if { j l , . . . , jp} # {i l , . . . , ip}

il ... i, - 1 if j l , . . . , jp is an even permutation of i l , . . . , ip

( - 1 if jl , . . . , jp is an odd permutation of ii , . . . , ip

then 6tA.'!' ... I , are tlie components of a tensor in al1 coordinate systems.

HINTS AND ANSWERS.

(i) w is determined if w (X) is known for al1 X, and uice versa. (iii) Given w (with w(p) # O for al1 p], there are everywhere linearly independent vector fields X1, . . . , X,-1 which span ker w at each point. (This is true only locally. For example, on x IR ihere is an w such that ker w(p, í) consists of vectors tangent to S2 x {í}.) (vi) For T : V x V + IR, let Tt(v, w) = T(w, u). Tlien T + T' is determined by S(v) = T(v, u). For, T(u + w, u + w) = T(v, u) + T(u, w) + T(w, v) + T(w, w). Similarly, T(v, u) = O for al1 v implies that T + T' = 0. (vii) Give~i w (with w ( p) # O for al1 p] , clioose Y complementary to ker w at al1 points. If o ( 2 ) = T(Y, Z) , then T (2, 2 ) = w (Z)o(Z)/w (Y) for all vector fields 2. (ix) T : V x V + W corresponds to T : V + V* [where T(v)(w) = T (v, w)]. The rank of T may be defined as the rank of T (coiisider the matrix of T with respect to bases U , , . . . , u n and v*i, . . . ,u",). (xii) Let V = Mp*. If T : V + V and p E V* and T(v) = av for aii u E kerp, there is a y complementary to ker ,u such that

(Begin by clloosing yo complementary to ker ,u and writing v uniquely as vo+cyo for u0 E ker p.)

(xiii) Define S: v x . . . x v x v * x - - ~ x v * + R --

p times p times

6. (a) Let iv : V + V ** be the "natural isomorphism" i v (v)(l) = l (v). Show that for aily linear traiisformation f : V + W, tlie following diagram commutes: v iv )y**

h) Show tliat there do not exist isomorphisms iv : V + V* such that the following diagram always commutcs.

2 v v-v*

Hint: There docs not even exist an isomorphism i : R + R* which makes tlie diagram commute for al1 linear f : R + R.

7. A covariant fuílctol- from (finite dimensional) vector spaces to vector spaces is a function F wliich assigns to every vector spacc V a vector space F ( V) and to evcry linear transformation f : V + W a limar uansformation F( f ) : F( V) + F(W), sucli that F ( lv ) = 1 ~ ( v ) and F ( g 0 f ) = F(g) o F(v).

(a) Tlie "identity functor", F(V) = V, F( f ) = f is a functor. (b) The "double dual functor", F ( V) = V **, F (f ) = f ** is a func tor. (e) The "Tk fuiictor", F(V) = &(V) = Tk(v*) ,

is a functor. (d) If F is any fuiictor and f : V + W is an isomorphisrn, tlien F( f ) is an isomorphism.

A contrava?iatii futictur is defined similarly, except tliat F ( f ) : F( W) + F( V) and F ( g o f ) = F ( f ) o F(g). Functors of moi-e tlian one argument, covariant in some and contravariant in otliers, may also be defined.

(e) The "dual functor", F(V) = V*, F( f ) = f * is a contravariant functor. (f) The "T' fuiictor", F(V) = T~ (V), F ( f ) = f * is a contravariant functor.

8. (a) k t Hom( V, W) denote al1 linear transformations from V to W. Choos- ing a basis for V and W, we can identify Hom(V, W) with the m x n matrices, and consequently give it the metric of Rnm. Show that a dfierent choice of bases leads to a homeomorphic metric on Hom(V, W). (b) A functor F @ves a map from Hom(V, W) to Hom(F(V), F(W)). Cal1 F mntinuous if this map is always continuous [using the metric in part (a)]. Show that if e = ?r : E + B is any vector bundle, and F is continuous, then there is a bundle F(e) = ir': E' + B for which ?ir-' (p) = ~(?r" ' (p)) , and such that to every trivialization

r : a - ' (u ) + U x Wn

corresponds a trivialization

r i : ? r ~ - ~ (U) + U x F(Rn).

(c) The functor TI (V) = T1 ( V*) = V** is continuous. (The bundle (TM) is just a case of the construction in (b).) (d) Define a continuous contravariant fuiictor F, and show how to construct a bundle F(6). (e) The functor F(V) = V* is continuous. (The bundle T*M is a special case of the construction in (d).)

Geiierally, tlie same construction can be used when F is a functor of several argumeii ts. The bundles T~ (M) are al1 special cases. See the next two problems for otlier examples, as well as an example of a functor which is no1 continuous.

9. (a) Let F be a functor from V", tlle class of tz-dimensional vector spaces, to vk. Given A E GL(n, W) we can consider it as a map A : Rn + Rn. Then F(A) : F(Rn) + F(RN). Cl~oose, once and for all, an isomorphism F(Rn) + Rk. Tllen F(A) can be considered as a map h(A): Rk + IRk. Show that iz : GL(t1, R) + GL(k, R) is a homomorphism. (b) How does tlle llomomorpllism h depeild on tlle initial choice of the isomorphism F(R") + IRk? (c) Let v = (VI , . . . , v,) and w = (wl, . . . , w,) be ordered bases of V and let e = (e l , . . . , e,) be the standard basis of R". If e + v denotes the isomorphism taking ei to vi, sllow that the following diagram commutes

where A = ( a i j ) is defined by

j - i

After identifying F(Rn) with Rk, this means that

also commutes. Tliis suggests a way of proving the following.

THEOREM. If h : GL(n, R) + GL(k, R) is any homomorphism, there is a functor Fh : V" + V' such that the homomorphism defined in part (a) is equal to h.

(d) For q,qt E Rk, define (v , Y ) - (w, Y ' )

if q = h(A)qt where wi = E = , ajivj.. Show that -- is an equivalencc relation, and that every equivalence class contains exactly one element (v , y ) for a @ven v. We will denote the equivalence class of ( Y , y ) by [v, y]. (e) Show that the operations

are well-defined operations making the sct of al1 equivalcnce classcs into a k-dimensional vector space Fh ( V ) . (f) If V , W E V" and /: V + W, choose ordered bases v, w, define A by

n / ( v i ) = Ij=, Llji W j , and define

Sliow that this is a well-defined liiiear transformation, tllat FA is a functor, and that Fh(A) = h ( A ) when we identify Fh(Rn) with IRk by [e,q] H y . (g) k t a : R + R be a non-continuous homomorpllism (compare page 380), aiid let b : GL(n,R) + GL(I ,R) = R be h ( A ) = a(det A). Then Fh: V" + VI is a noil-continuous functor.

10. In classical tensor analysis there are, in addition to mixed tensor fields, other "quantities" which are defined as sets of functions which transform according to yet other rules. These new rules are of the form

A' = A operated on by h (S) For exarnple, assignments of a single function a to each coordinate system A-

such that the function a' assigned to x' satisfies

are called (even) scalar densities; assignments for which

a '= det - - a l (:;)I are called odd scaIar densities. The Theorem in Problem 9 allows us to construct a bundle whose sections correspond to these classical entities (later we will have a more illuminating way):

(a) Let h : GL(n7 B) + GL(1, R) take A into multiplication by det A. Let Fh be tlie functor @ven by tlie Theorem, and consider the 1-dimensional bundle Fh ( T M ) obtained by replacing each fibre Mp witl~ F], (Mp) . If (x, U ) is a coordinate system, then

is non-zero, so every section on U can be expressed as a a, for a unique f function a. If xf is anotller coordinate system and a a, = a axl, show that

axi o' = det (-.) u .

(b) If, instead, i7 takes A into multiplication by 1 det Al , show that the corre-

(c) For tliis i7, show tllat a non-zero element of F/i ( V ) determines an orientatian for V. Conclude tll at the bundle of odd scalar densities is not trivial if M is not orientable.

(d) \Ve catl identify q k ( R n ) ,th IRnk+' by taking

Recall that if f : V + V, we define T~ ( f ) : T~ ( V) + qk ( V ) by

Ghen A E GL(Iz, R), we can consider i t as a map A : Rn + Rn. Then A : ( I R ) + T~ (IRn) determines an elemeilt qk (A) of GL(IZ~+' , R). Let h : GL(>I,R) + G L ( ~ ~ + ' , R ) be defined by

h(A) = (det A ) ~ ~ ~ ( A ) w an integer.

Tlie 1,undle F (TM) is called tlie bundle of (even) relative tensors of type (f ) and weight w. For k = 1 = O we obtain tlie bundle of (wen) relative scdars of weight w [tlie (even) scdar densities are tl-ie (even) relative scalars of weigl-it 11. 1f (det A)w is replaced by 1 det A l W (w any real number), we obtain the bundle of odd relative tensors of type (f) and weight w . Sliow that tlie ti-aiisformation law for the components of sections of these bundles is

(or tlie same formula with det(axi/axti) replaced by 1 det(axi/axfj)l). (e) Define

+ I if il , . . . , i, is an even permutatjon of 1,. . . , n -1 if i l , . . . , i, is an odd permutation of 1,. . . , T I gil ... in - O if i, = ib for some a # p .

Sliow tliat there is a covariant relative tensor of weight - 1 witli these compoiients in every coordinate system. Also sliow tliat E'' .-.'M = Ej, are the components in every coordinate system of a certain con travariant relatjve tensor of wcigh t 1 . (See Problem 7-1 2 for a geometric in terpretation of these relative tensors.)

CHAPTER 5

VECTOR FIELDS AND DIFFERENTIAL EQUATIONS

w e i-eíurn ío a more detailed study of the tangent bundle TM, and its sections, i.e., vector fields. kt X be a vector field defined in a neigh-

borhood of p E M. We would like to know if íhere is a curve p : (-E, E) + M hrough p whose tangent vectors coincide with X, thaí is, a curve p wiíh

Since tliis a local quesíion, we wish ío introduce a coordinate system (x, U) around p and íransfer the vector field X to x (U) c Rn . Recall that, in general, a, X does not make sense for Cm functions a: M + N. However, if a is a difTeomorpliism, then we define

It is noí liard to clieck (Problem 1) that a, X is Cm on a (M). In particular, we have a vector field x, X on x (U) c Rn. There is a funcíion f : x(U) + Rn with

i.e., (x* X), lias "componeiits" f * (y), . . . , f" (y). Consider íhe curve c = x o p. The condition

means that

hence

If ure use c' (í) to denote the ordinary derivative of the Rn-valued function c, tl-ien this equation finally becomes simply

This is a simple example of a differential equation for a function c : R + Rn, wliich may also be considered as a system of n differential equations for the functions c*,

e* (i) = f * (cl (i), . . . , cn ( 1 ) ) i = I, . . . , n.

We also want tl-ie "initial conditions"

Solving a differential equation used to be described as "integrating" h e equation (tlie process is integration when h e equation has the special form cf(i) = f (í) for f: R + R, a form to which our particular equations never reduce); solutions we1.e consequently called "integrals" of the equation. Part of tliis termiliology is still preserved. A curve p: (-E, E) + M with

is called an integral curve for X with initid condition p(0) = p. Similar ter- mi~iolom is applied, of course, to tl-ie differential equations one obtains upon introducirig a coordinate system. For quite sonle time, we will work entirely iil Euclidean space, ar-id for a wliile x, y, etc., will denote points of Rn. If U c R* is oper-i and f : U + Rn, tlien a curve c : (-E, E) + M with

is called an integral curve for f with initíd condition c(0) = x

E c t o ~ Fields and Dguenlial Equalioíls 137

Before stating the main theorem about the existence and uniqueness of such integral curves, we consider some special cases.

The equation for a curve c with range IR,

which would be written classically in terms of a function y : R + R as

is the special case f ( a ) = -a2. The standard method of solving this equation is to write

Thus the curves 1

are supposed to be solutions. This can be checked directly if you don3t believe the above manipulations. (Thcy really do make sense; the equation in question asserts that y' = f o y , so

hence, if F' = i/f, then

( F o y)' = I

F ( y ( x ) ) = x + C for some C.) To obtain tlie initial conditions c(0) = a, we must take

1

Tllis works in al1 cases except a = O. ln this case, the correct solution is

c ( i ) = O for al1 i

(wliich we missed by dividing by y). In terms of vector fields, the curves c are the integral curves of

Notice that no integral curve, except c(i) = 0, can be defined for al1 r, even tllougll X is defined on al1 of R . It miglit be thought tliat this somehow reflects the fact that X(0) = 0, but this has nothing to do with the case. For a > 0, the curve c (1) = 1 / ( r + 1 /a) is defined for al1 large i, and as r + cm it approaches, but never reaches, 0. On the other hand, as r + - 1 / a the curve escapes to infinity because the vector field gets big ~ O O fast. This will continue to be true even if we modify tlie vec tor field near O so tbat it is never O.

Ano tlier phenomenon is illustrated by the equation

ct(r) = c(1)2/~,

written classicaily as

There are two dfirent solutions with the initial condition c(0) = O, namely

(1) c(i) = 0 for al1 r, 1 3

C(i) = zi for al1 r . In this case, the function f, @ven by f (a) = a2i3, is no! differentiable. Unicjue- ness wiI1 always be insured when f : U + Rn is C ' , but it can also be obtained with a rather Iess stringent condition. We say that tlie function f satisfies a Lipschitz condition on U if there is some K such that

I f (x) - f1Y)I 5 Klx - Yl for aiI S, y E U.

Notice that /(a) = a2i3 is not Lipschitz; in fact, there is no K with

lf - f (011 5 Klx-l

for x near O, since

A Lipschitz function is clearly continuous, but not necessarily dfirentiable (for rsample, f (x) = 1s 1). On the other hand, a C ' function is locaiiy Lipschitz, ~ h a t is, it satisfies a Lipscliitz condition in a neighborhood of each point-this Sollows from Lemma 2-5. A Lipschitz function is also clearly bounded on any bounded set.

The basic existente and uniqueness theorem for differential equations depends on a simple Iemma about complete metric spaces.

l . THEOREM (THE CONTRACTION LEMMA). Let (M,p) be a non- empty complete metric space, and Jet f : M + M be a "contraction", that is, suppose there is some C < 1 such that

P (f (x), f ( ~ ) ) 5 CP (x, Y for aii x ,y E M.

Then there is a unique x E M such that f (x) = x (the function f has a unique " h e d point").

PROOF. Notice that f is clearly continuous. Let xo E M and define a sequence {S,) inductively by

Xn+l = f (xn) ,

n times

Then an easy induction argument shows that

Thus

Since C < I , the sum xEo Cn converges, so C" + e m .+c"~- ' + O as n + m. Thus the sequence {x,) is Cauchy, so there is some x with

x = Iim x,. n 3 w

Continuity of f then shows that

f (x) = lim f (xn) = Iim xn+i = x. n-+w n-+m

We are going to apply the Contraction k m m a to certain spaces of functions. Recall that if (M, p) is a metric space and X is compact, then the set of al1 continuous functions f : X + M is a metric space if we define the metric o by

If M is bounded, then we do not even need X to be compact. Moreover, if M is complete, then h e new metric space is also complete; this is basically jusí the theorem that the uniform limit of continuous functions is continuous, plus the fací that each lim fn ( S ) exisís since M is complete. In particular, if M is a

x-fm compací subset of Rn, then the set of aii continuous functions f : X + M is complete with the metric

Our basic strategy in solving difirential equations will be ío replace difirentiable functions and derivatives by continuous functions and integrals. If U c Rn and f : U + Rn is continuous, then a continuous function a : (4, b ) + U, defined on some iníerval around O, clearly satisfies

if it satisfies the integral equation

where the integral of an Rn-valued function is defined by integraíing each component function separately. Conversely, if a satisfies (1), then a is differentiable, hence continuous; thus a' - f o a is continuous, so

For the proof of the basic tlieorem, we need only one simple estimate. If a continuous function f : [a, b] + Rn satisfies 1 f 1 5 K, then

To prove this, we note tliaí it is true for consíaní functions, hence for step functions, and thus for co~~tinuous functions, wliicl~ are unjform limits on [a, b] of step functions.

2. THEOREM. Leí f : U + R" be any function, where U c R" is open. Let xo E U and leí a > O be a number such that the closed ball 8 2 Q ( ~ ~ ) , ~f radius 2a and center xo, is contained in U. Suppose that

Choose b > O so that

Then for each x E 8,(xo) there is a unique a, : (-b, b) + U such that

axt(r> = f (a, ( r ) ) a, ( O ) = x.

PROOF. Choose x E & (xo), which will be fixed for the remainder of the proof. Let

M = {continuous a : (-b, b ) + BZa (xo)}.

Then M is a complete metric space. For each a E M, define a curve Sa on

Sa ( r ) = x + lo f (a (4) d1.4

(the integral exists since f is continuous on B20(x0)). The curve Sa is clearly continuous. Moreover, for any r E (4, b) we have

Since (x - ,yo( 5 a , it follows that ( S a ( i ) - xo( < 2a, for al1 i E (4, b), $0

Thus S : M + M. Now suppose a , p E M. Then

sup la(u) -B(u)l -bcucb

by (2)

= bK ((a - fi (1.

Since we chose bK < I (by (4.11, íhis shows that S : M + M is a coníracíion. Hence S has a unique fixed point:

Tliere is a unique a : ( -b , b ) + (xo) wiíh f

a(r) = x +l / ( a ( u ) ) d u .

This, alas, is noí quite wliat íhe ílieorem síates. Having used íhe eleganí Con- traction Lemma, we pay for ií by finishing off wiíh a finicky d e t d :

Tlie map a is the unique B : (4, b ) + U saíisfying r f

Reuson: We claim that any such B actudly lies in &(xO), in fact, in BZa (x.0).

Consider first numbers r > O. We have already seen (statement (*)) that for each í with O 5 1 < b,

(**) p ( 1 ) = x + lo f ( B ( U ) ) d~ is in BzU (xo) [the op ball]

provided that

so certainly if

B ( U ) E B2,(xO) for dl u with O 5 ti 5 r

We can now use a simple Jeast upper bound argument. Let

Let a = sup A . Suppose a < b . We cIearIy have B (u ) E B2, (xo) for O 5 u < a . So B (a) E B2, (xO), by (**l. Tliis clearly implies that p (a + S ) E B2, (xo) for sufficiently smdl s > 0, wbicli contradicts tlie fact tllat a = sup A . So it must be tllat sup A = b . A similar armgument works for -b c 1 5 0.

To sum up, the ui-iique fixed point a, of tlie map S is the unique curve with tlle desired properties. +:+

~ I O T Fields and Dfl~eniiQZ i7quations

Notice íliat solutions of the dfirential equaíion

remain solutions under additive changes of parameter; that is, if

then B ' ( 0 = ~ ' V O + 1 ) = f ~ ( I O + 1 ) ) = f ( p ( ~ ) ) .

TJiis remark allows us to exíend the uniqueness parí of Theorem 2.

3. THEORFM. Suppose f : U + IRn is Jocally Lipschitz, that is, around each point there is a bail on which f satisfies condition (2) of Theorem 2 for some K (and Ilence also condition (1) for some L). Let x E U and Jet al , a2 be two maps on some open intervai I with al (1), a 2 ( I ) c U and

ait(i) = ( í ) )

ai ( O ) = x

Then al = a2 on 1.

PROOF. Suppose al (lo) = a2 (10) for some io E 1. If we define

then tlie functions pi satisfy the same difirentiai equation, p i r ( i ) = f ( p i ( i ) ) ,

and have the same initial condition pi ( O ) = al (lo) -- a2(iO) E U. Hence pi (1 ) = p 2 ( i ) for sufficiently small i, by Theorem l. Thus the set

is open. It is clearly aiso closed and non-empty, so it equals 1. 43

We now revert to the si tuation in Theorem 2. \ e will write a, ( i) as a (1 , x ) , so íhaí we have a map

satisfying

a ( O , x ) = x d -a(i,x) = f (a(1,x)) di

[i.e., Dla(r, A-) = f (a(i , x)) , but we will frequently use d / d í or d / d i in this

discussion]. This map a is called a local flow for f in (4, b) x 3, (xo). To picture this rnap a, the best we can do is to draw the images of the integral

cuntes a,. If y = a,(i0), tllen the integral curve a, with the initial condition a,(O) = x differs from t l~e integral curve a, with initial condition ay(0) = y only by a cl~ange of parameter, so the two images overlap. For each h e d x, the rnap i I+ a(r ,x) for -b < i < b lies alongpart of the curve through x. O n the other hand, if we fix L, then the rnap

gives the result of pushing each A- along the integral curve tlirough it, for a time intentd of r . To focus attention on tliis map, we denote it by #*:

This rnap #r is dways continuous. In fact, tlle whole flow a is continuous (as a function of both i and x):

4. THEOREM. If f : U + Rn is Iocally Lipschitz, then tlie flow

given by Theorem 2 is continuous.

PROOF. Let us denote t11e rnap S defined in tlie proof of Theorem 2 by S,, to indicate explicitly the role of x. Tlien

Recall that

- SP I I 5 bKlla - B II. lf S; denotes the n-fold iterate of S,, then

Recall also tliat in Theorem 1 the fixed point a, of S, is the limit of SFa for any a. Hence a, = lim S~a,, so we obtain

t 1 4 00

Since Ila, -ayll = sup la([, A-)-a([, y)I, this certainly proves continuity of a. 43 1

If additional conditions are placed upon tlie map f, then further smoothness conditions can be proved for a. In fact,

Unfortunately, this is a very hard theorem. A clean cxposition of the classical proof is given in Lang's introductiotl to L)@mntzable ManifoldF (2nd ed.), and a recently discovered proof can be found in Lang, Real and Functional AnalysLs (3rd ed.), pp. 371-379. In order to read this liigh-powered proof, you must first Jearn the elements of Banach spaces, including the Halin-Banach theorem, and then read about differential calculus in Banach spaces, including the inverse and implicit function theorems (Real and Functional Annbsis, pp. 360-3651, but this is probably easier than reading the classical proof (and, besides, when YOU'PZ fin- ished you'll dso know &out Banach spaces, and differential caIculus in Banach spaces).

We wilI just accept h is fact. Notice that tlle maps #I are consequently Cm if f is Cm.

Since tlie map a : (-b,b) x B,(xo) + U

satisfies a (O, x ) = x, we llave

a : {O} x B.,2(xo) ' &/2 ( ~ 0 ) c Ba ( ~ 0 ) -

Continuity of u and cornpactness of {O) x BUj2(xO) irnply that there is some E > O S U C ~ that

[If x E Baj2(xa), then the integral curve with initial condition x stays in B, (xo) for 11 1 c E.]

So if Jsl < E , and x E Bn /2 (~O) , then h e point a(s , x) E BU(xo), so we can also define

y ( ) = ( , ) ) I i I c E *

This satisfies

We have also noted that

satisfies

Consequendy, p ( r ) = a(i,a (S, x)) for 11 1 c E. In other words,

if J s ~ , J t ~ , J s + t ~ < E , then a ( i , a ( s , x ) ) = a ( s + t , x ) .

If we now Jet #t : Bal2(x0) + Hg" be #t (x) = a(i, x) for x E Ba/2(xO), we can say :

Roughly speaking, #t+s = #r o = O #t. This shows, in particular, that for Isl c E each #$ is a diffeomorphism, with invene 6,-' = #-s. Everything we llave said, since it is local, can be resaid, without requiring any more proof, on a manifold.

k t o r Fiel& and Dzferential Eqt~ations 147

5. THEOREM. Leí X be a Cm vector field on M, and let p E M. Then there is an open ser V containing p and an E > O, such íhat there is a unique col- Iection of difkomorphisms #t : V + #f ( V ) c M Mor Ii 1 < E with íhe following properties:

(1) #: ( - E , & ) X V + M, defined by #(i, p) = & ( p ) , is Cm.

(3) If y E V, ttlien Xg is íhe tangení vector at i = O of the curve i K- # f (q ) .

The exarnples Fven previously sliow íhat we cannoí expecí #f to be defined for al1 i, or on al1 of M. In one case however, this can be attained. The support of a vector field X is just the closure of { p E M : X, # 0).

6. THEOREM. If X Iias compact support (in particular, if M is compact), íhen íliere are dfiomorpI~isms #t : M + M for al1 i E R wiíh properíies (11,

(21, (3)*

PROOF. Cover support X by a finite number of open sets V , , . . . , Vn given by TIieorem 5 wiíli comespoiiding E I , . . . , E , and dfiomorphisms #:. Let E = m i n ( r l , . . . , E ~ ) . Notice thaí by uniqueness, # ( ( y ) = #{ ( y ) for y E Vi n 5. So we can define

if y E Vi

if y # support X .

Clearly # : ( - E , & ) x M + M is Cm, and #fct=F = #1 o#s if IiI,IsI, Ii + S I < E,

and each #f is a dfiomorphism. To define #, for Ii 1 >_ E, write

r = k ( ~ / 2 ) + I' with k an integer, and 17'1 < ~ / 2 .

# E / 2 O m m O # E / 2 O #r [#E/2 iíeraíed k times] for k 1 O #f = {

# - E / 2 O - . - o #-&/2 O #r iterated -k times] for k < O.

It is easy to check t1iat this is the desired ( $ 1 ) . *:+

The unique collection { # 1 ) @ven by Theorem 6, or more precisely, the map r I+ #t from R to tlie group of alI difkomorphisms of M, is called a 1 -parameter group of diffeomorphisms, and is said to be generated by X. In the JocaI case of Theorem 5, we obtain a "local 1-parameter group of local difXeomorphisms". The vector field X is sometimes called the "infinitesimal generator" of { # t )

(vec tor fields used to be called "infinitesimal transformations"). Condition (3) in Theorem 5 can be rephrased in terms of the action of X,

on a Cm function f : M + R. Recall that

Thus, to say that Xg is the tangent vector at i = O of the curve r H #I (q) arnounts to saying tliat

( X f ) ( q ) = X, f = lim f (#h ( Y ) ) - f ( Y ) h h+O

Tliis equation will be used very frequently. The first use is to derive a corollary of Theorem 5 whicli allows us to simplify many caicuIations involving vector fields, and wliicli also Jias important tlieoretical uses.

7. THEOREM. Let X be a Cm vector field on M with X ( p ) # O. Then there is a coordinate system ( x , U ) around p such that

n

PROOF. It is easy to see that we can assume M = R" (with the standard coordiiiate system r', . . . , r f l , say), and p = O E R". Moreover, we can assume that X(O) = a / a t l 1,. Tlie idea ofthe proofis that h a neigliborhood of O there is a uiiique integral cunfe tlirougli eacli point (0, a2, . . . , a"); if lies on the integral

curve through this point, we will use a2 , . . . , a" as ihe last n - 1 coordinates of q íuid the time intervd it takes the curve to get to q as the first coordinate. To do ihis, Jet X generate #* and consider the map x defined on a neighborIiood of O in Rn by

1 x(a ,. . . , an) (0,a2,.. . ,an) .

W compute that for a = (a1,. . . , a"),

1 = lim - [ ~ ( ~ ( a ' + h , a2,. . . , a")) - f (x(Q))]

k+O h

Moreover, for i > 1 we can at Jeast compute

1 = Jim -[f(x( O, ... ,h ,..., O)) - f(0)]

114 0 11

1 = lim - [ f (O,. . . , h , . . . ,O) - $(O)]

h 4 0 h

Since X (O) = d/a i l l o by assumption, this shows tliat X,O = I is non-singular. Hence x = X-' may be used as a coordiilate system in a neighborhood of O. This is the d e s h d coordinate system, for it is easy to see tliat the equation ~ * ( a / d r ') = X O x, which we have just proved, is equivalent to X = d/axl.

The second use of the equation

is more co~nprelieiisi\le. The fact tjiat Xf can be defined totally in terms of the diffeon~orphisms #iI suggests that an aciioii of X on other objects can be

obtained in a similar way, To emphasize the fundamental similarity of these notions, we first introduce the notation

L x f for X f .

\Ve cal1 Lx f tlle (Lie) derivative of f with respect to X ; it is another function, wliose value at p is denoted variously by ( L x f ) ( p ) = L x f ( p ) - ( X f ) ( p ) = Xp ( f ). Now if w is a Cm covariant vector field, we define a new covariant vector field, tlie Lie derivative of w with respect to X , by

Tliis is the limit of certain members of M,*. Recall that if Xp E M,, then

A fairly easy direct argument (Problem 8) shows that this limit always exists, and that tlie newly defined covariant vector field L x w is C m , but we will soon compute this vector fieId explicitly in a coordinate system, and these facts will then be obvious.

If Y is another vector field, we can define the Lie derivative of Y with respect to X ,

1

Tlie \rector field #h+Y appearing here is a special case of the vector field a,Y defined at the beginning of the chapter, for a : M + N a difkomorpliism and Y a vector field on M . Tlius (#/, ,Y), = #h* (Y$-,, (,)) is obtained by evaiuating Y at ( p ) = #-A ( p ) , and then moving it back to p by #/,*.

integral curve r $1 ( P )

of X through p

Tlic definition of L x Y can be made to look more closely anaiogous to L x f and Lx w in tlie follotving way. If a : M + N is a dfiomorphism and Y is a

Tr,tor l~iclds and Dzfenntial Equatians 151

vector field on the range N, thni a vector field a*Y on M can be defined by

Of course, a* ( Y ) is just (&-') ,Y. Now notice that

1 yp - (#h*Y)P 1 Jim - Y )p - Yp] = lim = lim - [Yp - ( $ L ~ * Y) , ] h+O /Z h+O -h k+O k

Nevertheless, we will stick to the original (equivalent) definition. We now wish to compute Lx o and Lx Y in a coordinate system. The cal-

culation is made a lot easier by first o b s e ~ n g

8. PROPOSITION. If Lx Y; and Lx wi exist for i = 1,2, then

If LxY and L x w exist, then

(3) L x f Y = X f . Y $ f . L x Y ,

(4) Lx f - o = X f . w + f . L x w .

FjnaIIy, if o ( Y ) deiiotes tlie function p I+ o ( p ) (Yp) and L x o and LX Y exist, then

PROOF. (1) and (2) are trivial. The remaining equations are al1 proved by the a m e trick, t l ~ e oiie used iii finding ( f g ) ' ( x ) . We will do number (3) here.

+ Jim [ f ( p) - { (m-h ( P ) ) 1 #/r* y+-,, ( p ) . / 1 4 0

The first limit is c'learly f ( p ) Lx Y ( p ) . In ihe second limit, the term iii brackeis approaches

lim f ( P ) - f (#k ( P ) ) k+O -k = Xf ( P ) ,

while an easy argument sliows that #h,Y+-,,(p) + Yp. *3

Mre are now ready to compute Lx in terms of a coordinate system ( x , U) on M . Suppose X = zyEI a i d / a s i . We first compute L x ( d x i ) . Recall (Prob- lem 4-1) that if f : M + N and y is a coordinate system on N, then

We can apply this to #h*, where y is s . Then

1 LX (dx') ( p ) = lim - [(#h*)dxi ( p) - dx' ( p ) ]

h+O h

Now the coeffrcient of d d ( p ) is

I ["'d"' - $1 = lini - [ lim - l 1 4 o h Ir40 h axj

{ibis step wiil be justified in a moment)

To justify (*) we note that the map A(h,q) = xi(#h(q)) is Coo from IR x M to IR; thus a 2 ~ / d h d x j = d 2 ~ / d x j d h , wliich is wliat the intercliange of limits amounts to.

It now follows tllat n aai

LX dx' = - d x j . axj

We couId now use (2) and (4) of Proposi tion 8 to compute Lx w i n general, but

we are really iiiterested in computing L x Y. To compute (d/axi) we could imitate the calculations of Lx dx'; but there would be a complication, because #h* on \rector fields iilvolves one more composition thari #h* on covariant vector fields. Tlie trick needed to deal with this complication has already been used to prove (3), (41, and (5) of Proposition 8, and we can now use (5) to get the answa immediately;

O = L. 6; = L x [dxi (a)] = (Lx dx') (5) + dx' (LX S). a x j

thus,

Using (3) we obtain

Sumniing over j and tlien iiiterclianging i and j in tlie second double sum we obtain

This somewhai complicated expression immediately leads to a much simpler coordinate-free expression for Lx Y. If f : M + N is a C" function, then Y f is a function, so XYf = X(Yf) makes sense. Clearly

Tlie secoiid partial derivatives which arise here cancel those in the expression for Y ( X f ), and we find tliat

L x Y = X Y - Y X , alsodenotedby [X,Y].

Often, [X, Y] (which is called the "bracket" of X and Y) is just defined as XY - YX; note that this means

A straightfonvard verification shows that

so tliat [X, YIp is a derivation at p, and can therefore be considered as a member of M,.

We are now in a very strange situation. Two vector fields LxY and [X, Y] liave both beeii defined independendy of any coordinate system, but they have Leen proved equal using a coordinate system. This sort of thing irks some people to no end. FortunateIy, in this case tlie coordinate-free proof is short, though hardly obvious.

In Chapter 3 we proved a Iemma wllich for the special case of W says that a Cm fu~iction f : (-E, E) + W with f (O) = O can be wntten

for a Cm fuiiction g : (-E, E) + W with g(0) = f '(O), namely

This lias an immediate generalization.

9. LEMMA. lf f : (-E, E) x M + W is Cm and f (O, p) = O for al1 p E M, tlien there is a Cm function g : (-E,&) x M + W with

PROOF. Define

1 0. THEORFM. If X and Y are Coo vector fields, theii

PROOF. Let f: M + R be Cm. k t X generate #,, Ii 1 < E. By Lemma 9 tliere is a family of Cm functions gf on M such that

Then

Tl-ie equality Lx Y = [X , Y ] = XY - YX reveals certain facts about LxY which are by no means obvious from the definition. Clearly

[X , Y ] = -[Y, X] , so [X , X] = o.

Consequently, LxY = - L y X , so LxX = 0.

Since we obviously Iiave Lx (a Yl + by2) = aLx Y, + bLx Y2, it follows immediately tliat L is also linear with respect to X:

Finally, a straiglitforward calculatioil proves tlie "Jacobi identity":

Tliis equation is capable of two interpretations in terms of Lie derivatives:

(a) Lx[Y, 21 = [LxK ',l + [Y,Lx',],

(b) as opei-ato1.s on Coa functions, we have

L[x, = Lx o L y - L Y o Lx (whicli miglit be written as [ L x , L Y]).

Finally, note that Lx Y is linear over constants oilly, not over tlie Coo functions .F. In fact, Proposition 8, or a simple calculation using the definiíion of [X, Y], shows that

[ f X , gY1 = fg[X, Y] + f f(Xg)Y - g(Yf )X-

Thus, the bracket operation [ , ] is not a íensor-that is, [X, Y'Jp does not depend only on Xp and Yp (wliich is noí surprising-whaí can one do ío two vectors in a lector space excepí íake linear combinations of íhem?), buí on íhe vector fields X and Y. In particular, even if Xp = O, it does not necessarily foIIow that [X, Y'Jp = O-in the formula

t11e first term Xp ( Y f ) is zero, but the second may not be, for X f may have a non-zero derivative iil tlle Yp direction even t l iou~h (X f ) (p) = 0.

The bracket [X, Y], although not a tensor, pops up in the definiíion ofprac- tically a11 otlier tensol-S, for reasons that \vil1 become more and more apparent. Before procecdii~g to examine its geomeíric interpreta tion, we wil1 endeavor to become more aí ease witIl the Lie derivative by taking time out ío prove directly from tlle definiíioli of Lx Y two facís whic11 are obvious from íhe definition of [X, Y].

(1) LxX = o. lf X generates #t, it certainly suffices to sl~ow íhat (#n, X)p = Xp for a11 h.

Recall that (#h, X)p = #fd+ X4,/, (p). Now X4-, (p) is just h e tailgent vector at

time r = -h to t11e curve r I+ #I (p), and tlius tlie tangent vector, at time I = 0, ío the curve

YP) = 41-/i(P).

TIius @/l* X$-l,(P) is tlie íangení vecíor, aí time I = O, to the curve

But this tangení vecíor is jusí Xp .

k t o r I;ieZh and 1lz&i-entiaZ Equations 157

(2) If Xp and Y, are both O, then LxY (p) = 0.

Since Xp = O, tlie unique integral curve c wit1i c (O) = p and dcldi = X (c (1)) is simply c (i) = p (an integral curve S tarting at p can never get away; conversely, of course, an integral curve starting at some other point can never get to p). Then Yp = O and

so LxY(p) = o. To develop an iiiterpretation of [X, Y] we first prove two lemmas.

1 1 . LEMMA. Let a : M + N l ~ e a dXeomorphism and X a vector field on M wliicli generates {#[l. Then a,X generates {a o #t o a-'}.

PROOF. We have

12. COROLLARY. If a : M + M, tlien a, X = X if and only if $ t o a = for al1 L.

13. LEMMA. Let X generate {#t) aiid Y generate {Srf}. Then [X, Y] = O if and only if #l o = $, o #t for al1 S, í .

PROOF. If #t o = o #t for al1 S, tlien #,,Y = Y by CoroIlary 12. If this is true for aII r , theii clearly Lx Y = 0.

Convei-sely, suppose that [X, Y] = O, so that

1 O=Iim-[Yq-(#h*Y)q] forallq.

h+o lr

Giveii p E M, consider tlie cuive c : (-E, E) + Mp @ven by

For h e derivative, ct(r), of this map into the vecíor space M, we have

= O ) using (*) with = 4-1 (P)

= o.

Consequently c(í) = c(O), SO #t,Y = Y. By Corollary 12, #r o = o #[ for alI s,r. 4 4

We kave already shown tliat if X(p) # 0, then íhere is a coordinate system x with X = a/dx l . If Y is another vector field, everywhere linearly independeiit of X, then we mighí expect to find a coordinaíe sysíem with

However, a sllort caIcuIation immediately gives the result

so íliere is no Jlope offi~iding a coordinaíe system satisfying (*) unless [X, Y] = 0. Tlie remarkable fact is that tlie condition [X, Y] = O is su#cient, as well as necessary, for the existente of the desired coordinate system.

14. THEOREM. If X,, . . . , Xk are IinearIy independent Coo vector fields in a neighborhood of p, and [X,, XB] = O for 1 - < a,B 5 k, then tliere is a coordinate system (s, U) around p such that

PROOF. As in the pmof of Theorem 7, we can assume tllat M = Rn, thaí p = 0, and, by a linear cl-iange of coordinates, that

X, (O) = - a = 1, . . . , k.

If X, generates (4," ), define x by

k4-1 x ( a l , . . , a ) = # , ( # * ( . . ( # ( , . . . , , a , . . . , a n ) ) . . * ) ) .

As in the proof of Theorem 7, we can compute that

Thus x = x-' can be used as a coordinate system in a neighborhood of p = 0 . Moreover, just as before we see that

Nothing said so fal. uses the l~ypothesis [X , , XB] = O . To make use of it, we appeal to Lemma 1 3; it shows that for each a between I and k , the map x can also be wri tten

and our previous argument then shows that

We tlius see tliat the bracket [X, Y ] measures, in some sense, the extent to which the integral curves of X and Y can be used to form the "coordinate lines" of a coordinate system. There is a more complicated, more diEcult to pi6ove, and less important result, which makes this assertion much more precise. If X and Y are two vector fields in a neigl~borhood of p, then for sufficiently small h we can

(1) follow the integral curve of X through p for time h ;

(2) starting from that point, follow the integral curve of Y for time h;

(3) then follow the integral curve of X backwards for time Ir;

(4) tlien follow the integral curve of Y backwards for time Iz.

If iliere happens io be a coordinaie sysiem x with x ( p ) = O and

then these steps take us to points with coordinates

(1) ( j , O , O , . . . , O )

(2) (h ,h ,O, . .. ,O)

(31 (O,h,O ,..., O )

(4) (0,OY 0, . - , O),

so tliat this "parallelogram" is always closed. Even when X and Y are (linearly iiidependent) vector fields with [ X , Y] # 0, the parallelogram is "closed up to first order". Tlie meailiiig of tl-iis phrase [an exiension of the terminology "c = y up to first order at O", which means tliat c'(0) = yf(0) j is the following. Let c ( h ) be tlie poini whicl-i step (4) ends up at,

c(/z) = * - f f ($4 (*h (#h ( O ) ) )

Tlieii tlie cunJe c is tlie constant curve p up to first order, thai is,

15. PROPOSITION. ~ ' ( 0 ) = O.

PROOF If we define

al ( i , /z) = * r (#h ( P ) ) a2 ( i , 11) = 4-l (*h (#f i ( P ) ) )

a3 ( i l h, = Sr-f (#-h (*k(#h (P)))),

Moreover,

kctot. JieZd~ and L)z&ien tial Equat ions

and for any Cm function f : M + R,

while

Consequently, repeated use of the chain rule gives

Wlienever we have a curve c: (-E,&) + M with ~ ( 0 ) = p and ct(0) = O E

Mp, we can define a new vector c"(0) or d2c/d121, by

A simple calculation sliows, wi?g he arsun@ioti ~ ' ( 0 ) = 0, that this operator ctt(0) is a derivation, ct'(0) E Mp. (A more general construction is presented in Prob- Jem 17.) It turils out tllat for tlle curve c defined previously, the bracket [X, YIp is r'elated to tliis "second arder" derivation. Until we get to Lie groups it wiI1 not be clear how anyone ever thought of the next tlieorem. The proof, which eiids tlle cliapter, but can easily be skipped, is an liorrendous, but clever, calculation. It is followed by an addendum containiilg some additional important points about differeiltial equations which are used Jater, and a second addendum coiicerning linearly independent vector fields in dimension 2.

16. THEOREM. c"(0) = 2[X, YIp.

PXOOF. Using the iiotation of the previous proof, since (f o c ) (1) = (f oa3) (I, I )

we have

Now

(2) 2D2,l (f a3)(0?0)

= 2D1 (-Yf o a3) by (e)

= 2[D1 (Yf 0 a21 (O? 0)

+ 0 2 ( Y f o a2) (0, o)] by (b) and the chain rule

= 2XYf (p) - 202(Yf 0 a2)(0,0) by (4 = 2XYf (p) - 2[Di (Yf o al)(O,O)

+ D2 (yf O a1 ) (0, o)] by (a) and the cliain rule

= 2 X Y f ( p ) - 2 Y Y f ( p ) - 2 X Y f ( p ) by(c)and(f).

Since (b) gives

Trector fieldi and L)$uentiaZ Equariotls 163

Finally, from

Substituting (1)-(4) in (*) yields the theorem. +:+

ADDENDUM 1

DIFFERENTI AL EQUATIONS

Al thougli we have dways solved dxerential equatioiis

with the initial condition a(0,x) = S ,

we could just as well kave required, for some io, that

To prove this, one can replace O by io everywliere in tlie proof of Theorem 2, or else just replace a by i w a(í - lo, x).

Ano ther omissioli in our treatmen t oí' dflerential equations is more glaring: tlle differentid equations ai(r) = f (a(t)) do not even include simple equations of the form a' (i) = g(i), Jet alone equatioiis Iike ai(i) = ía (i). In general, we would Iike to solve equations

where f : (-c, c ) x U + Rn. One way to do t his is to replace f (a (i, x)) by f (1, a( i , s)) wl~erever it occurs iii the proof. There is also a clever trick. Define

by f ( s , s ) = (1 , f (s,x)).

Then tliere is a flow (&',E2) = E: ( -b ,b ) x W + R x R" with

For tlie f i i ~ t coinponent í'unction &' this means that

a 1 -6 (i, S, S) = 1 al

thus &'(i ,s,x) = s + i .

For the second component á2 we have

= f ( á l (i, S, s ) ,á2( t , s , s ) )

= f (S + 1, ii2 (1, S, x)).

Then ~ ( I , x ) = a 2 ( i , o , ~ )

is the desired Aow with

p (O, x ) = x.

Of course, we could also llave arranged for B (ro, x) = x (by first finding á with á (io,s, X) = (S, x), not by considering the curve r I+ (i - io, S)).

Finally, consider the special case of a linear differential equation

wliere g is an n x 12 mati-ix-valued function on (a, b). In this case

If c is any n x n (constant) matrix, then

so c .a is also a solu tion of the same differeiitial equation. This remark allows us to prove an important property of linear differential equations, distinguishing tliem from general differential equations a' (i) =: f (r, a (i)), which may have solutions defined onIy o11 a small time interval, even if f : (a, b) x Rn + Rn is Cm.

J 7. PROPOSITlON. If g is a continuous n x 11 matrix-valued function on (u , b), tlien tlie solutioi~s of the equation

can al1 be defined on (a, b).

PROOF. Notice tliat continuity of g implies that f (í, x ) = g(í) . x is locally Lipschitz. So for any ío E (a, b) we can solve the equation, with any given initial condition, in a iieighborhood of ío. Extend it as far as possible. If the extended solution a is not defined for al1 r wit h ío 5 í 4 b, Iet r 1 be the Ieast upper bound of tlie set of í ' s for which it is defined. Pick S with

TJien f l (r*) # O for r * < i l close enough to í l . Hence there is c with

By uniqucncss, c coiricidcs witli a oii tlie iiiterval whe1.e they are defined. I l tus a may be extended past í l as c - f l , a contradiction. Similarly, a must be

defined for al1 í with a c r 5 10. +3

ADDENDUM 2 PARAMETER CURVES IN TWO DIMENSIONS

If f : U + M is an immersion from an open set U c IWm into an 17-dimensional manifold M, the curve í H f (al , . . . , aj-l,r, aj+l,. . . ,an) is caIIed a parameter curve in the ith direction. Given 7t vecior fidds Xi , . . . , Xn defincd in a neiglilorhood of p E M and Jinearly indepaident at p , we know that there is usuaily no immersion f: U + M with p E f (U), whose parameter cures in the ith direction are the integral curves of the Xi-for we might not have [Xi, Xj] = O. However, we mighi hope to find an immersion f for which the parameter culves in ihe ith direction lie along the integral curves of tlie Xj, but have different parameierizations. A simple example (Problem 20) shows tliat even this inodest kope cannot be fulfilled in dimension 3.

O n the oilier Iiand, in the special case of dimension 2, such an imbedding can be found:

18. PROPOSITION. Let X1, X2 be Iiiiearly independent vector fields in a neighboi.hood of a point p in a 2-dimensional manifold M. Then there is aii imbedding f: U + M, where U c IR2 is open and p E f (U), whose ith parameter Iines Iie aIong the integral curves of Xi.

PROOF. We can assume tliai p = O E IR2, and that Xi(0) = (ei)a. Ewry poini y in a sufficiently small iieighborhood of O is on a unique integral curve of Xl thiough a point (0,x2(y))-we proved precisely this fact in Theorem 7. Similarly, y is oii a unique iiitegral curve of X2 through a point (xi (y),O).

The map y c (xl (y),x2(y)) is Cm, wiih Jacobian equal to 1 at O (ihese facts also foIIow from tlie proof of Theorem 7). Its iiiverse, in a sufficieiitly small neighborhood of O, is tlie required dfiomorphism. +:+

\ e can always compose f with a map of the form ( x , y) I+ ( a @ ) , f l(y)) for dXeomorphisms a and f l of W , which @ves us considerable flexibility IIf, for example, C c W 2 is the graph of a monotone function g , then the map

(.Y, y ) H ( x , g ( y ) ) takes the diagonal { ( x , A-)) to C . Moreo\ler, for any particulai- paiiimeterization c = ( e i , c 2 ) : W + W 2 of C , we can further arrange that c ( r ) maps to ( c ( r ) , c ( r ) ) , by composing with ( x , y) H c x ) y ) . Coiisequently, we can state

19. PROPOSITION. Let X I , X2 be linearly ii~dependent vector fields in a ileighboi-hood of a point p in a ?-dimensional manifold M, and let e be a curve in M with c ( 0 ) = p and ~ ' ( 1 ) never a multiple of Xi or X2. Then there is an imhcdding f : U + M, where U c W 2 is open and p E / ( U ) , whose ith parameter liiies lie along the i i ~ tegral curves of Xi, and for which f (r, 1 ) = c(r ) .

PROBLEMS

1. (a) If a : M + N is Cm, then a, : T M + TN is Cm. (b) If a : M + N is a diffeomorphism, and X is a Cm vector field on M, tlien a, X is a Cm vector field on N. (c) If a : R + R is a(r) = r ', tlien there is a Cm wcior field X on R such that a,X is not a Cm vector field.

2. Fii~d a now11e1-e O vector field on R such tl~at al1 integral curves can be defined only on some interval around O.

3. Fii~d ail example of a complete metric space ( M , p) and a function f : M + M such that p( f ( S ) , f ( y ) ) < p(x, y ) foi- al1 x, y E M, but f has no fixed point.

4. Let f : ( - e , c) x U x V + R" be Cm, where U, V c R" are open, and let (xo, yo) E U x V. Pro1.e tl~at there is a neíghborhood W of (xo, yo) and a number b > O such that for each ( x , y ) E W tl~ere is a unique a = : (4, b) + U wit11 af ( t ) E V for t E (4, b) and

Moi-eover, if we write a(,,,,) ( t ) - a( t , x, y) , then a : (4, b) x W + U is Cm. Hi726: Consider tlie system of equations

5. \Ve soinetimes have to solve equations "depending on parameters",

wl-ierc f : (-c, c) x V x U + R", for opeil U c R" and V c Rm, and we are solving for a(,,,,) : (4, b) + U for each initial condition x and "parameter" y. For example, the equation

is such a case.

(a) Define j : (-C,C) x v x U + R" x W~

by T u , Y, x) = (O, f(r, y , x)).

If (E', ¿i2) = ¿i : (4, b) x W + R" x Rn is a flow for j in a neighborhood of (YO, XO), SO that

show tliat we can write

for some a , and coiiclude that a satisfies (*). (b) Sliow that equations of the form

a (**) a;a(r, X) = f (1, X., Mt, 4 )

can lze i-educed to equatioiis of tlie foi-m (*) (aiid tl-ius to equations

a -41, x.) = f (a(!, x)), at

ultimately). [Wlien one pi+oves that a C k function f: U + W" has a Ck flow a : (-b, b) x W + U, tlie hard part is to proi7e tliat if f is C i , tlien a is differei~tiable ~ t i t l i respect to tlie arguments in W, aiid tliat if tlie dei-ivative wi tli 1-csl~ect to tliese ai-gumeiits is denoted by D2a, tlien

(***) DI D2a (r, s) = D2 f (a(t, x)) - Dza(t, x)

(a i-esult wliicli follows directly fi-om tlie original equatioii

if f is c 2 , silice DI D? = D2D1). Since (***) is an equation for Dza of tlie foi-m (**), i t follows that D2a is diffei-eiitiable if Dr f is C 1 , i.e., if f is C2. Differential~ility of class ck is tlien proved similarly, by induction.]

14ct0r fields at2d Ihjiroztial Equnr ions

6. (a) Consider a linear dfierential equation

where g : R + R, so that we are solving for a real-valued function a. Show that al1 solutions are multiples of

where Jg(r) dr denotes some function G with Gt(r) = g (one can obtain al1 podive multiples simply by changing G). The remainder of this problem investigates the extent to which similar results hold for a system of linear dserential equations. (b) Let A = (u i j ) be an rz x 12 matrix, and let IA 1 denote the maximum of al1 lai, l. Show that

(c) Conclude tliat the infinite series of n x n matrices

converges absolutely [in the sense tliat ihe (i, j)& entry of the partial sums converge absolu tely for each (i, j ) ] and uniformly in any bounded set. (d) Show that

~ X ~ ( T A T - ' ) = T(exp A)T-'.

(e) If AB = BA, t l~en

exp(A + B) = (exp A)(exp B).

Hint: Write C (A + N

p=o P! p=o p=o

and show that l R ~ l + O as N + m. (f) (exp A)(exp -A) = 1, so exp A is always inverti ble. (g) The map exp, considered as a map cxp : lRn2 + R"', is clearly differentiable (i t is even analytic). Show that

expt(0)(B) = B (=exp(O). B).

(Notice thai for IA 1, tlie usual norm of A E lRnL, we have IA 1 5 IA l 5 n lA l. )

(h) Use the limit established in part e) to show that expf(A)(B) = exp(A) m B if AB = BA. (i) Lec A : R + Rn2 be differentiable, and lec

If B3(r) denotes t l~e rnatrix wliose enti-ies are the derivatives of the entries of B, show that

B' (t) = Ar(r) - exp(A (r)),

plovided dznt A(r) A*(r) = At(r) A(r). (Tliis is clearly true if A(s) A (r) = A(r) A (S) for al1 S, r .) (j) Show that tlie linear differential equation

lias the solution

~wit ided [ /nt g(s)g(r) = g(r)g(s) for al1 s,r. (This certainly happens when g(r) is a constant matrix A, so every system of linear equations with constant coeficients can be solved explicitly-the exponential of i, g(s) ds = r A can be found by put ting A in Jordan canonical form.)

7. Clieck tllat if tlie cooidinate system x is x = X-', for x : R" -t M? then X = a/axi is equivaleni lo x, (alar l) = X 0 x. 8. (a) Let M and N be Cm manifolds. For a Cm function f : M x N + R aiid y E N, let f ( - , q) denote the funciion from M to R defined by

If (x, U) is a coorditiate systeni oii M, sliow that tlie function a f /axi, defined bv

is a Cm function on M x N. (b) If # : (-E, E) x M + M i~ a 1-parameter group of diffeomorphisms, show tliat for every Cm function f: M + R, the limit

exists, and defines a Cm function on M. (c) If #, : ( - E , E ) x TM + TM is defined by

S ~ O M ~ tliat #, is Cm, and conclude tliat for every Cm vector field X and covariant vector field w on M , tlie limit

exicts and defines a Cm function on M. (d) Treat L x Y similarly.

9. Give the argument to sliow tliat #h*Y4-,, ( P ) + YP in the proof of Proposi- tion 8.

10. (a) Prove that

(b) How ~vould Proposition 8 have to be changed if we had defined ( L x l ' ) ( p ) as

11. (a) Show that # * ( d f ) ( Y ) = Y ( f o#).

(b) Usirig (a), sliow direc~ly from tlie definition of Lx tliat for Y E Mp,

and conclude that L x d f = d ( L x f ) .

The formula for Lx ds' , dciived in tlie text, is jiist a special case derived in an uiiiiecessarily clumsy way In the iiext part we get a much simpler proof that L x Y = [ X , Y 1 , usirlg the techxiique whicli appeared in the proof of Proposi- tion 15. (c) Let X and Y be \rector fields on M , and f : M + IR a Cm function. If X generates (#,), define

/ t ) = yd-r(p) ( f o #h)-

174

Show tliat

Conclude that for c(h) = a(h, k) we llave

12. Check t1ie Jacobi identity.

13. On R 3 let X, Y, Z be tlie vector fields

(a) Show that tlie map

is an isomorpliism (íiom a certain set of vector fields to IR3) and tliat [U, VI H

the cross-product of tlie images of U and V. (11) Sllow tliai tlie flow of uX + b y + c Z is a rotation of R3 about come axis througli 0.

14. If A is a tensor field of iype (:) on N and # : M + N is a diffeomorpliism, we define #*A on M as follows. If v i , . . . , vk E Mp, axid A l , . . . , hr E Mp*, then

(a) Check tliat under tlie identificatioxi of a vector fie1d [or covariani vector field] witli a tcnsor field of type ( y ) [or type (i)] this agrees with our old #* Y. @) If the vector field X on M generates ( # t ) , and A is a tensor field of type ( f ) on M, we define

Sliow that

(so that

in particular). (c) Show that

Lx,+x,A = Lx, A + Lx,A. Hini: IVe already know tliat it is tme for A of type (:), (:), (:). (d) Let

r k - I C : ( v ) + J ( V )

he any contraction

= contraction of

Show tliat Lx(CA) = C(Lx A ) .

(e) Noting that A(Xi , . . . , Xk, wl , . . . , wI) can be obtained by applying contractions repeatedly to A @ XI @ . @ Xk @ w~ @ . @ wl, use (d) to show that

n j l ... j , .

( f ) If A has compoiients A,[ .,., in a coordinate system x and X = o'a/axi, i= l

show tliat the coordinates of Lx A are $ven by

I n

+ E E ...4 . .. . - ... la - ] ila+] ... ik

a-1 i i l

15. Let D be an operator taking tlie Cm functions F to F, and the Cm vector fields V to V, sucli that D : .F + F and D : V + V are linear over W and

(a) Show that D has a unique estension to an operator taking tensor fields of type (f) to tliemselves such tliat

(1) D is linear over W (2) D ( A @ B ) = DA @ 3 + A @ DB

(3) for any coatractioxi C, DC = CD.

If we lake D f = X.f and D Y = Lx Y1 tliea tliis unique exteasion is Lx. (b) Let A be a teiisoi. field of type (i), so tliat \vc cari consider A ( p ) E Elrd(Mp); then A ( X ) is a vector ficld for each vector field X . Show that if we define DA f = 0, DAX = A ( X ) , tlien DA has a uni-que extension satisfying (1), (2), and (3). (c) Show that

( D A # ) ( / ) ) = - A ( P ) * ( # ( P ) ) -

(d) Show that

LfX = f LX - DX@df

Hillf: Clieck this for functions and vector fields first. (r) If T is of type (i), slio~r tliat

a-l "-1 u-1

Generalize to tensors of type e). 16. (a) Let f : W + W satisfy f ' ( O ) = O . Define g ( t ) = f (A) for t 2 O. Sliow tllat the right-hand derivative

g'+ ( O ) = lim ~ ( 1 2 ) - g(0) f"(0) =: -.

fz+ O+ 11 2

(Use Taylor's Tlieorem.) (b) Giw.11 c : W + M witli c'(0) = O E M,, define y ( t ) = c ( & ) for t 2 0. SIIOW tllat tlic taiigeri t vectoi- cl'(0) defined by c"(0) ( f ) = ( f o c)"(O) can also be desci-ibed by c"(0) = 2y t (0) .

17- (a) Let f : M + R have p as a critical poiat, so that f,, = O. Given rY P.. P..

vectors X,, Yp E Mp, clioose vector fields X , Y witli Tp = Xp aild Y, = Y,. Define

f**CXp, Y,) = .",(O 1. Using the faci that EX, Y],( f ) = O , SIIOMJ that f,,(Xp, Y,) is symmetric, and conclude that it is well-defined. (b) Sliow that

(c) Tlie rank of ( a 2 f /as iax j ( p ) ) is independent of tlie coordinate system. (d) Let f : M + N llave p as a critical point. For X,, Y, E M and g : N + R define

P.. P..

f**(X, Y ) ( g ) = Xp(Y(g 0 f )). Show that

$5.: Mp x M, + N/(,)

is a well-defined hilinear map. (e) If c : R + M lias O as a critical poixit, show that

takes ( l o , l o ) to the taiigent vector c"(0) defined by ctr(0)( f ) = ( f o c)"(o).

18. Let c he tlie c u n ~ of Theorems 15 and 16. If A- is a coordinate system around p wiih x ( p ) = O, and

sliow that x i (c ( t ) ) = uir + o(t 2 ) ,

where o( t2) denotes a function such that

lim o ( t2 ) / t 2 = O. r+ O

19. (a) If M is conipact and O is a regular value of f : M + R, then there is a neighborhood U of O E R such that f - ' ( U ) is dineomorpliic to f-'(O) x U,

by a diReomorphism #: f - ' ( O ) x U + f - ' ( U ) with f ( # ( p y t ) ) = t . Hini: Use Theorem 7 and a partition of unity to construct a vector field X on a neighborhood of f -' (O) such that f* X = d/d t . (11) More geiierally, if M is compact and q E N is a regular value of f : M + N: then there is a neighborhood U of q and a diffeomorphism # : f -' ( q ) x U + f- ' ( U ) with f ( # ( p , y')) = q'. (c) Ii follo\r:s from (b) tliat if al1 points of N are regular values, tlien f - ' ( q l ) and f -' ( q 2 ) are diffeomorphic for qi, q2 sufficiently close. If f is onto N , does jt follo~r thai M is diffeomorphic to f -' ( q ) x N ?

20. In let Y and Z be unit wctor fields always pointing along tlie y- and r-axes, respeciively, and let X \vi11 be a vector field oiie of whose integral curves is the s-axis, while certaixi other integral curves are parabolas in the planes J' = constant, as shown in ihe first part of the figure below. Using the second part of tlie figure, show ihai Propositioxi 18 does not hold in dimensjon 3.

CHAPTER 6

INTEGRAL MANIFOLDS

PROLOGUE

A matlieinatician's reputation rests oii Beauty is the first test: there is no tlie i-iuinbei. of bad pi-oolq he has giveii. permaneiit place in the world for [Pioxxeer woik is clumsy] ugl); mat iiematics.

A. S. Besicovi tch, quoied iii J. E. Li~lewood, A A,faílze?nnlicin?z's Adircellaniy

1 n ihe previous chapter, we have seen thai ihe integral curves of a vector field on a inanifold M may be defixiable only for some small time interval, even

thougli the vector field is Cm on al1 of M. 14re will now vary our quesiion a little, so that global results can be obtaiaed. Iastead of a vector field, suppose ihat for eacli p E M we have a 1-dimensional subspace Ap c M p . TIie f~~nciioxi A is cdlcd a 1-dimensional distribution (this kind of distrihution has xiotliixig ~rhatsoever to do with the disiribuiions of analysis, whicli iiiclude sucll things as the "6-function"). Tlien A is spanned by a vector field ZocuZ~; that is, wc can choosc (in niaay possilJe ways) a vecioi. field X sucli tllat O # Xq E Aq for al1 q ixi some open set around p. 14fe cal1 A a Cm distrihution if such a vectox. field X can be chosen to be Cm in a ncigliborliood of each point.

For a 1-dimexlsional distx-ibution tlle iiotioa of an i~iiegral curve niakes no sense, I ~ u i wc dcfine a (1-dimeiisioiial) subnianifold N of M to be an integral manifold of A if for every p E N we Iiave

i*(Np) =: Ap whei-e i : N + M is the ixiclusion map.

For a givcn p E M, we can always f i~id a11 iiitegral manifold N of a Cm distribution A with p E N; we just choose a vector field X with O # Xq E Aq for C] ir) a iieighboi-liood of p , find arr iiitegral curvc c of X with iiiitial condition c(0) = p, and theri forget about the ~mrameterization of c, by defining N to be (c ( t ) j . Tliis al-qrinent actually slrows that for cvery p E M there is a coordinate system (x, U) sucli tliat for each fixed sei of iiunibers u 2 , . . . ,un, the set

is an integra1 manifold of A on U: and that tliese are the only integral manifolds in U.

TIris is still a local resuli: but because Mte are dcaling with submanifolds, rathei- thar-i cuntes ~irith a particular parameter-ization, wfe can join over1appirig iritcgral s~il~mariifolcls together. The entire manifold M can he written as a disioint uriion of con1 lec ted integral submanifolds of A, which locally look like

(1-ather tlian like

or sonlething e1fer-i more coniplicated). For esample, there is a distribuiion o11 thc IOI-LIS IVIIOSC iritegral 1iinriifo1ds al1 look likc tlre densc 1-diii-ierisioriaI

submariifcitc1 pici~irccl in Cllaptcr. 2. O11 t1ie other llaiicl, tliere is a disiribution o11 tlrc ioriis \,.liiclr 138s OIIC coiii~~act coririectecl integral riianifold, arid a11 otlier

ir-iicgi-al niai-i ifb1cIs riorr-compact. It liíippcris tIiat thc iritcgral mariifoIc1s of tI-iesc, twfo clistril-i~iiions are also tlie in tesral cunres for certain vector fidds, but on thc

h4d1ius strip thei'e is a distributiori \vliicli is sgarined by a \rector fie1d only locally

IVe are leaving out tlie details invohred in fitting together these local integral mariifo1ds because \ve ~rill everitualty do tliis olrer again in tlie Iiigher dimensional case. For the niornent Mre \vil1 investigate higIier- dimensioiial cases onIy locally.

A k-dimensional distribution oii M is a fiiiiction p H A,, ~t l iere A, c M, is a k-dimerisiorial subspace of M,. For any p E M tliere is a neighborhood U arid li vcctor fields X1, . . . , Xk such that XI ( q ) , . . . , Xk ( q ) are a basis for Aq, for eacll q E U. IVe cal1 A a Cm distributioii if it is possible to choose Cm vector fields XI, . . . , Xq witli this property, iri a iicighborhood of eacli point p. A (k-dimensional) submanifold N of M is calIed aii integral manifold of A if for every p E N we have

i,(Np)= A, where i : N + M is theiriclusionmap.

Aliliough tlic dcfiiiitions givci-i so fal. al1 look the same as the 1-dimensional case, the results \vi11 look verv different. In general, integral manifolds do ~ i o f

ex&!, even locally. As tlie simplest cxarnple, consider the 2-din-iensional distributioii A in IR3 for

ivl-iich A, = A(,,b,,) is spanned by

l f we icler,tify TIR' witli IR" IR3, tlien A, conrists of al1 (r, S, br-),. Thus A, n1ay bc ~~icturccl as tlie plaiie with the equation

The figure be1ow s h o ~ s Ap for ~ o i n t s p =: (o, b, O). The plane through (u, 6, c.) is just parallel to the one through (o, b,0).

If you can pict~rrc tl-iis disti-il~ution, you can probably see that it has no integral manifolds; a proof can be @ven as fo1Iows. Suppose there were an integral manifold N o i A witll O E Al. The intersection of N and ((0, y, r ) ) would he a cunfe y in the (y,:)-plane tlxrouglx O whose taiigent vectors wou1d Iiave to lie ir1 the intersection of Ala,,,=) and tlie (y, z)-p1ane. The only such vectors have tljird component O, so y 111ust 13e tl-ie 17-axis. Now corisider, for each fixed yo, the iritersection N n ((A-, ?lo, c ) ). This will be a curve in the p1ane ((A-, yo, z)) t11rou~l-i (O, yo, O), with a11 tangent vectors Iiavirig slope yo, so it must be the linc ((x, yo, yox)). OUI. iiitcgi-al nianifold would have to look like the following picture. R ~ i t this submanifold cloes not urork. For example, its tangent space at ( 1,0,0) contairis \fcctoi-s wi th tl-iird componcn t non-zero.

To see in greater detaii what is happening liere, consider the somewhat more general case where A(,,b,,) = Ap is

geometrically, Ap is the plane with the equation

Z - C r i r f ( f l , b ) ( ~ - f f ) +g( f f ,b ) (y - b).

As in tlie first exarnple, the plaiie through (o, b,c) wiil he paralle1 to the one through (u, b, O ) , since f and g depend only ori a and b.

M'e riow ask wlleri tlic distribution A has an integral manifold N through each poiiit. Since Ap is never perpendicular to the (-Y, y)-plane, the submanifold is giveii locally as tlie grapli of a function:

No\%: tlie tangent space at p = (a, b, @(u, b)) is spanried by

Tlicse tangenr vectors are in Ap if and oiily if

So \ve need to fiild a funct~on a: R2 + R with

a@ (4 -=f, - " g. ax

It is wcll-kr~ot~n tliat this is not a1ways possible. Ry usirig the equality of mked partial deri\?atives, \ve find a necessary condition on f and g :

In our previous exampIe,

so tliis iiecessary condition is not satisfied. It is also well-kriown that tlie neces- hary co~~diiion (a*) is st@riFltl for the existente of the function cr satisfying (a) in a ~i~igllborliood of any point.

O. PROPOSITI ON. If f , g : IR2 + IR satisfy

in a neig1iborhood of O, and zo E IR, tlien tliere is a f~inction cr, defined in a ncigliborliood of O E IR2, such tliat

cr (O, O) = -'o

PXOOE IVe fjrst define ~ ( x , O) so that @(O, O) = :o and

riamely, wc define X

Tlleii, for eacll x, we define a (x , y ) so that

namely, we define

This cor~structioi~ does aot use (w), and always provide us with an a. satisfying (2), aa./ay = g. IVe claiin that if (**) holds, tlieii also aa/ax =: f. To prove tliis, concider, for each fixed x, the function

This is O for JI = O by (1). To prove that it cquals O for al1 y, we just have to S ~ I O M ~ that its derivative is O. Rut its derivative at y is

IVc ar-e IIOW ready to look at essentially t11e rnost general case of a 2-dimensional distr-ibution in R3:

where f, g : R3 + R. Suppose tliat

N == ((x, y,z) : 6~ = a.(x,y))

is an integral manifold of A . The tangent space of N at p = (a , b, a(a, 6) ) is spanned, once again, by

These tangent vectors are in Ap if and on1y if

aa f (a,b, d a , 6 ) ) = - ( a , b), a~

aa g(nl b, a(o, b) ) = - (a , b).

ay

In order to ohtain neccssary conditions for the existente of such a function a , we again use the equality of mixed partial derivatives. Thus (*) and the cllain ru1e imply that

Tliis condition is not very usefu1, since it still involves the unknown function a , l~u t we can substitute from (*) to obtain

-(a, b,a(a, b ) ) + - (n,b,a(a, b ) ) g(a,b,@(a,b)) ay a z

Now wc are looking for conditions whicli will he satisfied by f and g when there is an integral niailifoId of A Iltroqk euu-poinf, which means tliat for eacli pair (u , b) these equations must ho1d no matter what a(a, b) is. Thus we ohtain finally the necessary coiidi tion

In tliis more genera1 case, the necessary condition again turns out to be sufficient. In fact, there is no need to restrict ourselves to equations for a single function defined on R2; we can treat a system of partial difGerentia1 equations for n functioils o11 Rm (i.e., a partial dflerential equation for a function from R" to R"). In the following theorem, we wdl use I to denote points in Rm and x for points in W"; so for a function f : Wm x R" + Rk we use

as - . for Dm+i f. axi

1. THEOREM. Let U x V c Rm x R" be open, where U is a neighborhood of O E R"', ar-id let J i : U x V + Rn be Cm functions, for i = 1 , . . . , 1 1 1 . Then for every A- E 1/, there is at most one function

defined in a neiglil~orhood W of O in Rm, satisfying

(hllore precisely, ariy two sud1 furictions a1 arid a*, defined on WI and W2, agree on tlie conil~oilerit of Wi n W2 wliic1i contains O.) Moreover, such a function exists (arid is autoriiatically Cm) in some lieighborhood W if and only if there is a iieighborhood of (O, x) E U x V on which

I'HOOF. Uiliqueriess ~ l i l l be obvious from tlie ~ r o o f of existence. Necessity of tlie conditioris (34) is lcft to the readei- as a siinple exercise, and we will concern oursch~es with provirig existerice if these coriditions do liold. Tlie proof will be like tliat of Proposition O, 14th a different twist at the end.

IYe f i l ~ t waiit to define ~ ( r , O,. . . , O) so that

a(0, o,. . .,O) = x

To do this, we corisider the ordinary differential equation

Tliis equation has a unique solution, defined for 111 < E I . Define

Then (1) holds for 111 -= EI . Now for eacli fixed r l with 1 1 ~ 1 < EI, consider thc equation

7'liis Iias a unique solution for sufficiently srnall 1. At tliis point the reader must refer hack to Theor.ern 5-2, arid veri+ the following assertion: If wre choose ~1

sufficiently srnall, then for 11'1 < E I tlie solutiorls of tlie equations for P2 witli tllc initial conditioiis p2(0) = Q ( I ~ , 0,. . . ,O) will eacli be defined for 111 ~2 for sor-ile ~2 > O. We then define

(2) aot 1 1

- ( f l , l , O ,..., O) = f * ( f ,1,0 ,..., O,ot(1 ,1,0 ,..., O)) a12

11'1 ( E l , 111 <E2.

l4'c claim that for eacli fixcd 1' witli 1 < E I wTe also Iiave, for al1 1 witli 111 < ~ 2 ,

a@ l 1 1 (3) o = g(1) = +f ,1, o, . . - ,O) - fl (1 ,1,0,. . a , O,ot(1 ,f ,O,. . . , O)).

81

Note first that

l e IIOW derive an equation for gf(1). In the following, a11 expressioris iilvoh/- ing ot are to be n~aluated at (I', 1, 0,. . . ,O) and all expressions involving fi are to be evaluated at (rr,1,0,. . . ,O,a( l l , l , 0,. . . ,O)). l4re llave

and thus

a/z " a/zaak as, n ah k = - + C T ~ - ~ - C ~ ~ ~ al1 by (2) again

k -1 ax al

k=i

by definition, (3)

Now equation (5) is a dxerential equatiori 1vit1i a uiiique solution for each iiiitial conditioli. The soIution with initial coridition g(0) = O, given by (41, is cleai-ly g(l) = O for al1 1. So (3) is true.

It is a simple exercise to continue rhe defiiiition of cr until it is eventually defiiied on ( - E ~ , E,) x x (-E,, E,) and satisfies (*). 43

Tlieorem 1 csseiitial1y solves for us the problem of deciding which distributions have iiitegi-a1 mailifolds. Our investigation of the prob1em so far illustrates one hasic fact about theox-ems in dxerential geometry:

T\J aiiy of tlie fu iidamen tal theorems of differen tial geometry fdl in to oiie of two classes. Tlie first kind of theorem says that if one has a certain xiice situat ioii (e.g., a distrihution ~ 4 t h integral submanifolds thi-ough every poiiit) theii certaiil othei. conditions hold; these con- ditioils are obtaiiied by settixig mixed partials equal, and are called "integrahility coilditions". The second kind of theorem justifies this terminoIo0, by diowiiig that the "integx.ahility coxiditions" are sufficient for recovering the nice situation.

The reniaiiliiig parts of oui. investigation, iii which we will essentially begin aiiew, illustratcs axi eicii inoiae impoi.tant fact about the theorems of dxerential geometry:

Tliere are a1ways iiici-edibly concise axid elegarxt ways to state the integrahility conditions, aiid prove tlieir sufficieiicy, without ever even meii tioiiiiig partid derivatives.

LOCAL THEORY

If f : M + N is a Cm function, and X and Y are Cm vector fields oii M and N, respectively, we say that X and Y are f-related if f*p(Xp) = Y'(p) for each p E M. If g : N + R is a Cm function, then

so IYg) 0 f = X I f og).

Conirersely, if this is true for al1 Cm functions g : N + R, then X and Y are f-related.

Of course, a given vector field X may not be f-related to any vector field Y, nor must a given vector field Y he f-related to any vector field on M. Iii one case, the latter condition is fulfilled:

2. PROPOSITION. Let f : M + N be a Cm function such that f is an immersion. If Y is a Cm vector field on N with

Yf(p) f fP*IMP)? then there is a unique Cm vector field X on M which is f-related to Y.

PROOF. Clearly we must define Xp to be the unique element of Mp with Y/(,) = fp,Xp. To prove that X is Cm, we use Theorem 2-lO(2): tliere are coordiiiate systems (x, U) arouild p E M and (y, V) around f (p) E N sud1 that

1 1 n y o f o r - ' ( o ,...,un) = (o ,..-, o ,O ,.-., O). This is easily seen to imply that

a fp* (a,) = d,,,

Thus if " a y=Crui- i=l

ay' where a' are Cm fuiictions, then

q

where a' o f = j?'. This implics that the functions j?' are Cm (Problem 3). 4+ The moct iinportant property of f-relatedness for us is the following:

3. PROPOSITION. If Xj aiid Yj are j-1-elated, foi. i = 1,2, then 1x1, X2] aild [Yl, Y,] are f-related.

PHOOI;. If g : N + iR is Cm, then

by ( I ) , with g replaced by Y2g and Yig, respectively

= XI (X2k o f)) - X2(Xi(g o f)) by (1) = Exi,X21(g ~ f ) . 6

Now coiisider a k-dimensional distribution A . M7e wi1l say that a vector field X belongs to A if Xp E Ap for al1 p. Suppose that N is an integral inaiiifold of A , aiid i : N + M is t1ie indusion map. If X and Y are two vector fields which belong to A , then for al1 p E N there are unique y,, E N, such that - -

Xp = i,Xp, Yp = i*Yp.

In otlier words, X and are i-related, and Y and I are i-related. Proposition 2 shows tliat F aiid y are Cm vector fields on N, and Proposition 3 then shows that IR, 81 and [X, Y] are i-related. Thus

- - i* Ex, Ylp = [X, Ylp.

- - Here EX, YIp E Np; tliis therefore shows tliat EX, YIp E Ap. Consequendy, if tliere is ail integra1 nlaiiifold of A through every point p, then EX, Y] ah belongs lo A.

For a moment look back at tlie distribution A in IR3 given by

The vector fields a x = - a ax +f- a z

bcloiig to A . Usiilg the foi.mula on page 156, we see that

This beloiigs to A only wlien tlle expressioil i11 pai-ei~tl~eses is 0, which is precisely the condition for A to have an integral nianifold through every point.

In general, A is called integrable if EX, Y ] beloilgs to A whenever X and Y belong to A . This condition can be checked fairly casily:

4. PROPOSITION. If Xl,. . . , Xk span A in a iieighborhood U of p , theii A is integrable on U if and only if each EXi, Xj] is a linear combination

k

[X;, Xj] = E CGX. rr=l

for Cm functions CG.

PXOOI;. Such functions clearly exist if A is integrable, since EXi, Xj]q E As, . liich is spaiiiled ly tlie X,(q). Coilversely, suppose such functions exist. If X

aiid Y belong to A we caii clearly write

To prove EX, Y] belongs to A , it obviously suffices to treat each [j;.Xi,gj Xj] sepai+ately. Since we have

clearly [ f X, g Y] bcloiig~ to A if X, Y and EX, Y] do. *:*

We are now ready for tlie maiii t-lieorem. It is equivalent to Theorem 1; in fact, Theoi-em 1 can be derived from it (Problem 7). Rut the proof is quite diffei-en t.

5. THEOREh4 (THE FRORENIUS INTEGRARILITY THE0REh.I; FIRST VERSION). Let A be a Cm iiitegrahle k-dimensional distribution oii M. For evcry p E M tliere is a coordinate system (A-, U) with

x(p) = 0

x(U) = (-E, E) x . - x (-E, E),

siicli tliat for each ak+', ,. . , a n with al1 lail e E, tlie set

( q E U : x ~ + ' ( ~ ) = nk+' , . . . , xn(q) = n")

is an iiitegi-al manifold of A. An): connected integral maiiifold of A i-estricted to U is contained in one of

tliese sets.

PX001;. We can clearly assume that we are in IR", with p = O. Moreover, we can assume tliat Ao c Wno is spanned by

Let s : IR" + IRk be projection onto the first k factors. Then rr, : A. + Wko is an isomorphisin. Ry contiiiuity, s. is one-one oii A, for q near O. So near O, we can clioose unique

Xi(q), - . ,xk(q) E Aq

so that

r*x i (q ) = 2 i = I , ..., k. a

Tlien the vector fields X j (oii a neigliborliood of O E W") and a/ar (on IRk) are rr -related. Ry Propositioii 3,

Rut, EXi, Xj]q E A, by assuinption, aiid rr, is one-one on A,. SO EXi, Xj] = 0. Ry Tlieorem 5-14, tliere is a coordinate system x such that

k+i Thc sets {q E U : xk+' ( q ) = n , . . . , xH(r/ ) = 0") are clearly integral manifold~ of A, siiice tlieii- tangeiit spaces are spaniied by the a/axi = Xi for i = 1, ..., k.

If N is a coniiected in tegi-al manifold of A restrictcd to U, with iriclusion map i : N + U, coilsider d(xn' o i) for k + I 5 rn 5 n. For any tangent vector X, of N, we have

c/(sm o i) (X,) = X, (xm o i) = i* X, (xm)

= o,

sirice i. Xq E A,, wliicli is spanned by the a/axjl, for j = 1 , . . . , k. Thus d(x-"' o i) = 0, which implies tliat sm o 1 is constaiit on tlie conriected manifold N.

GLOBAL THEORY

In order to express the global results succinctly, we introduce the following terminoloa

If M is a Cm mailifold, a (usually disconnected) k-dimensional submanifold N of M is called a foliation of M if every point of M is in (some compoiient of) N, and if arouild every point p E M there is a cooi-dinate system (A-, U ) , with

x ( U ) = ( - E , E ) X - X ( - E , E ) ,

such that the components of N n U are the sets of the form

k+l { q E U : xk++' ( q ) = n , . . . , x n ( q ) = nn) 10'1 < E.

Eacli compoiierit of N is called a folium oi- leal of the foliation N. Notice that two distinc~ componcnts of N n U might belong to tlle same leaf of the foliation.

6. THEOREh4. Let A be a Cm k-dimeilsional integrable distribution 011 M. Then M is foliated by an integral manifold of A (each component is called a maximal integral manifold of A).

PROOI;. Usirlg Tlieorem 1-2, we see tllat we caii cover M by a sequerice of coordinate systems (x ; , U j ) satisfying the coiiditions of Theorem 5. For such a coordinate systein ( x , U ) , let us cal1 eac11 set

{ q E U : xk+l ( q ) = nk+I , . . , , x N ( q ) = 0")

a s l i ~ t of U . It is possible for a single slice S of U; to intersect U, in more than one slice

of U j , as shown bclow. Rut S n Uj has at most countably many components,

and each component is contained in a single slice of Uj by Theorem 5, so S nUj is contained in at most countably many slices of Uj.

Given p E M, cl~oose a coordinate system (xo, Uo) with p E Uo, and let So be the slice of U. contaii~ing p. A dice S of some Ui wil1 be ca11edjoined to p if there is a sequence . . O = r o , ~ ~ , . . . , i ~ = i

and corresponding dices

with

Sia n S,+, # 0 a =O,. . . , I - l .

Since ttiere are at most countably many such sequences of slices for each sequence io,. . . , i l , and only countably many such sequences, there are at most countably many slices joiiled to p. Using Problem 3-1, we see that the union of al1 such slices is a submanifold of M. For q # p, the corresponding union is either equal to, or totaUy disjoint from, the first union. Consequently, M is foliated by t1ie disjoint union of a11 such submai~ifolds; this disjoint union is clearly an integral manifold of A . +:+

[If we are allowing non-metrizable manifolds, the proof is even easier, since we do not have to find a countable number of coordinate systems for each leaf, and can merely describe the topology of the foliation as the smallest one which makes each slice an opei-i set. In this case, however, the discussion to follow will not be valid-in fact, Appendix A describes a non-paracompact manifold whic11 is foliated by a Iower-dimensional conneckd su bmani fold.]

Notice that if (x, U) is a coordinate system of the sort considered in the proof of tlie theorem, then ii~finitely many slices of U may belong to the same folium.

However, a/ rizos/ coun1ab3, mang slices can belong to the same folium; othemise

tliis folium would contain an uncountable disjoint family of open sets. This allows us to apply a proposition from C hapter 2.

7. THEOREh4. Let M be a Cm manifold, and Mi a folium of the folia- ti011 determiiied by some disti-ibution A. Let P be aiiother Cm mai~ifold and f: P + M a Cm function with J ' ( P ) c Mi. Then f is Cm considered as a map into M I .

PROOI;. According to Proposition 2-1 1 , it suffices to sllow tliat f is continuous as a nlap iilto M I . Given p E P, choose a coordiilate system ( x , U ) around f ( p ) such that tlie slices

( q E U : ,yk+' ( q ) = ak+], . . . , x n ( q ) = a")

are integral maizifolds of A. Now f is coi~tinuous as a map into M, so f takes

some neighborhood W of p into U; we can choose W to be connected. For k + 1 5 i 5 t i , if we had x i ( f (P ' ) ) # ni for any p' E W, iheii xi o f would take on al1 values between n' and xi( f ( p t ) ) , by continuity. This would mean tliai f (W) contained points of uncountably many dices, contradicting the faci t l~at f ( W ) c M1

Consequei~tly, ~ ' ( f ( ~ ' ) ) = ni for al1 p' E W. In other words, f ( W ) is contained in t l~e single dice of U whicl~ contains p. This makes it clear that J' is continuous as a map into M I . +3

PROBLEMS

1. (a) Let { = ír: E + 3 be an n-plane bundle, and {' = ír': E' 4 3 a k-plane bundle such that E' c E. If i : E' + E is the inclusion map, and Ig : 3 + 3 the identity map, we say that {' is a subbundle of { if (i, I B ) is a bundle map. Show that a k-dimensional distribution on M is just a subbundle of TM. (b) For the case of Cm bundles { and {' over a Cm manifold M, define a Cm subbundle, and sliow t l~at a k-dimensional distribution is Cm if and only if it is a Cm subbundle.

2. (a) In the proof of Theorem 1, check the assertion about cl~oosing E I sufficiently small. (b) Supply tlie proof of the uiiiqueness part of the theorem.

3. (a) In the proof of Proposition 2, show tliat

(13) Complete the proof of Proposition 2 by showing t l~at if

with oii o f = s i , then tlie fui~ctions B i are Cm

4. In the piaof of Proposition 4, show that the functions CG actually are Cm.

5. Let Al , . . . , Al, be integrable distributions on M, of dimeiisions d i , . . . ,di,. Suppose tliat for eacli p E M,

Show that tllere is a coordinate system (x, U ) arouiid each point, sucli t l~a t Al is spaiined by a /axl , . . . , a/axdl, etc.

6. Prove Theorem 1 from Theorem 5, by considering the distribution A in Rm x R" (with coordinates I, x), defined by

Notice that everl when tbe fi do not depend on x, so that the equations are of the form

with the integrability conditions

we neirertheless work in Rm x R", rather than Rm. This is connected with the classical technique of "introducing new independent variables".

7. This problem outlines another method of proving Theorem 1, by reducing tbe partial dfirential equations to ordinary equations along lines through the origin. A similar tec hnique will be very important in C hapter 11.7.

(a) If we want a (u t ) = p(u, I ) for some function 8 : [O, E) x W + Vy show tliat p must satisfy the equation

We know that we can solve such equanons (we need Problem 5-5, since the equatiori depends on the "parameter" I E Rm). One has to check that one E

can be picked wl-iich works for al1 I E W. (b) Sllow that

p(u, VI> = ~ ( u u ? 1).

(Show tliat both functions satisfy tl-ie same differential equation as functions of u , witli tl-ie same ii-iitial condítion.) Ry sl~rinking W, we can consequently assume that E = 1. (c) Conclude that

(d) Use tl-ie integrability condition on f to show that

satisfy tlie same diffei.eiitia1 equation, as functions of v. Use (c) to conclude tliat the two functions are equal. (e) Define a(r) = P(1,l). Noting that a(vr) = P ( v , f ) , sliow that a satisfies the desired equation.

8. This pi-oblem is for tliose who know something about complex analysis. Let f : @ x @ + @ be complex analytic. If we denote the coordinate functions in @ x C by -1, z2 = X] y y lY x2, y2, then f = u + iv satisfies the Caucliy-Riemann equations

Use Theorem 1 to p r o i ~ tliat we can solve the equations

aa2 - =

aal ax v(x,y ,a l (x , y ) ,a2(x , y)) = - -

ay

iii a neighborl-iood of O E @ (or of ariy poil-it zo E C), aiid coiiclude that the differen tial equation

#'(d = f (z, #(E))

(iii ~ h i c h ' denotes the coinplex derii~atiire) has a solution in a neigl~borhood of ro, with any giireii iiiitial condition #(zo) = wo.

CHAPTER 7

DIFFERENTIAL FORMS

w e turri our attention once more to teiisor fields, but we will be concerned witli a special kind of tensor field, the discussion of which requires some

more algebraic preliminaries. Let V be aii n-dimensional vector space over R. An element T E T k ( V ) is

called alternating if

If T is alternatiiig, then for aiiy vi,. . . , vk, we have

Tlierefore, T is skew-symmetric:

O f coui.se, if T is skew-symmetric, tlien T is also alteri~ating. [This is not true iii the special case of a vector spacc ovei. a field ivhcre I + I = O; i i ~ this case, skew-symmetry is tlie same as symmetry, and the condition of being alternating is the strongei. one.]

W will denote hy Rk(V) the set of al1 alternatiiig T E T k ( V ) . It is clear tliat Rk(V) c 7 " ~ ) is a subspace of T k ( V ) . Moreover, if f : V + W is a linear traiisfoi.niatioii, tlien 1' : 7k (W) + 7 ' ( V ) preserves tliese subspaces- f * : ( W + R k ( . Notice t l~at R1(V) = T 1 ( V ) = V*, so R1(V) has dimension j7. It is also convenient to set RO(V) = 7 O ( V ) = R. At the moment it is iiot cleai 12~liat tlie dirnension of Rk (v ) equals for k > 1, but one case is 1ve11-kiiowii. The most familiar example of aii alternating T is the determinant fuiiction det E T"(Rn), c.onsidered as a function of the n rows of a matrix- wc shall sooii see that tliis fui~ction is, iri a certain seiise, tlie most general al tei-iiatiiig furiction- hdost discu~sioiis of tlie determiiiant begin by s howing tliat of any t~vo alterriatiiig 17-linear fuiictions oii Rn, one is a multiple of the

other; ii i other ivords, dim a" (R" ) 5 1. Then one proves dim S2"(Rn) = I bv actually constructing the non-zero function det (it follows, of course, that dim 52" ( V ) = I if V is any 11-dimensional \rector space). The construction of det is usua1ly bv a messx explicit formula, which is a special case of the definition to follow.

Let Sk denote the set of all permutations of (1,. . . , k); an element E Sk is a function i H ~ ( i ) . If ( U ] , . . . , uk) is a k-tuple (of any objects) we set

This defiriition has a built-i11 confusion. On tlie riglit side, the first element, for example, is t l~e o(I)" of tlie u's on t11e Ieft side; if these u's Iiave indices 1-unning in some order ofhm than 1,. . . , k, then the first e1ement on the right is tior

iiecessarily tliat u ~rliose iiidex is o(]) . The simplest way to figure out somet11ing like .(u3, u2. U] , . . . ) is to renanie tliings: vs = W I , u2 = wl, U I = w3,. . . . Tlius warried, we computc

by setting

so that

Thus

No\v foi- aiiy T E T'(v) we defiiic tlie "alternati011 of T"

wliei-e sgn is + I if a is an even permutation aiid -1 if o is odd.

1. PROPOSITION.

(1) If T E Tk(v), then Alt(T) E Qk(v) .

(2) If o E Qk(v), then Al to = o.

(3) If T E Tk(V), then Alt(Alt(T)) = Alt(T).

PK001;. Left to the reader (or see pp. 78-79 of Cafcuius on Manifofds). +3

Mle riow define, f o ro E Qk(V) and fi E Q'(v), an element o ~ f i E Qk+'(v), the wedge product of w and fi, by

The funily coefficient is not esseiitial, but it makes some things work out more nicely, as we sliall soon see. It is cIear that

(1) A is bilinear: (01 + 0 2 ) A f i = 0 1 A f i + 0 2 A f i

(2) f*(w A fi) = $*o A f '7.

Moreover, it is easy to see that k l (3) A is "anti-commutative": o A fi = (- I ) fi A o .

111 particular, if k is odd then

FinaIIy, associativity of A is proved in the following way

2. THEOREM.

(1) If S E Tk(v) and T E T'(v) and Alt(S) = O , tlien

Alt(S @ T ) - N t ( T @ S ) = 0.

(2) ~ i t ( ~ l t ( o @ @ e ) = A I ~ ( W @ fi @ e) = ~ l t ( ~ @ ~ l t ( ~ @e)) . (3) If o E Q k ( v ) , fi E Q2(V), 0 E Qm(V), then

= E sgno - ( S 8 T ) . ( a - ( v ~ , . . . ~ v k + l ) )

= E sgn o S ( ~ U ( I ) , . . . Y v~(k) ) T(v~(k+i) , . . . , vo(k+I)). ufiSk+r

NOMI let G c Sk+l coiisist of all o ivliicli leave k + 1 , . . . , k + I fixed. Tlieri

E sgno - S(U,(I), . . . , ~o(k) ) + T(vo(k+~), 3 VU(~+I)) UEG

Suppose noMt tliat o0 $ G. Let ooG = (a0or : a' E G). Tlien

= sgn o0 . sgriof . (S @ T)(o l . (oo . ( V I , . . . , vk+]))) .by (*). UIEG

M7e baile just slio~lri tliat tliis is O (since a0 ( v i , . . . , vk+l) is just some other. (k + 1)-tuple of vectors). Notice tliat G n aoG = 0, for if o E G n ooG, then o = oOal for sorne o' E G, so a0 = o(of)-' E G, a contradiction. We can theii coiitiiluc iil tliis way breaking Sk+] up into disjoint slibsets, tlie sum over eacli l~eiiig 0. Tlie relatioii Alt(T @ S) = O is proved similarly

(2) Clearly

Alt(Alt(l7 @ O ) - q 63 O) = Alt(q @ O ) - Alt(77 @ O ) = 0,

tlie otlier equality is proved similarl.

(k + I + m)! ( # A q ) A @ =

(k + I)! m! Alt((w A q) @ 0)

+ +

(k + I + m)! (k + I)! (k + I)! n7! k! l!

M t ( o @ q @ $1.

Tlie othei- equality is proved similarly. S+

Notice tliat (2) just states tliat A is associatii;e eiren if we had omitted tlie factor (k + I)!/ k! I! in t l ~e defiilitioii. On tlie otlier liand, tlie fac.tor l/k! in tlie definition of Alt is essential-without it, we would ilot have Alt(Alt T) = Alt T, and the first equation in tlie proof of (2) would fail. [If we had defined just like Alt, but iwithout the factor l/k!, then A could be defined by

Tliis inakes sense, E I ~ C ~ oum njeId ofjnik clla~ackklic, because each term in tlic sum At(o @ q)(vl,. . . , occurs k!I! times (siiice o and q are alternating), and l /k! I! can be interpreted as meaning that tliese k!l! terms are replaced by just one.] Tlie factor (k +l)!/k!I! Iias been iilserted into the definition of A

for the folloiving reason. If u , , . . . , u,, is a basis of V, and #], . . . , #, is the dua1 basis, thcii

( 1 + . + I)! #] A * . . A#, = 1!.*-1!

W # l @ . @ #,)

= sgno a ( # ] @ . . - @ # , J o o . S,,

In particular, ($1 A . A #n)(v1, . . . , un) = 1 .

(So if V I , . . . , v, is tlie standard basis for Rn, tlieil #] A . . . A #, = det .) A basis for Slk(v) can iiow be described.

3. THEOREh4. Tlie set of al1

is a basis for R' (v) , wliich tlierefore Iias dimension

(In particular, Slk(V) = (0) for k > n.)

PKOOI;. 1f o E R'(v) c T'(v), we can write

Eacli At(#il 8 8 #,) is either O or = =t(l/k!)#jl A A #jk for some

jl c . c jk, SO tlie elenlents #jl A . . A #jk for j i < < j k span f ik (v) . If

the i~ applying b0t11 sides to (vil, . . . , vik ) $ves uil ...jk = 0.

4. COROLLARY. If w,. . . ,wk E f i l (V) , tlieii 01,. . . ,wk are ljnearly independent if a i~d only if

O] A . * . A O k # O .

PHOOI;. If 01 , . . . , wk are liiiearly independent, there is a basis vi,. . . , VA,. . . , v, of V sucli tl~at the dual basis vectoi-s $1 , . . . , #k, . . . , satisfy #j = W j foi' I 5 i 5 k. Tlien 01 A . . . A wk is a basis elemciit of R ~ ( v ) , so it is not O.

On t l~e other hand, if

then

To abbi-eviate foi-mulas, it is coi~vciiieiit to let 1 deiiote a typical "multi-index"

(ii, . . . ,ik), and let #I ddeote #jl A . A #i,. Tlien cveq elcmeni of f ik(V) is

Notice that Throi-cm 3 implies tliat nrci-y o E f i k ( ~ " ) is a lineal- combiiiatioii of tlie functions

Oile more siiiiple theoi-em is iii ordei; befoi-e we proceed to apply oui- coii- structiorl to manifolds.

5. THEOREM. Let vi,. . . , u, be a basis for V, let w E Qn(V), and let

j l i

Then o(w], . . . , wn) = det(aij) a w ( v ~ , . . . , U,).

PR00E Define 17 E T"(Rn) by

Theii clearly 7 E S2"(Rn), so 17 = c det for some c E R, and

6. COROLLARY. If V is 11-dimensional and O # ¿o E Qn(V), tlzen there is a unique orientation p for V such that

[ut, . . . , U,,] = ,u if and only if VI, . . , un) > 0-

Witlz our izew algebraic coiistructioiz at lzaizd, we are ready to apply it to \rector buildles. If < = ?r : E + B is a \lector buiidle, we obtain a new bundle Qk(<) by replacing cach fibre rrwl(p) witli ~ ~ ( a - ' ( ~ ) ) . A section w of ak(<) is a function with w(p) E fik(?rwi (p)) for eacli p E B. If 17 is a section of al(<), tlien we caii defiiie a section w A 17 of QkU(<) by (w A q)(p) = w(p) A q(p) E

nk+'(?r-l (p ) ) . In particulai; srctioiis of R ~ ( T M ) , wliicli are just alternating covariant tensor

fields of 01-der k, al-e called k -forms on M. A 1 -form is just a covariaizt vector field. Siiice Qk(TM) can obviously be rnade into a Cm vector bundle, we can speak of Cm forms; al1 forms will be uizdei-stood [o be Cm forms unless tlze con trary is explicitly stated. Remember tlzat covariant tensors actually map contravai-iaiztly: If J': M + N is Cm, aizd o is a k-form on N, tlzen f * w is a k-forin oii M. iJe caii also define wi + o 2 aiid o A v. The following properties of k-forms a i r obvious fronz tlie coi-1-espoiidiizg properties for nk(V):

If (x, U) is a coordinate system, then the dxi(p) are a basis for M,*, so the dxil (p) A . A dx'a (p) (i] < . . < ik) are a basis for nk (p). Thus every k-form w can be ~ ~ i t t e n uniquely as

or, if we denote dxil A . . . A dx'" by dxx' for the multi-index I = (i], . . . , ik),

The problem of finding the relationship between the wl and the functions o'! when

w = E 01 dx' = E w ' ~ dyJ I I

is left to tl-ie reader (Problem IG), but we will do one special case liere.

7. THEOREM. If f : M + N is a Cm fui~ction between n-manifolds, (x, U) is a coordinate system arouiid p E M, aild (y, V) a coordinate system around q = f ( p ) E N , then

f*(g c¡yl A - . A dy") = (g o f ) . det ("2") dxl A . A d.',

PROOI;. It suffices to sliow that

Now, by Problem 4-1,

= dyl (q) A . -

8. COROLLARY. If (x ,U) and (y, V) are two coordinate systems on M and

then

11 = g det

PKOOI;. Apply the tlieorem with f = identjty map. *t.

[This coro11ary S hows tliat 12-forms are the geometric objects corresponding to the "even scalar densities" defined in Prob1em 4-10.]

If t = is : E + 3 is an 12-p1ane bundie, then a nowlrue zuo section o of 52" ( t ) has a special significance: For each p E 3, the non-zero w(p) E Rn(x- '(p)) determines an orientation pp of is-'(p) by Corollary 6. It is easy to see that the collection of orientations (pp ) satisfy the "compatability condition" set forth in Chapter 3, so that p = (pp) is an orientation of t . In particular, if there is a nowhere zero n-form o on an 11-manifold M, then M is orientable (Le., the bundle TM is orientable). The converse also holds:

9. THEOREM. If a Cm manifold M is orientable, then there is an n-form w on M which is nowhere O.

PKOOI;. By Theorem 2-13 and 2-1 5, we can choose a cover O of M by a collection of coordinate systems ((x, U)), and a partition of unity (#u) subordinate to O. Let p be ail oriei~tation of M. For eacll (A-, U) cl-ioose an n-form wu on U sucl-i that for v i , . . .,u, E Mp, p E U we have

wu(vi,. . . , u,) > O if and only if [u],. . . , u,,] = pp.

Now let

Then w is a Cm rz-form. Moreover, for every p, if v i , . . . , v, E Mp satisfy [VI , . . .,U,] = pp, then ea4

and strict iiiequality holds for at least one U. Thus o ( p ) # O. +

Notice that the bundle Qn(TM) is I-dimensional. We have shown that if M is orientable, then Qn(TM) has a nowhere O section, which implies that it is trivial. Conversely, of course, if the bundle Qn(TM) is trivial, then it certainly has a nowhere O section, so M is orientable. [Generally, if is a k-plane bundle, tlien Qk(<) is trivial ifand only if < is orientable, provided that the base space B is L ' p a r a ~ ~ m p a ~ t ' ' (every open cover has a locally-fini te refinemen t) .]

Just as QO(V) has been introduced as another name for R, a O-Form on M will just mean a function f on M (and f A o will just mean f o). For eilery O-form f we have the I-form df (recall that df (X) = X( S)), which in a coordinate system (x, U) is given by

If o is a k-form

I

then each d o r is a 1-forin, and we can define a (k + 1 )-form d o , the diflerential of o , by

It iurns out t l~at t11is definition does not depend on the coordinate system. Tllis can be 131-oved i i i several ways. The first way is to use a brute-force computation: comparing the coefficients otl in the expression

1

witli the o]. Tlie second method is a lot sneakie~ Mfe begin by finding some properties of

do (sti1l defined witll respect to this particular coordinate system).

1 0. PROPOSITION.

(1) d (o l + os) = do, + do2 .

(2) If wl is a k-form, then

PK001;. (1) is clear. To prove (2) we first note that because of (1) it suffices to consider only

01 = f d x I

3 0 2 = g dx .

Then wl A y = fg dx' A dx3 and

d(o l A 02) = d( fg ) A dx' A dx3

= g d f ~ d x ' ~ d x ' + f d g ~ d x ' ~ d x J

=dw l A w 2 + ( - 1 ) ~ f d x 2 ~ d g ~ d x 3 k - dwl A m2 +(-1) m1 A dw2.

(3) It cleal-ly sufices to consider only k-forms of the form

w = f dx ' .

Then

In this sum, the terms

cancel in pairs. *:+

iVe riext note t l~at t11ese propei-ties characterize d on U.

1 1 . PROPOSITION. Suppose d' takes k-foi-ms on U to (k + 1)-forms on U, for al1 k, a i ~ d satisfies

(1) d'(w1 + w2) = d'wl + d1w2.

(2) d'(w1 A w2) = d'wl A w2 + (- 1)'wl A dtw2.

(3) d'(df f ) = 0.

(4) d'f = (the old) d j :

Then d' = d on U.

PROOF. It is clearly enough to show that d'w = dw when w = f dx'. Now

by (21,

d'( f dx' ) = d y A dx' + f A dr(dx')

= df ~ d x ' + f A d'(dxr) by (4).

So i t sufices to show that dr(dx') = 0, where

We will use induction on k . Assuming it for k - 1 we 11ave

dt(dx') = d'(d'wii A . . . A dt~- 'k )

= d'(d'xil) A d'Xi2 /\ . . . A d'x*

- d ' r i i A d'(d'xii A . . A d'xi*) by (2) = O - O, by (3) and the inductive hypothesis. 6

12. COROLLARY. Tl-iere is a ui-iique operator d from the k-forms on M to the ( k + 1)-forms on M , for al1 k , satisfying

and agreeii~g wit1-i the old d on functions.

PROOI;. Foi- each coordinate system (x , U ) we have a unique du defined. Given t l~e form o , and p E M , pick any U with p E U and dcfii~e

Tlie thii-d way of proving tl~at the defii~ition of d does not depend on the coorclinate swem is to give an invariant definition.

For dw, as for any form, we have

It is clear from (*) that dw(i3/axffl,. . . , a/axffk+1) = O

Since the a's are increasing, this happens only if

in which case

which isjust the old definition. +:+

This is our first real example of an invariant definition of an important tensos, and our first use of Theorem 4-2. We d o not find dw(p) (v l , . . . , vk+]) directly, but first find dw(Xl , . . . , Xk+]), where Xi are vector fie1ds extending vi, and then evaluate this fuilction at p. By some sort of magic, this turns out to he iiidependent of t1ie extensions Xl , . . . , Xktl. T11is may not seem to be much of an improvement over using a coordinate system and checking that the definition is iildependent of the coordinate system. Rut we can hardly hope for anything better. After al], although dw(Xl , . . . , Xk+])(p) does not depend on the values of Xi except at p , it dom depend oii the values of o a t points other tllan p- this must enter into our formula someliow. O n e otller feature of oui- definition is common to most iiivariant definitions of teiisors-the presence of a term involviiig brackets of various vector fields. This term is what makes tlie operator

I j~ear over the Cm functions, but ii disappears iii computations in a coordinate system.

In the particular case where w is a 1-form, Theorem 13 Gves the following formula.

TIiis enables us to state a second version of Theorem 6-5 (The Frobenius Inte- grability Tlieoreni) in tei-nis of dxereiitial forms. Define the sing R (M) to be the direct sum of the rings of 1-forms on M, for a11 I. If A is a k-dimensioi~al distribution oii M, then $(A) c R (M) ~rill denote the subring generated by ihe set of al1 forms o wiili the property that (if o has degree i)

o(X,, . . . ,XI) = O wlieiiever Xl, . . . , XI be1ong to A.

It is clear that ol + o1 E $(A) if q,o1 E $(A), and that q A o E $(A) if w E $ (A) [tiius, J(A) is aii ideal iii the riiig R (M)]. Locally, tlie ideal $(A) is generated by n - k independent 1-forms wk+l, . . . , wn. In fact, around any point p E M we can choose a cooi-diiiate system (x, U) so that

Then d ~ ' ( ~ ) A . . A d ~ ~ ( ~ ) is non-zero on AP.

By coniiiiuity, tlie same is true for q sufficiently close to p, which by Corol- lary 4 implies that dxl (q), . . . , dxk are linearly independent in A,. There- fore, tliere are Cm functions fa such that

k

dx"(q) = E fa (q) dxB(q) reskkkd lo A, a = k + l , ..., n. ,g= 1

We can therefore 1ei k

14. PROPOSITION (THE FROBENIUS INTEGRABILITY TMEOREM; SECOND VERSlON). A distributioi~ A on M is iiitegrable if and on1y if d($(A)) c $(A).

PXOOI;. Locally we can dioose 1-forms w', . . . , w" which span Mq* for each q sudi that ok" , . . . ,m" generate 4 (A). Let Xl , . . . , X, be the vector fields with

Then XI , . . . , Xk span A. So A is integrable if and oiily if tliere are functions C: with

k

Now doa (Xi , Xj) = Xi (m' (Xj)) - Xj (w' (X;)) - wa ([Xf, XJ 1).

For 1 5 i, j < k and a > k , the first two terms o n the right vanish. So dwff(Xi, Xj) =; O if and only if wff([Xi, XjJ) =: O . But each wlY([Xj, Xj]) = O i f and only if each [Xi, Xj] belongs to A (Le., if A is integrahle), whi1e each dwa(Xi, X,) = O if and only if dw' E 4(A). $+

Notice that siiice tlie wi A W J (i j) spaii R 2 (Mq) for eacli q, we can always wri te

= E 0; A w J for certain forrns 0;. j

If a > k , and io, jo 5 k are distinct, we have

so we can write the condition d(4(A)) c 4(A) as

dw' = E 0; A o B .

0 1 c e we have iiiti-oduced a coordinate sysiem (A+, U) such that the slices

are integral sul~maiiifolds of A, tlie forms dxk+', . . . , dx" are a hasis for 4(A), SO wk+l , . . . , wfl must lrie linear combinatioiis d tliem. i47e therefore llave the fol1owing.

15. COROLLARY. If o'+', . . . ,on are linearIy independent 1-forms in a neighhorliood of p E M, then there are 1-forms 19; (a, /3 > k) with

if and oiily if t here are functions f g , g B (a, B > k) with

Nthough Theorem 13 warms the heart of many an invariant lovet, the cases k > I will hardly ever he used (a very significant exception occurs in the last chapter of Volume V). Prohlem 18 @ves another invariant definition of dw, using induction on the degree of o, which is much simp1er. T h e reader may refiect on the difficulties which would be involved in using the definition of Theorem 13 to prove the following important property of d :

16. PROPOSITION. If f : M + N is Cm and o is a k-form on N, then

f * ( d o ) = d ( f *o).

PH001;. For p E M, let ( x , U ) he a coordinate system around f ( p ) . )Ve can assume

= g d x " A . . . A dx'k.

We wil1 use induction on k. For k = O we Iiave, tracing through some definitions,

f * ( d g ) ( X ) = & ( f * X ) = [ f * X l ( g ) = X ( g o f ) = d ( g 0 f ) ( X )

(and, of course, f * g is to he in te r~re ted as g o f ) . Assuming the formula for k - 1, we have

d ( f 'o) = d (( f * g dxi l A . - A dxh-l) A f *dxik ) = d ( f * ( g d x i l A ... A dx ih1) ) A f*dxik + O

since d f *dxi' = d d ( x i k o f ) = O

= f * ( d ( g dxil A . . . A dx'k-1)) A f *dxlL

by t1ie inductive liyposthesis

= f ' ( d g A dxil A . . . A dxk-1) A f *dx"

f * ( d g A dxil A . . . A dxih-] A dxik)

= f * ( d o ) . *t*

One propert)' of d qualifies, by the criterion of the previous chapter, as a hasic theorem of differential geometry The relation d 2 = O is just an elegant way of stating that mixed partial derivatives are equa1. There is another set of terminology for stating the same thing. A form o is called closed if d o = O and exact if o = dq for some form q. (The terminology "exact" is c1assical- djffei-ential forms used to he called simply "differentjals"; a dxerential was then called "exact" if it actua11y was the differential ofsomething. The term "closed" js hased on aii analogy with chains, which will he discussed in the next chapter.) Since d 2 = 0, every exact form is closed. In other words, d o = O js a necessary condition for soIving o = dq. If w is a 1-form

then the condition dw = O, i.e.,

is necessary for solving o = df, i.e.,

a f -- axi - mi'

Now we know from TIxeorem 6-1 that these conditions are aIso sufficient. For 2-forms the situation is more complicated, however. 1f o is a 2-form on IR3,

then o = d ( P d x + Q d y + R d z )

if and only if

The necessary corzdition, d o = O, is

In general, we are dealing with a rather strange collection of part id dRerentia1 equations (carefully selected so that we can get integrahility conditions). It turns out thai these necessary conditions are also sufficient: if o is closed, theii it is exact. Like our results ahout solutions to differential equations, this result is true only locally. T h e reasons for restricting ourselves to local results are iiow somewhat different, however. Consider the case of a closed 1-form o on R2:

a f ag o = f dx + g dy, with - = -

a y ax We know how to find a function or on al/ of R2 with o = da, namely

O n the otller lland, the situation is ver\; different if o is defined only o n R2 -(O) . Recall that if L c R2 is [O,w) x (O), then

defined iil Chapter 2, is Cm; in fact,

is the inverse of the map

(u, b) H (u cos b ,a sin b),

whose derivative at (u, b) has determinant equal to u # O. By deleting a dXerent ray L1 we can define a different function B1. Then I9i = 8 in the region Al and Oi = S+2x in tlie region A2. Consequently d19 and dol agree on tlleir common

domain, so that togetlier they define a 1-form w on IR2 - (O) . A c ~ m ~ u t a t i o n (Problem 20) shows that

The 1-form w js usually denoted by do, but this is an abuse of notation, since w = d8 only on IR2-L. In fact, w is not d f foray C' function f : IR2-(O) + IR. Indeed, if o = df, then

so d ( f - 8 ) = O on IR2 - L, which implies that a f / a r = aO/ax and 8 f / a y = ;iO/aj) aiid hence f = 8+ constant on IR2 - L, wliicli is impossible. Nevertheless, dw = O [the two relations

clearly imply that tliis is so]. So w is closed, but not exact. (It is still exact in a neigliborliood of any point of IR2 - (O).)

Clearly w is also not exact jn any small region contajning O. This example sliows that i t is tlie shape of the region, rather tliaii its sizc, that determines whether os not a closed form js necessarily exact.

A mariifold M is callcd (smoothly) contractible to a point po E M if tliere is a Cm ftinction

H : M x [O, 11 + M

such tliat

For example, IR" is smoothly contractible to O E IR"; we can define

H : IR" X [O, 11 + IRn

bl1oi.e gerierally, U C IR" is coiitractil~le to po E U if U lias the property that

p E U implies po + I ( p - po) E U for O 5 I 5 I (such a region U is called star-shaped with respect to po).

Of course, niaiiy other regions are also contractible to a point. If we tliink

of [O, 1 1 as represeriting time, tlieii foi- each time I we Iiave a map p I+ H ( p , I ) of M into itself; ai time 1 this is just t1ie identity map, and at time O it is the Collstari t map.

l e will show ihat jf M is sinoothly contractible to a point, then every closed forrn o11 M is exact. (By tlie way, this result and our investigation of the form d0 pi-ove the iiituitively obvious fact that IR2 - {O) is l i o ~ contractjble to a point; the same result Iiolds for R" - {O), but we will not be in a position to prove this uiitil tlie iiext cliapter.) The trick in proving our result is to analyze M x [O, 11 (for ariy manifold M), and pay liai-dly any attention at al1 to H.

For I E [O, 1 1 we define

il : M + M x [O, 11

147e clairn that jf w is a foi-m oii M x [O, I ] ~ 7 i t l i do = O, tlieii

i i*w -io*w is exact;

we wil1 see later (and you may try to convince yourself right now) that the t1ieorem follows trivially from this.

Consider first a 1-form w o n M x [O, I ] . We will begin by working in a coordinate system on M x [O, I ] . There is an obvious function I o n M x [O, 11 (namely, the projection ~ s on the second coordinate), and if (x , U) is a coordinate system on M, while ~ S M is the projection o n M , then

is a coordinate system on U x [O, 11. We will denote x i o B M by X z , for conve- nience. I t is easy to check (os should he) tllat

where mi( . , a ) denotes thc function p H (p, a ) .

Now for w = wi d ~ ' + f dr we have

n n ami - -a

dw = [tesms not involving dr] - - d.Zi A dr + - d r z A dr. i=l

a r i=I

azz

So d o = O imrrilies that 1 ami a f

- = - al a s '

I f w e define g : M + R by

then

?f * ( p ) =/ $ P , I ) df. axi o

Equations (1) and (2) show that

* il o - io*w = dg.

Now although we seem to be using a coordinate system, the function f, and l~eilce g aiso, is really independent of the coordinate system. Notice that for the tangent space of M x [O, I] we have

(4 ( M x [O, Il)(p,r) = ker x, $ ker XM*.

ker rr,

If a vector space V js a direct sum V - VI $ V2 of two subspaces, then any o E R1(V) can be written

o = w 1 + w 2

where

Applyjng this to tlle decomposjtion (*), we write the I-form o o n M x [O, I] as o] + 0 2 ; there is then a unique f with o2 = f df.

In general, for a k-form o, it is easy to see (Problem 22) that we can write w uniquely as

o = o1 + (df A 7)

where o] (VI , . . . , vk) = O jf some vi E ker x ~ , , and 7 is a (k - 1)-form with the

analogous property. Define a (k - 1)-form Iw on M as follows: r 1

1% claim tliat d o = O implies tliat i i * o - io*o = d ( I o ) . Actually it is easier to find a formula for i l * o - io*o that holds even when d o # O.

17. THEOREM. For any k-form o on M x [O, 11 we have ' * l ] w - io*w = d ( I q ) + l ( d w ) .

PROOE Since lo is already iiivariantly defiiied, we can just as we11 work in a cooidi~iate system (3' , . . . , T", 1 ) . The operator I is clearly linear, so we just liavc to consjder two cases.

( 1 ) o = f d2'1 A - . . A ~ T ~ R = f d2'. Tlien

i t js easy to see that

Sjilce lo = O, ihis proves the resuli in this case. * (2 ) U = j 'dr A f/ , f i l A a a A dx*-1 f dr A dx'. Theii j i*o = io o = 0,

Now

and

18. COROLLARY. If M is smoothly contractible to a point po E M, then every closed form o on M is exact.

PKOOF. We are given H : M x [O, I] + M with

H(P> 1) = p for al1 p E M.

H(p,O) = Po

Thus

H o il : M + M is the identity

H o i o : M + M is the constantmappo.

Bui

o - O = i lS(H*o) - io*(H*o)

= d( l (H*o)) by the Theorem. +:+

Corollary 18 is called the Poincaré Lemma by most geometers, while d 2 = O is called the Poiiicaré Lemma by some (I don't evexl kiiow whe tlier Poincaré had aiiything to do with it.) In the case of a star-shaped open subset U of Rnl where MT have an explicjt foi-mula for- H, we can fiizd (Problem 23) an explicit formula for I (H*o) , for eveiy form o on U. Since the new form is given hy an integral, we can solve tlie system of partial differeiiiial equations o = dq explicitly in terms of integi-als. There are classical tlieorems about vector fields in R3 whicli can l ~ e derjved fi-om thc Poiricaré Lemma and its converse (Prohlem 27), and orjginally d was introduced in order to obtain a uniform generalization of al1 these results. Everi though tlie Poincaré Lemma aiid its converse f i t very nice1y into our pattern for basic thcorems about differential geometry, it has always heen somethiiig of a mystery to me just why d turns out to he so important. A11 aiiswer io tliis questiori is provided by a thcorem of Palais, Nutulril O@utiotis 011 Dijil-e?lfiaI Forms, Trans. Amer. h4ath. Soc. 92 (1 959), 125-141. Suppose we Iiavc any opei-ator D from k-forins to I-forms, such that tlie foUowing diagram

commutes for evcry Cm map f: M + N [it actually suffices to assume that the diagram commutes only for diffeomorphisms f J.

f* k-forms on M - k-forms on N

PaIais' tlieorem says that, with few exceptions, D = O. Roughly, these excep- tiona1 cases are tlie followjng. If k = f, then D can be a multiple of the identity map, but ~iothing else. If I = k + 1, then D can only he some multiplc of d. (As a corollary, d2 = O, since d2 makes the abow diagram commute!) There is only one other case ivhere a non-zero D exists-when k is the dimensjon of M and f = 0. In ihis case, D can be a multiple of "integrationn, which we discuss in the next chapter.

(d) Formula (c) can be used to give a definition of o] A o 2 by induction on k + I (which works for vector spaces over any field): If A is defined for forms of degree adding up to < k + E , we define

Show that with this definition o1 A o 2 is skew-symmetric (it is only necessary to check that interchanging vl and v2 changes the sign of the right side). (e) Prove by induction that A is bihnear and that wl A 0 2 = (- 1)"w2 A ol.

( f ) If X is a vector field on M and o a k-form on M we define a (k - 1)-form x ~ o b y

( X J o ) ( p ) = X ( p ) J W ( P ) .

Show that if wl is a k-form, then

X J ( W ] A w ~ ) = ( X J W ] ) A W + ( - I ) ~ w , A ( X J o 2 ) .

5. Show that n funcrions 5,. . . , S, : M + R form a coordinate system in a iieighborhood of p E M if aiid only if dfi A - A dS, ( p ) # O.

6. An element o E R k ( v ) is called decomposable if o = #] A - - A#A for some #i E V* = Q 1 ( v ) .

(a) 1f dim V 5 3, then every w E Q 2 ( v ) is decomposable. (b) If #;, i = 1 , . . . , 4 are independent, then w = (#l A $2) + (#3 A #4) is not decomposable. Hinl: h o k at o A o.

7. For any o E Q k ( v ) , we define the annihiiator of o to be

Ann(o) =: (# E V* : # A o =: 0) .

(a) Show that dim Ann(o) 5 k ,

aiid tliat equality 11olds if and only if o is decomposable. (b) Every subspace of V* is Ann(o) for some decompasable o, which is unique up to a multiplicative constant. (c) If o, and w;! are decomposable, then An?i(wl) c Ann(w2) if and only if 0 2 = o] A q for some v. (d) lf are decomposable, then Ann(oi) í l Ann(w2) = ( O ) if and only if wl A o 2 # O. In this case,

Ann(oi) + Ann(w2) =: Ann(ol A w2) .

(e) 1 f V Iias dimensioii n , tlien any w E Qn-' ( V ) is decomposable. (f) Sji~ce vi E V can be regarded as elements of V**, we can consider vl A - . - A

ve E R k ( v * ) . Reformulate parts (a)-(d) in terms of tliis A product.

8. (a) Let o E f i 2 ( V ) . Show that there is a basis #l,. . . ,#, of V * such that

choose #] invol\?ing $ 1 , $3,. . . , .Srn and $2 involving Ilrz,. . . , $, so that

where or does not involve $ 1 os (b) Show that the r-fold wedge product o A . A o is non-zero and decomposable, and that the (Y + 1)-fold wedge product is O. Thus r is well-determined; it is called the rank of o. (c) If o = xicj u i j q i A $ j , show that the iank of o is the rank of the ma- ~h ( u i j ) .

n 9 . If v i , . . . , v, is a hasis for V and wi = xj,, ajivj, show that

10. Let A = (ui,) he an ri xri matrix. Let 1 ( p 5 n be fixed, and let q = n - p . For H = 11, < < hp and K = k i < < kq , let

(a) If V I , . . . , ün is a basis of V and

show that

(b) Let H' = (1, . . . , 1 1 1 - H (arranged in increasing order). Show that

K # H' V R VK =

~ H , N I vi A . . A vn K = H',

wliere e R , ~ l is the sign of the ~ermutation

(c) Prove "Laplace's expansion"

det A = e H , n , B H C H 1

11. (Cartan's Lemma) Let # i , . . . ,#k E V* be independent aild suppose tliat $ i , . . . , $k E V * satisfy

Then k

12. In addition to forms, we can consider sections of hundles constructed fi-om T M usirig SI aild otlier operations. For example, if ,$ = n : E + 3 js a vector bundle, we can consider f ik (t*), the bundle whose fibre at p is f ik ([n-' ( p ) ] ' ) . Siiice we can regard

a - as an element of (Mp)**, axl

aiiy section of Q n ( T * M ) can be writtei~ IocalIy as

(a) Show tliat if

thcn

This shows that sections of Rn ( T * M ) are the geometric objects corresponding to the (even) relative scalars of weight - 1 in Prohlem 4-10.

@) Let Tktml ( v ) denote the vector space of all multilinear functions

V x ... x V x V * x ... x V * + R m ( V ) . -- k rirncs I times

Show that sections of T ~ ~ " ' ( T M ) corres~ond to (even) relative tensors of type

(:) and weight 1. (Notice that if vi , . . . , u. is a basis for V , then elements of 51" ( V ) can be represented by real numbers [times the element u* A . a A u*,] .) (c) If 7/tm1(v) is defined similarly, except that Rn ' (V) is replaced by Rm(V*) ,

k show tliat sections of 5jill ( T M ) correspond to (even) relative tensors of type (,) and weigh t - 1. (d) Sliow tliat the covariant relative teiisor of type (:) and weight I defined in Problem 4-10, with components E'' ...'?I, corresponds to the map

v * x x v* + R " ( V ) - n times

given by ( $ 1 , . . . ,#,) t+ #, A A #,. Interpret the relative tensor with com- ponen ts Eil ..,ir, similarly. (e) Suppose R " ; w ( V ) denotes a11 functions q : V x x V + R which are of the form

q ( u ~ , ~ . ~ , u n ) = [ o ( ~ ~ , ~ - - , ~ n ) l ~ w an integer

for some o E Q n ( V ) . Let T ,k [" 'w l (~ ) be defiiied like 7/kin1, except that R R ( V ) is ,k[n;w] replaced by Rn'W(V) . Sliow tliat sections of J , ( T M ) correspond to (even)

relative tensors of type (f) and weight w. Similarly for 7,il:wl. (f) For tliose who kiiow about tensor products V @ W and exterior algebras hk ( V ) , these resulta can al1 be restated. We can identif~ qk ( V ) with

k times I times

Siiice R t " ( v ) = Am ( V * ) = [Am ( V ) ] * , we can identify

( v ) with V* @ @' V @ An'(V*).

Consider, more generally

Noting that An(V) @ - @ An(V) is always 1-dimensional, show that sections of -k[n;wI k

I (TM) and T~~ ; ,~ (TM) correspond to (even) relative tensors of type ( / ) and weigl~t w and -w , respectively

13. (a) If V has dimension n and A : V $ V is a linear transformation, then the map A * : Qn(V) + Qn(V) must be multiplication by some constant c. Sliow that c = det A. (Tbis may be used as a definition of det A.) (b) Conclude that det A 3 = (det A)(det 3 ) .

14. Recall that the characteristjc polynomial of A : V + V is

x ( I ) = det(h1 - A)

= In - (trace A)A"-' + - + ( - 1)" det A

= hn - cl hn-' + + . + ( - 1 ) " ~ ~ .

(a) Show tliat ck = trace of A*: Qk(v ) + Qk(v). (b) Conclude that ck (AB) = ck (BA). (c) Let s~'"'. be as defined in Pmhlem 4-5(xiii). If A: V + V has a ma- ... lk

trix (u!) (with respect to some hasis), sliow that

Thus, if S is as deíined on page 130, and A is a tensor of type (i) , then the function p H ck(A(p)) can he defined as a (2k)-fold contraction of

k times

15. Let P(Xij) he a polynomial in n2 variables. For every n x n matrix A = (u,) we then have a numher P(uij). Cal1 P invariant if P(A) = P(BAB-') for a11 A and al1 invertihle B. This prohlem outlines a proof that any invariant P is a polyi~omial in the polynomials ci , . . . , c, defined in Prohlem 14. We will

need the algebraic result that any symmetric poIynomial Q(yi , . . . , y,) in the n variables yl, . . . ,y,, caii be written as a polynomial in 01,. . .,o,,, where oi is the ih elementary symmetric polynomial of y,, . . . ,y,. Recall that the q can he defined by the equation

Thus, they are the coefficients, up to sign, ofthe polynomial with roots yl , . . . , y,, Since the eigenvalues Al,. . . , h, of a matrix A are, by definition, the roots of the polynomial ~ ( h ) , it fol1ows that

We M ~ I first consider matrices A over the complex numbers C (the coefficients of P may also be complex).

(a) Define Q ( y l , . . . , y , ) to he P(A) where A is the diagonal matrix

Then there is a polynomial H such that

The polyiiomial 8 has real coeficients if P does. (b) P(A) = H(ci (A), . . . , c, (A)) for al1 diagonalizab1e A. (c) The discriminant D(A) is defined as n i f j ( L i - A,)', where hi are the eigenvalues of A , Show that D(A) can be written as a polynomial in the entries of A. (d) Sliow that P(A) = K(cl (A), . . . , c, (A)) whenever D (A) # O. ConcIude, by continuity, that the equation holds for a11 matrices A over C. (This last conclu- sion folIows even if C is replaced by some other field, since the set where D # O is Zaris ki-dense; this is " the principal of irrelevance of algebraic inequalities", compare pg. V 375.)

Now suppose that the coefficients of P are real and that P(A) = ~ ( 3 ~ 3 - ' ) for al1 real A and real invertible B. (e) The same equation holds for complex A and complex invertible B. (Regard the equation as n2 polynomial equations in the uij and bij.)

16. (a) Let vi, . . . , v, be a basis for Vy and let WI, . . . , wk E V be given by

For o E a k ( v ) show that

where al is the determinant ofthe k x k submatrix of (a,) obtained by selecting rows i],.. . ,ik. (b) Generalize Theorem 7 and CoroIIaqr 8 to k-forms. (c) Check directly from (b) that the definition of d does not depend on the coordinate system.

17. Sliow tliat d ( x i , % j dxi A dx j ) = O if and only if

18. In PI-oblem 5-1 4 we defined Lx A for any tensor field A .

(a) Show that if o is a k-form, then so is Lx o. (b) Show that

(c) Using 5-14(e), show that

(d) Deduce the foUowing two expressions:

(e) Show that X J dw = L x w - ~ ( X J W ) ,

(This may be used to give an inductive definjtjon of d.) (f) Using (e), show that d ( L x w ) = Lx(dw) .

19. Let uij be n2 fuiictions on Rn with oij = uji. Show that in order for there to b e fuilctions ul , . . . , u, in a neighborhood of any point in R" with

it is necessary and sufficient that

a2uij a2uik azul, a2ulk for al1 i, j , k, l . - - - a s k a x l axi a x f axkaxi axjaxi

Hini: First set up partial dxerential equations for tlie finctions hk = auJ /axk - a u k / a ~ j , and use Theorem 6-1.

20. C o m ~ u t e that

(At niost places 6 = arctan y / x [ + a constant] .)

21. (a) If o is a 1-form f dx on [O, I] with f ( 0 ) = f (1), show that there is a unique number h such tliat w -h d x = dg for some fuiiction g with g(0) = g ( I ) . Mni: Iiitegrate the equation w - h d x = dg on [O, 11 to find h. (b) Let i : S' + R2 - ( O } be the inclusion, and let o' = i ' (d0). If C : [O, 11 + S' is

c ( x ) = (COS 2x A-, sin S x x ) ,

show that C* (u') c 2x dx-.

(c) If w is a closed 1-foi-m on Si sliow tliat there is a unique number h such that w - 10' is exact.

22. (a) Show that every w E R ~ ( V ~ $ V2) can be written as a surn of forms ol A w;, where o1 has degree oc and 0 2 has degree B = k - oc and

(b) If dim V2 = 1, and O # A E V2*, then o can be witten uniquely as o1 + (w A A), where wl is a k-form and w;, is a (k - 1)-form such that

23. Let U c IR" be an open set star-shaped with respect to 0, and define H : U x [O, 11 + U by H(p , 1) = fp, If

on U, show that

24. (a) Let U c IR2 be a bouiided open set such that IR2 - U is connected. Show that U is ditfeomorphic to IR2, and hence smoothly contractible to a point. (The converse js proved in Problem 8-9.) Hinf: Obtain U as an increasing union of sets, the kth set being a finite union of squares containing the set of points in U whose distance from boundary U is 5 I/k.

(b) Find a bounded open set U c IR3 such that IR3 - U is connected, but U is not contractible to a point.

25. Let U c R" be an open set star-shaped with respect to O. 1s U homeomorphic to R"? (It would certainly appear so, hut the "obvious" proof does not work, since the length of rays from O to the boundary of the set could vary discontinuously.)

26. Let ( , ) be the usual inner product on Rn,

(a) If v i , . . . , U n - ] E Rn, show that there is a unique vector vi x + . . x v,-1 E Rn with

@) Show that x . . x E R n - l ( R " ) , and express it in terms of the e*i , using the expansion of a matriu by minors. (c) For R3 show that

3 2 1 2 2 1 v x w = ( v 2 w 3 - v w , v 3 w 1 - v 1 w 3 , v W - u W ) .

(First find al1 ei x ej .)

27. (a) If f : R" + R, define a vector field grad f , tlie gradient of f' on R" by

" af a n

gradf = E G . - a = E D j f . -

i- i axi

i- 1 a x i

Introducjng the formal symhoIism

we can write grad f = Vf. If (grad f ) (p) = w p , show that

wl~ere D, f (p) denotes tlle direciional del-ivaiive in the direction u at p (or simply v,( f), if we regard vp E RP). Conclude that Vf (p) is the direction in which f is changing fasiest at p. (h) If X = EL1 a'a/axi is a vector field on W", we define the divergente of X as

n aa!

div X = - axi

(Symbolically, we can wriie div X = (V, X).) We also define, for n = 3,

Define forms

S how iliat

c/f = Wgrad f

d(0x) = rlcurl x c l ( v ) = (div X) dx A dy A dz.

(c) Conclude that

curl grad f = O

div curl X = 0.

(d) If X is a vector field on a star-shaped opeii set U c Rn and curl X = 0, ilieii X = grad J' for- some fuiiciioii f : U + R. Similarly, if div X = O, iheii X = curl Y for some vector field Y on U.

CHAPTER 8

INTEGRATION

T he basic concept of this chapter generalizes Iine and surface integraIs, ~tllich first arose fiom very pIlysical considerations. Suppose, for example,

iliat c : [O, 11 +- R2 is a curve and o = f dx + g dy is a 1-form on IR2 (WIICIF f , g : R2 + R, and x and y denote tlle coordinate functions on R2). If we choose a pai-tition O = to < < r, = 1 of [O, 11, then we can divide the curve c into >, pieces, tlie ith piece going fiom c(li-1) to When the difirences ti - ri-1 ai-e smaII, eacll sucll piece is approximateIy a straight segrnent, witll

Iioiizoiital pi-ojection C' (li) - C' (li-1) and vertical projection c2(ti) - ~ ~ ( l ~ - ~ ) . \Ve can clioosc poiiits ~ ( 6 ~ ) on eacli piece by clioosing points Ci E Eh-1 , li]. For eacli parti tiori P aild each sucli cl-ioice 6 = ((1 , . . . , Ct1), consider ihe sum

If tIlesc sums appi-oac11 a limit as tlie "mesli" 11 PIJ of P approaclles O, that is, as tl-ie inaxin~um of ti - r i W l appi.oacl-ies 0, then ihe limit is denoted by

(TIlis is a con111Iicated limit. To be precise, if J J PJI = max{Ii - ri-l), then the 1

equaiion P

lim S (P ,C)= llPlI+O

means: for al1 E > O, there is a 6 > O such that for al1 partitions P with 11 PJ1 < S, we have

for al1 choices 6 for P.) The limii which we have just defined is called a "line integral"; it has a natural

plysical interpretation. If we consider a "force field" on R2, described by the vector field

then S ( P , 6 ) is tI-ie "work" inwlved in moving a unii mass along the curve c in tl-ie case wl-iere c is actually a straiglii Iine between ti-1 and ti and f and g are coilsiani along tIiese straigIit line segrnenis; tlie limit is ihe naiuraI definition of tl-ie work done in tl-ie general case. (In classical terminology, ihe differeniial -f dh- + g c/-v would be desci-ilxd as ihe work done by tlle force field on an "ii-i- liilitelv small" displacement wit11 components dx, cIJ); the integral is ihe "sum" of iliese infinitely small displacements.)

Rcfore worrying about how to compute tllis limit, consider the special case

1 1 2 2 In tllis case, c ( 1 ; ) - c ( t i - , ) = l j - l j - 1 , while c ( t i ) - c ( l i - l ) = O, SO

Tl-iese sums approach

lc f d x + g dy = f(x,yo) dx. 1- On the other hand, if

then C' ( l i ) - C' ( l i-1) = ( b - a)(tj - l i - l ) , SO

These sums approach

In general, for any curve c , we have, by the mean value theorem,

A soinewhat messy argumei~t (ProbIem 1) shows that these sums approach what it looks like they should approach, namely

PI-iysicistsY notation (or abuse thereof) makes it easy to remember this resuli. Tlie components c l , c2 of c are denoted simply by x and y [;.e., x denotes

x o c and y denotes y o c; tllis is indicated classically by saying "let x = x ( t ) ,

y = y ( t ) " ] . The above integral is then written

In preferente to this physical interpi-etation of "line integralsu, we can introduce a more geometrical interpretation. Recall that d c l d t ( 6 , ) denotes the

tangent vector of c at time ti. Then the sums

cIearly also approach

Consider the special case wl-iere c goes witli constan1 velocity on eacll ( t i , l , t i ) .

If we cIloose any ti E ( t i - l , t i ) , then

de Iength of - ( C i ) = tlie constant speed on ( I ~ - ~ , l i )

dl - - Iength of the segment from c ( l i W l ) to c( l i )

l j - ti-1 Y

( t i - t i - ] ) = Iength of segrnent from c ( / ; - i ) to c ( / ~ ) . dl

In this case,

is tIle leilgth of c , and tIle limit of sucli sums, for a general c , can be used as a definitioii of the length of c . The Iine integral

o = limit of the sums (*)

can be tliouglit of as the "leilgth" of cy wlien our ruler is cllanging continuously iil a way specified by o: Noiice tl-iat the restriction of w ( c ( t ) ) to the 1-dimeiisional subspace of I R 2 , ( , ) spaniled by dcldt is a constant times "signed le11gt1-i". The natural way to specify a contiiluously changing Iength along c is to specify a Iength oii its taiigeiit vectors; this is tIie modern couilterpart of tl-ie classical conception, whereby the curve c is divided into infinitely small parts, the infinitely smaIl piece at c ( / ), with compoiients d x , dy , having length f ( ~ ( 0 ) dx + g ( c ( / ) ) dl) .

Refore pushiilg this geomeirical inierpretation ioo far, we sIlouId note that tliere is no 1-form o on I R 2 such that

LO = Iength of c for al1 curves c

It is true ihat for a giveii oile-oiie curve c we can pi-oduce a form o which works for c; we choose o(c (1 ) ) E 52i(R2c(,)) so that

(choosing ihe heme1 of o arbitrarily), and tlieii exteiid o to IR2. But if c is

not one-one this may be impossible; for example, in the situation shown beloy

tliere is no element of Ri(W2,(,)) which has the value 1 on al1 three vectors. 111 general, giveii any o on JR2 wllich is everywliere iion-zero, t l ~ e subspaces A, = ker o ( p ) form a 1-dimensional distribution on Ik2; any curve contained i11 aii ii-itegral submanifold of A will have "length" O. Later we wilI see a way of cii.cumventing this dificulty, if we are intei-ested in obtaining the ordinary leilgth of a cunte. For ilie preseilt, we note that the sums (*), used to define this gcileralized "length", make sense even if c is a curve in a manifold M (where iliei-e is iio notioil of "leiigth"), aiid o is a 1 -form on M, so we can define Jc o as the limit of these sums.

One property of liiie iilregrals sliould be meiitioiied now, because it is ob- viouc wiih our 01-igiiial definition and mercly true for our new definitioii. If p ; [O, 11 + [O, 11 is a one-oi-ie iiicreasiilg funciion from [O, 11 onto [O, 11, then the cume c o p : [O, 11 + M is called a reparameterization of c-it has exactly the same image as c, but ti-ansverses i t at a dfiereiit rate. Every sum S ( P , 6 ) for c.

is clearly equal to a sum S(P', 6 ' ) for c 0 p , and con\tersely, so ii is clear froin oui first definition that foi- a cuive c: [O, i ] + JR2 we I-iave

("tlie integral of o ovcr c is iildepcndent of tlic parameterization"). Tliis is ilo loiigci- so cleai. wlieii we consider the sums (u) for a curve c : [O, 11 + M, iioi. is it cleai- eveii for a cunre c : [O, 11 + W2, but iii this case we can proceed right to tlie iiliegral tl-iese sums approacli, narnely

The 1-esult tlien folluws from a calculation: tlle subsiiiution I = p(u) @ves

= 1- [ f (e o p(u))(c o p)If(u) + g(c 0 p(u))(c o p12'(u)] du.

For a curve in Rn, and a 1-form o = E:=, mi dxi, there is a similar calcula- iion; for a general maiiifold M, we can iiitroduce a coordinate system for our calculations if c([O, 11) lies in one coordinate sysiem, or break c up into severa1 pieces otherwise. Mle are being a bit sloppy about al1 this because we are about to iilti-oduce yet a t1iii.d defiiiition, which will e\tentually become our formal choice. Consider oiice again the case of a 1-foi-m on IR2, wliere

Notice tIiat if 1 is the standard cooi-dinate system on R, tlien for tlie map c : [O, 11 + R2 we have

c*( f dx + g dy) = (f o c)c*(dx) + (g o c )c*(d~)

= (f o C) d(x o C) + (g o c) d (yoc )

= (f O c)cl' dr + (g o c)c2' dr,

so iliat formally wc just integrate c*( f dx + g dy); to be precise, we write c.*(fdx + g dj)) = h dr (in the unique possilJe way), and take the integral of h on [O, 11.

Everyiliing we Iiave said foi- cuwes c : [O, 11 + IR" could be generalized to fuiictions c: [O, 112 +- R". If x and y al-e tlle coordinate functions on IR2, let

For a paii. of pariitioi-is so < - . - < S,, aiid lo < . - - < E , of [O, 11, if we clioose

1. PROPOSJTION. Let c : [O, 11" + IRn be a one-one singular 11-cube witll det c' > O on [O, 11". Let w be the 11-form

w = f d x ' A . . . A ~ x " .

PROOF. By definition,

= / ( f o c ) J det c'J d x ' A A dx" by assumption [o,-l]l1

= / f by the change of variable formula. 4 3

c(I0,' P')

2. COROLLARY. Let p : [O, 11' + [O, 11' be one-one onto with det p' 2 0, let c be a singular k-cube in M and Iet UI be a k-form on M. Then

I - =Lpo. PROOF. We llave

J o = J ( C 0 p ) * o = J P*(c*o) C O P

[O, ' lk [O, ' jk = / c*(w) by the Pmposition, since p is onto

The map cop : [O, l lk + M is called a repararneterization of c if p : [O, 1 l k + [O, 11' is a Cm one-one onto map with det p* # O everywhere (so that p-' is also Cm); it is called orienta tion preserving or orientation reversing depending on whetller det p' > O or det p' < O everywhere. The corollary thus shows independence of parameterization, provided it is orientation preserving; an ori- entarion reversiiig repai-ameterization clearly changes the s i p of the integral. Notice that there ~ f o u l d be no such result ifwe tried to define the integral over c of a Cm fuilcrioil f : M + IR by the formula

For example, if c : [O, 11 + M then

1 f ( ~ ( 1 ) ) dl is generally # f ( c ( p ( 1 ) ) ) 0 1 o

o

From a formal point of view, differential forms are tlie things we integrate because they transform coi.rectly (i.e., in accordance witll Theorem 7-7, so that tlle change of variable formula will pop up); funcrions on a manifold cannot be integrated (we can integrate a function f on the manifold R' only because it gives us a form f dx l A A dxk).

Our definition of the integral of a k-form UI over a singular k-cube c can iinmediately IIC generalized. A k-chain is simply a formal (finite) sum of singular k-cubes multipIied by integers, e.g..

Tlle k-cllain 1c1 = 1 cl will also be deiioted siinply by c l . We add k-cllains, and multipIy them by integers, purely formally, e.g.,

Morcovcr, wc define the integral of over a k-chain c = xi aici in t11e obvious way :

The 1-easoil Toi. iilti.oduciilg k-chaiiis is that to evei-y k-chain c (wl~icli may bc just a singulai A--cube) we wisll to associate a (k - 1 )-cliain ac, ~}hich is called tl-ie boundary of c , aild which is sul~posed to be t11c sum of the various singular

(k - 1)-cubes around the boundary of each singular k-cube in c. In practice, it

is convenient to modify this idea. The boundary of í2, for exarnpk, 4 1 not be the sum of tlie four singuIar 1-cubes indicated below on the left, but the sum,

witIl the indicated coefficients, of tlie four singular 1-cubes shown on the right. (I\'otice tliat tliis will not cliange the integral of a 1 -form over a l2 . ) For each i witli 1 5 i 5 n we first define two singular (n - 1)-cubes ICgo) and II;,,) (the (i, O)-face and (i, 1)-face of í") as foIlows: If x E [O, 11"-' , then

I:,~)(X) = ~ " ( x ' , . . . , xi-i ,O,X i y . . . y ~ n - l ) 1 = (x , . . .y xi-l y o, 2,. . . y xn-l),

i - 1 i I;.,,)(x) = í " ( x l , . . . , x , 1, S , . a . ,xn-l) 1 i-l i = (S ,..., x , l , x , . . m , A-"-1)

The (i, a)-face of a singular n-cube c is defined by

Now we define n

FinaIly: the boundary of al1 n-cllain Ci aici is defined by

These definitions al1 make sense only for n >_ 1 . For the case of a O-cube e : [O, 11' + M, wliicli we will usually simply identify with the point P = c(O), we define ac to be tlie number 1 E IR, and for a O-chain xi aicj we define

Notice that for a 1-cube c : [O, 11 + M we have

Mre also Iiave, for a singular 2-cube e : [O, 112 + M'

From a picture it can be cl-iecked that this also Iiappens for a singular 3-cube, a good exercise because this involves figuring out just what tfle boundary of a 3-cube looks like. In general, we have:

3. PROPOSITION. If c is any n-chain in M, then a(&) = O. Briefly, a2 = 0.

PROOE Let i 5 j 5 i 7 - 1, and consider (I{,a))(j,B). For x E [O, 11"-~, we have, from the definition

(';,al )(jln<") = ':,al (x))

1 n = '(;,.)(x . Y xj-I, By xJ y . , , , r 2 )

i = ~ ~ ( x l ~ . . . , xi-l >CY,X ? . . . Y X J - ~ , by x j y . . . y x ' -~)

Similarly,

(';/+i ,a>)(i,a> = 'c+i .m(';:)(~)) 1 i-1 I = r;+, (A- , . . . y x ,ay x , . . . , x " -~ )

i-l I = rn(xIy.. . y x y CY, x * ? . . . y x J - l y p y ~ j y . . . , x " -~) .

Thus (':lct))( jla> = (';+I lg))(i,ct) for i 5 j 5 n - 1 . Jt follows easily for any singular 17-cube c ihat (C(~,~))(~,@) = ( ~ ( ~ + ~ , a > ) ( ~ , ~ ) for i 5 j 2 n - 1. NOMI

ri n-l

In this sum, ( j , ~ ) and ('( ,B))(~,~) occur witli opposite signs. Therefore al1 rei-nis cailcel in pairs, arld a(&) = O. Since tlle tlleorem is true for singular 71-cubes, it is clearly also irue for singular n-chains. 4%

Norice tliai for some 11-cl-iains c we I-iave not only a(ac) = O, but even ac = 0. For example, this is the case if c = c1 - c2, wliere c, and c2 are two 1-cubes

witli cl(0) = ~ ~ ( 0 ) and cj (1) = ~ ~ ( 1 ) . Jf c is just a singular 1-cube itself, then

ac = O precisely when c(0) = c(l), Le., when c is a "closed" curve. In general,

any k -chaii-i c is called closed if ac = 0. RecaII tl-iat a differei-itial form o with do = O is also caIIed "closed"; tliis

terminology has been purposely chosen to paralle1 the terminology for chains (on tlle otl-ier haild, a chairi of t I ~ e form ac is iiot desa-ibed, recipi.ocally, I q , tlie classicaI term of "exact", but is simply called "a boundary"). This paralle1 terminology \vas not cllosei~ mcrely because of tlie formal similarities between d and 8, expressed by tlie relatioiis d 2 = O and a 2 = O. The connection betweeii fo1.n-i~ and cliains goes muc11 deepcr tl-ian that. For example, we have seen that on IR2 - {O) tlier-e is a 1-form "de" wl-iich is closed but rlot exact. There is also a 1 -chaiii c ~ ~ I i i c h is closed but iiot a boundary, namely, a closed curve encircling

t I~e point O once. AIthough ii is irituitiveIy clear tIiat c is not the boundary of a 2-chain in IR2 - {O), the simplest proof uses the theorem which establishes the corinection between forms, chains, d , and a.

4. THEOREh4 (STOKES' THEOREM). If o is a (k - 1)-form on M and c is a k-chain in M, then

PROOF R4ost of tlie proof involves the special case where o is a (k - 1)-form oii IRk and c = rk . In tllis case, o is a sum of (k - 1)-forms of the type

and it suffices to prove the tlieorem for each of these. Mle now compute. First, a little ríotaiion translation shows that

k *(f dx' A . - . A ~ X ; ' A + ~ . A ~ X )

- 1,. /'dxl A * . . n d x l A - . . ~ d x ~

k

* ( f d x l A . . . n d x i A . . . A ~ A - ~ ) = (-])]+u J I ~ , " ) j = l a=0,1 [O, l]k-1

On the other hand,

d ( f d x - ' n m . .A 2 n m . . A c/xk)

By Fubiili's theorem and the fundamental theorem of caIculus we have

- - f ( x l , . . . ,O,. . . , x k ) ] d x l . . . dx t . . . clx k

Thus

For an ar l~i t raq sing~lar k-cube, chasiiig througl~ the defiilitions sliows tIiat

Therefore

The iheorein cIearly follows for k-cIiains also. $4

Notice that Stokes' Tlieorem not onIy uses tlie fundamental tIieorem of calculus, but actually becomes that theorem when c = 1' and o = f.

As an application of Stokes' Theorem, Mte show that the curve c : [O, 11 + IR2 - (0) defined by

althougI1 closed, is not ac2 for any 2-chain c2. Ifwe did have c = ac2, then we

But a siraightforward computation (wliicl-i will be good for the soul) shows that

[There is also a ilon-computational arguinenr, using the fact tbat "d6'" really is d6' for 6 ' : IR2 - ([O, cm) x (O)) + IR: We havr

and 8(1 -E) - @(E) + 21r as E + O.] Altliougl~ we used iliis calculation to s h o ~ t tIlat c is not a boundary, we couId

just as well llave used it to show that o = "cf8" is not exact. For, if we had o = df foi- some Cm fuilction f : IR2 - (O) + IR, then we would Iiaw

147e wei-e III-eviously able to give a simpler ai.gument to show that "de" is not exact, but Stokes' Theoi-em is the tooI \vIlicI1 ~till enable us to deal with forms o11 R" - (O). Foi- example, we will eventually obtain a 2-form w on IR3 - (O),

whicli is closed but not exact. For the moment we are keeping the origin of w a secret, but a sti-aightfonvard calculation shows that do = O. To prove tliat o is not exact we \vil1 want to integrate it over a 2-chain which "fills up" the 2-sphere S2 c IR3 - {O). Tliere are lois of ways of doing this, but they al1 turn out to give the same result. In fact, we first want to describe a way of integrating n-forms over ii-maiiifoIds. Tliis is possible orily wlien M is orientable; tlie reason will be clear from the riext result, whicli is basic for our definition.

5. THEOREM. Let M be aii 11-maiiifold witli an orientation p, aiid let ci, c2 ;

[O, 11" + M be two singular 17-cubes which can be extended to be difiomor- phisms iii a iieiglil~orliood of [O, 11". Assume tliat cl and c2 are botli on'enlalion preservilzg (with respect to the oiientation p on M, and tlie usual orientation on IR"). If w is an 11-form oii M sucli tliat

support o c ci ([O, 11") n c2([O, l]"),

tIieri

PROOF 14% waiit lo use Corollary 2, and write

TIic only problcm is tliat c2-' o cl is i ~ o t defined on al1 of [O, 11'' (it does satisfy det(cz""' o cl )' > O, since c.1 aiid ~2 are 13otli orientatioii preservirig). However, a glaiice at tlie proof of Corollary 2 will show tliat tlie result still follows, because of tlie fact tliat support w c cl ([O, 11") n c2 ([O, 11"). *:*

Tlie romiilon iiumber [ w, for singular ii-cubes c : [O, 11" + M with sup- J c

poi-t w c c([O, 11") aiid c 01-ieiitatioii 11i-eserving, wiI1 be derioted by

lf o Is aii al-l~itrary 11-form oii M, tlieii there is a cover 0 of M by open sets U. eacli coiitaiiied iii some c([O, l]"), wliei-e c is a sirigular ii-cube of tliis sort; if @ is a partitioii of uiiity subordinate to this cover, tlien

is defiiied for each 4 E @. We wish to define

14% wilI adopt tIiis definition only when o has compact support, in wliich case the sum is actually finite, since support w can intersect only finitely many of t11e sets { p : $ ( p ) # O), which form a locally finite collection. If we haw anothei. partition of unity Q (subordinate to a cover O'), tIlen

iliese sums are al1 finite, and the last sum can clearly also be written as

so tl-iat our definiiioii does not depeiid on the pai-tition. (1We really sl~ould denote tliis sum by

for tlle oi-ientation - p of M we clearly liave

Howevei; we usuaIly omit explicit mention of p.) Witli ininor modifications we can define iM o evcn if M is an i?-manifold-

witli-boundary. If M c IR" is aii 11-dimensional maiiifold-~4th-bouiidary and f : M + IR has compacr support, tlien

wllerc tl-ie 1-iglit haild side dellotes the oi.di1-iai-y ii~iegi-al. TIiis is a simpIe conse- querlce of Proposition 1 . Likewise, if f : M" + N" is a diffeomorphism onto: aiid o is aii 17-foi-m witli compact support or-i N, tlieii

r l o if f is orientation presewing

- ..- ii f is orientation reversing.

AItIlough n-forms can be integrated only over orientable manifolds, there is a way of discussing integratioil on non-orientable mai-iifolds. Suppose that o is a function on M such that for each p E M we have

4 ~ ) = Iqpl for some qp E Qn(Mp),

i.e., for any n vectors u],. . . , u, E Mp we have

Sud1 a furlction w is called a volume element-on each vector space it deter- milles a way of measuring n-dimeiisional volume (not signed vol ume). If (x, U) is a cooi.dii1ate system, then on U we can write

= f l dx ' ~ - . . ~ d x " l for f 2 O;

\ve cal1 o a Cm volun-ie element if f is Cm. One way of obtaining a vo1ume element is to begin witll an rr-form q and tllen define w(p) = Jq(p)i. Howeve~ not every volume elemei-it arises in this way-the form qp may not vary contii~uously with p. For exainple, consider the Mobius strip M, imbedded in W3. Siiice M, can be considered as a subspace of R3p, we can define

w(p)(up , wp) = area of parallelogram spanned by u and w.

lt is not Ilard to see thai w is a volume element; Iocally, w is of the form w = JqJ for an 11-form q. But tliis canilot Ile true on al1 of M, siilce there is no 11-form q on M which is everywhere non-zero.

T1-ieoi.em 7-7 Iias ail obviohs modification for volume elemeiits:

7-7'. THEOREh4. If J' ; M + iV is a Cm function I~etween rl-manifolds: (x, U) is a coordiiiate sysrem arourld p E M, aiid (y, V) a coordinate system arouiid q = f (p) E N , tlien for iion-llegative g: V + W we have

PROOF. Go tl-irougli tlie proof of Tlieorem 7-7, putting in absolute value signs in rhe right place. +f,

7-8'. COROLLARY. If (x, U) and ( y , V ) are two coordinate systems on M and

Idy' ~ . . . l i d y " J = bldxi A ~ ~ ~ A ~ X " ~ g , h 2 0

tlien

[Tliis coi-ollary sliows tliat volume elements are tlie geometric objects corresponding to tl-ie "odd scalar densities" defined in Problem 4-10.]

It is now an easy matter to iiitegrate a volume element o over any manifold. First we define

Tlien foi- an n-chain e : [O, 11" + M we define

Tlieoirm 7-7' sliows tliat Proposition 1 holds for a vol ume element o = f ldx ' A

A dx"] even if det e' is not 3 O. Thus Coi.ollary 2 liolds for volume elements eveii if det p' is iior 3 O. From this we conclude that Theorem 5 holds for volume elements o on aiiy manifold M, without assuming el, c2 orientatioii presen~ing (or even diat M is orientable). Coi~sequently we can define JM o for ai-iy voluiiie elcment o wit1-i compact suppor~.

Of course, when M is orientable these consider-ations are unnecessary. For, there is a nowliere zero 11-foi-m v on M, and consequently any volume element o can be written

= f Ivl, f 2 0.

1f we clloose an orientation p for M such that o ( v l , . . .,U,)) > O for vi,. . . ,U,

l~ositively oriented, tlien we can define

I~olume clements will be important later, but for tlie remainder of this chaptei. we are coiiccrned oilly ~ 4 t h integrating forms over oriented manifolds. In fact, our main resuli about iiltegi-als of forms over manifolds, an analogue of Stokes' Theor-em about the integral of forms over chains, does not work for volume elements.

Recall from Problem 3-16 that if M is a manifold-with-boundary, and p E

aM, then certain vectors v E Mp can be distinguished by the fact that for any coordinate system x : U + W" around p, the vector x+(v) E Wnf(pl points "out~~ards". 14% cal1 such vectors v E Mp "outward pointing". If M has an

01-ientation p , we define iIíe induced orientation ap for aM by the condition that [ u I , . . . , E ( 8 ~ ) ~ if and only if [w, vi,. . . , ~ ~ - 1 1 E pp for every outward poiiiting w E Mp. If p is the usual orientation of H", then for p = (a, O) E W n we have

Since (-en)p is aii outward poiiiting vector, tliis shows that the induced orieiitation on IRR-' x {O) = a W R is (- l ) n times t1le usual one, The reason for this choice is the follotiliilg. Let c be an 01-ieiitation preserving singuIar n-cube in (M, p) sucli thai aM n c([o, 11") = C ( ~ , ~ ) ( [ O , 11"-'). Then c(.,o): [O, 11"-' +

(aM, ap) is 01-ieilration pi-esenring for even n , and orientation reversing for odd n. If w is ai-i (n - 1)-foi-m on M whose support is contained in tlie interior

of tl-ie image of c (this interior contains points in the image of it follows ihat

But c(,,0) appears witll coefficient (- 1)" in ac. So

If it were not for this choice of ap we would have some unpleasant minus s i p s in the followiiig theorem.

6. THEOREM (STOKFS' THEOREM), If M is ail oriented n-dimensional manifold-with-boundary, aild aM is giveii the induced orieniation, and o is an (il - 1)-form on M with compact support, then

PROOF. Suppose first that there is an orientation preserving singular n-cube c

in M - aM suc11 rhat supporr o c interior of image c. TIien

= O since suppori o C interior of image c,

Suppose nexi that tlierc; is an orientation presenring singular n-cube c in M sucli tliat 8~ n c ([O, 11") = C(,,~)([O, 11"-'), and suppori o c interior of image c. Then once again

Jn geiieral: rIiere is an open cover O of M and a partition of unity @ sub- ordinaie io 13 such ihai for each $ E @ the form # . o is one of the two sorts already considered. \We have

Sii-ice o lias compacr suppori, tIiis is reaIIy a finite sum, and we conclude tliat

Therefore

One of ilie simplesr applicaiioris ofSrokes7 Theorein occurs wlien ilie orjeiiied iz-n~niiifold (M, p ) is coinpaci (so tliat every foim has compact support) and a& = 0. 111 iliis case, if is aiiy (u - 1)-form, ihen

Tlierefore we caii fii-id aii 11-fo1.m o on M wllich is t~of exact (eveii thougli i r iiiusr be closed, l~rcause al1 (12 + f )-firms on M are O), simply by findiiig aii o wi t11

Such a fui . i~~ o iil~vays exisis. Indecd we havc seen ihai il-iere is a fm-iii o sucl~ ihai foi- vi,. . . , u,, E M, we liave

If c. : [O, I]" + ( M , 11) is oi-ieii tatioii pi.esei\*iiig, trheri t l ~e foiam c*w o11 [O, 1 1 " is c.leaiiy

g dxi A - . A dx" for some g > O oil [O, I]",

so 1, w > 0. Ii follows that JM w > O. Tliere is: moreover, no need to choose a foi-m w with (*) holding eveiywhere-we can aIIow the > sign to be replaced by >_ . Thus Mte can even ob tain a non-exact 11-form on M which Iias support contained in a coordinate neighborhood.

This seemiiigly minor result already pi.oves a theorem: a compact oriented maiiifoId is not smoothly contractible to a point. As we have already empha- sized, it is tlie "shape" of M, rather than i ts "size", which de termines whether or not every closed form on M is exact. RoughIy speaking, we can obtaiii more informatioii aboui tlie shape of M by analyzing more cIoseIy the extent to ivhicli closed foorms are not necessarily exaci. In particular, we wouId now Iike to ask jusr Iiow many non-exact 17-foi-ms there are on a compact oriented 11-manifoId M. Naturally, if w is not exact, then tlie same is true for w + d q for any (n - 1)-form q , so we realIy want to consider w and w + d q as equivalent. Tliei-e is, of coui.se, a standard way of doiiig tliis, by coiisidering quotient spaces. 14% will apply tliis coiistruction not onIy to 11-forms, but to forms of any degree.

For eacli k, ihe collectioii z'(M) of a11 closed k-rorms on M is a vector space. Tlie space B'(M) OS al1 exact k-forms is a subspace (since d2 = O), so we can form tlie quotient vector space

this vecior space Hk (M) is cvlled iIie k-dimensional de Rham cohomoIogy vector space of M. [de Rlzam % T/zeore~n states tliat tliis vecior space is isomorphic to a ccrtain vectoi. space defiiied pureIy in ierms of the iopology of M (for any space M), called tile "k-dimensional coliomology gi-oup of M with real coefficients"; tlic noiatioii z', 3' is chosen to cori-espond to the notation used iii algebraic ropolog): wliere ihese groups are defined.]

An element of H k (M) is an equivalente class [m] of a closed k-form o, two cIosed k-foi.ms w1 aiid w2 beiiig equivaleiit if and onIy if their dserence is exact. Iii terms of these vector spaces, tlie Poiiicart Lemma says that H' (W") = O (the vecior space coiitaiiiing only O) if k > O, or more generally, H' (M) = O if M is contractible aiid k > 0.

To computc H0(A4) we iioie fiist iIiat BO(M) = O (there are no non-zero exacr O-hrms, since tliere are no non-zero (-1)-forms for them to be the dif- Serential 00. So H'(M) is iIie same as iIie vector space of al1 Cm functions .f : M + W ~ 4 t h df = O. If M is coiiiiected, the condition d f = O implies tliat / is constant, so H0( M) % R. (Iii general, tlie dimension of H'(M) is the iiuml~er of components of M.)

Aside from these trivial i-emarks, we 111-esently know only one other fact about H k ( M) -ir M is compact and oriented, tIien H n (M) Iias dimension >_ l . The furtlier study of H k ( M ) i.equires a careful look at spheres and Euclidean space.

O n S"-' C R" - {O) tliere is a naturaI choice of an (n - 1)-rorm a' with Js,l-l 0' > O: for VI)^, . . . , E S"-lp, we define

ClearIy iliis is > O if . . . , ( v , - ~ ) ~ is a positiveIy oriented basis. In fact, we defiiied ihe 01-ientation of S"-' iii precisely iliis way- tliis orientation is just tlie induced oiientation ~ t l i en S"-' is considered as the boundary of the unit ball { p E IW" : Ipl 5 1 ) ~ i i l i tlie usual oi-ieiiiaiion. Using the expansioii oT a rlctermiiiaiit by minors along tlie top row we see tIiat o' is tlie restriction to S"-1 of ilie form a on R" defiiied by

Tbe form a' on S"-' will now be used to find an (n - 1)-form on R" - {O) wliich is closed but not exact (thus sl~owiiig that H"-' (W" - (0)) # 0). Colisidei- the inap 1 . : IW" - {O) + S"-' defined by

ClearIy i0(p) = p if p E S"-'; otheiwise said, if i : S"-' + R" - {O) is tlie inclusion, tllen

r o i = identity of S"-'.

(Ii1 seneraI, if A c X aild 1.: X + A satisfies r ( a ) = a for a E A, tllen r ir; called a retraction of X onto A.)

Clearly, r*a' is closed:

Ho~t-vcr, i t is not exact, for ir r"a' - dq, then

11ui íve ki~ow that a' is not exact.

It is a worthwhile exercise to compute by bru te force tliat

* 1 x d y - y d x x d y - y d x forn=:2, r o = - - = d6'

x2 + y2 v 2

* I x d y ~ d z - y d x ~ d z + z d x ~ d y for n = 3, r o = (,Z + }'2 + $)3/2

1 = - [ x d y ~ d z - y d x ~ d z + z d x ~ d y ] .

u3

Since we ~ 7 i I l actually need to know r*ol in general, we evaluate it in anothei. way :

7. LEA4h4A. If o is the form on IR" defined by

-. o = E(- 1)'-lx' d x l A - A dxJ A A dx",

aiid o' is the restrictioli i*o of o to S"-', then

PROOF. Ai any poiii t p E IWn - (O), tlie tangent space IWP is spanned by pp aiid tlie vectors vp ili tlie tangent space oT tlie spliere S"-' (1 pl) of radius 1 pl. So ir sufices to check that botli sides of (*) give the same result when applied io ir - 1 vectors eacll of wliich is one of tl-iese two sorts. Now pp is the tangent vector of a curve y lyiiig aloiig ilie straight line tliiough O and-p; this curve is takcn to tlle single point r (p ) by r , so r+(pP) = O . O n the other hand,

So i t suffices to apply both sides of (*) to vectors in the tangent space of S"-' (1~1). Tlius (Pi-ol~lem 15), i t suffices to s l i w that for such vectors vp we liave

1

But tliis is almost obílious! since the [rector vp is t l ~e tailgent vector of a circle y lyins .... in S"-' ( lpl) , aiid the cuna r. o y líes in S"-' and goes I/lpl as far in the same time. *:*

8. COROLLARY (INTEGRATION IN "POLAR COORDINATES"). Ler f : 3 + IR, where

= { p E Ik" : 1pl 5 J ] ,

and define g ; S"-' + W b!.

Tlieii

~ f = ~ f d x ' ~ - - . ~ d x " = / Si,- t gorm

PROOF. Coiisider S"-' x [O, 1 1 and the two projectioiis

ni : S"-' x [O, I ] + S"-'

: S"--' X [O, 11 + [O, 1 1 .

Lei us use tlie abbreviation S"-'

a' A dt = ni*or A n2*di. a IT (y, U) is a coordiiiate system on S"-', \vi tli a corresponding coordinate system (J ' , t ) = ( y o T I , n2) on S"-' x [O, J ] , and ar = cr dy' A - - A dyn-' , theii clearly

o r h d r = & o n i d j l ~ . . . n d y " - ' ~ d t .

From rliis i t is easy to see tliat if we defiiie h : S"-' x [O, 1 1 + IR by

h ( p , u ) = un-' f ( u p),

theii = (- )"-l /zar A dt

SI!- 1 / / SI)- t x[o,IJ

NOM' í~re caii define a dfleonioi-pl?ism @ : 3 - { O ) + S"-' x ( O , J ] l y

Then

(- 1)"-' n - -

""+1 E ( X ' ) ~ d x l A m A ds" i=l

Hence

= ""-1 f . (- l)"-' Y"-1 d x l n - . A dx"

f dx l A . . . A dxn = (- I)'-' / $*(ha1 A d l ) 3-(01

= (- l)"--I J ha' h dt S"-' x ( 0 , l j

Stl- t

(This lasi siep requires some justificaiion, w11icIi sliould be supplied by the reader, since tlie rorms involved do not bave compact support on the mani- IoIds B - {O) aiid S"-' x (O, 1 ] where they are defined.) +:+

We are aboui i-eady io coinpuie H' ( M ) in a kw more cases. We are going to reduce our calcuIations to calculaiions within coordinate neighborhoods, which are sul~nianifolds or M, bui not compact. It is tllererore necessary to introduce ailotIier co11eciion of vector spaces, which are interesting in their own righi.

The de Rham cohomoIogy vector spaces with compact supports 13: (M) are

wliere Z: (M) is the vector space of closed k-forms with compact support, and B:(M) is tlie vecior space of al1 k-rorms dq where q is a (k - 1)-Torm with compact support. Of course, if M is compact, then H: (M) = Hk (M). Notice tliat ~ , k (M) is nol ihe same as tlie set of al1 exact k-forms with compact support. For example, on Rn, if f 2 O is a function with compact support, and f > O ai some point, then

o = f dx' A - . . ~ d x "

is exact (ever): closed form on Rn is) and 11as compact suppor t, but o is not dq for any form q with compact support. Indeed, if o = dq where q has compact support, tlien by Stokes' Theorem

Tliis example shows tliat HC(Rn) # 0, and a similar argument shows that if M is any orieniable maiiifold, ilien H;(M) # O. M7e are now going to sl~ow tl~at for any connected orientable manifold M we actually have

Tliis means ilia t if we clioose a fixed o wi tli JM o # O, then for any n-form o' witli compaci support there is a real number a sucli that w' -ao is exact. The number a can be described easily: if

then

ilie problem, of course, is sliowing ihat q exists. Notice that the assertion thai H:(M) R is equivalent to the assertion that

II

is an isomorpliism of H!(M) with R, i.e., to the assertion that a closed form o with compaci suppori is the dflerential of another form with compact support if JM o = O.

9. THEIOREM. If M is a coiinecied oi-ieniable ti-ma~iifold, then H," (M) i-- R.

PROOF. \Ve wi11 establish tlie theorem in three steps:

(1) The theorem is true for M =: W

(2) If the t11eorem is aue for (n - 1)-manifolds, in particular for S"-', then i t is true for W n .

(3) If t11e tlieorem is true ior IR", then it is true Tor any connected oriented 11-manifold.

S~efi 1. Le t w be a 1 -form on W wi th compact suppor t such that JR w = O. There is some fui~ction f (not iiecessarily with compact support) such that w = df. Since suppori o is conipact, df = O outside some interna1 [-N, N], so f is a

constan1 cl on (-m, -N) and a constant c2 on (N, m) . Moreover,

TIierefore ci = c2 - c aiid we Iiave

wIiere f - c Iias compaci suppori.

Skp 2. Let w = .f dxl A - A dx-" be an 11-hm witl~ compact support on W" such tliat IR,, w = O . For simplicity assume that support w c ( p E W" : Ipl < 1). Mre ki~ow r11ai tliere is an ( r t - 1)-form q on W n suc11 tliat w = dq. In fact, from Prol~lem 7-23, we tiave an expIicit formula for q,

Using tlie subsriru tion u = 1 pl l this becomes

Define g: S"-' + W by

0 i i tlie ser A = { p E Wn : 1 pl > 1) we I1a17e f = O, so on A we Iiave

h4oi~eover, 11y CoroIIary 8 hre Iiave Ioi ilie (11 - 1)-Iorm ga' on S"-',

= J.. o = o.

Tlius, Ily rlie I~yporliesis for S~cj i 2,

go' = d l Ior some (il - 2)-form 1 o11 S"-'.

I x t lr : R" + [O, I ] l ~ c aiiv CbO Iunctioli witli h = 1 011 A aiid h = O in a iirigliboi-Iiood of O. Tlieii jii-*A is a Cm form on W" and

the foi'm q - d(hlm*h) Iias compact support, since on A we have

Sley 3. Choose an n-form o such that JM o # O and o has compact support coiitaiiied iil an open set U c M, with U diffeomorphic to Rn. If o' is any otlier 12-Sorm with compact support, we want to show that there is a number c aiid a form q witli compact suppor t sucli tliat

Using a partiiion of unity, we can write

wliei-e eacli $;o' lias coinpact support contained iii some open set Ui C M with Ui diffeomorpliic to R". Jt obviously suffices to find ci and with $irn/ = t i a + dqil Sor eacli i . In otlier words, we can assume m' Iias support contained in some open V c M wliicll is difíeomorphic to Wn.

Usiiig tlie coiiiiectedness of M, ii is easy io see iliat there is a sequence of open sets

u = vi, ..., V, = if

diffcomoi-pliic to R", with I/i n # 0. Clioose Sorms w j witli support c

n and Jv. oi # O. Siiice we are assuiniiig the tlieorem for W" we haw I

W~ICIP al1 q j liave coiiipact support (C K). From tliis we clearly obtain the desired resuli. *3

The method used in tl-ie last step can be used to derive another resu1 t.

1 O. THEOREh4. IT M is any connected non-orientable n-manifold, then H ; ( M ) = O.

PROOF. Clioose an 11-Torm o ~ 4 t h compaci suppori contajned in an open set U difiomorphic to IR", such that Ju w # O (this integral makes sense, since U is orieiitable). It obviously suffices to sIlow that o = dq for some form q with compact support. Consider a sequence

u = v, , . . . , vr = v - 1 of coordinate systems (Vi, x i ) where cacIi xi o xi+l is orientation preserving.

Choose tIle forms mi in S1el3 so that, using the orientation of Vi which makes xi : 6 + IR" orieniaiion pi-esen~iiig, we llave iK iYii > O ; then also JVirl mi > O. Consequently, the numbers

c = w / o are positive.

It ~ O ~ I O W S that oi =: Cw + dq wliere c > 0.

Now if M is unorientable, tllere is such a sequei-ice where V, = VI but A-, 0x1-' is orieiitaiion rezielrnizg. Taking o' = -o, we llave

-w = cw + dq Tor c > O

*

(-c - ])o = dq Tor - c - 1 # O. +3

We can also compute H " ( M ) Toi- non-compact M

11. THEOREh4. If M is a connected non-compact n-manifold (orientable or not), ihen H n ( M ) = 0.

PROOF. Consider firsi an 11-form o witli support contained in a coordinate i~eigliboi-liood U wIiic1-i is diffeomorphic io IR". Sii-ice M is not compact, t11ere is an infiniie sequence

u =: U,,Uz,U3,U4 ,...

of sucli coordinate neighborhoods such that Ui n Ui+1 # 0, and sucli that the sequence is eventually in the complement of any compact sei.

Now clioose ir-forms mi witll compact support contained in Uj n Ui+i, such tliat Jui q # O. There are constants cf and forms with compact support c Ui sddi tliai

Since aiiy poiiit p E M is eventually in the complement of the UiYs, we have

where the riglit side makes sense since the Ui are eventually outside of any compact set.

Now i t can be sliown (ProbIem 20) that there is actually such a sequence U1, U2, U3,. . . whose union is al1 of M (repetitions are allowed, and Uj may intersect several Uj for j < i, but the sequence is still eventually outside of any compact set). Tlle cover 13 = (U) is tllen locally finite. Let ($u) be a partition of unity suboi.di~iare to O. If o is an 19-form oii M, then for each Ui we have seen that

$vio =: dqi wl~ere V i has support contained in Ui U Ui+1 U Ui+2 U - . - . Hence

SUMMARY OF RESULTS

(2) If M is a connected n-manifold, then

R ir M is orientable H ; ( M ) =

O ir M is non-orientable

e if M is compact

H n ( M ' - { ~ i f ~ i s n o t c o m p a c t .

\/Ve also know that Hn-' ( R n - ( 0 ) ) # 0, but we have not listed tllis result, since we will eventually improve it. In order to proceed furiher with our computations we nced to examine the behavior of the de Rham cohomology vector spaces under Cm maps f ; M + N . If o is a closed k-form on N, then f * w is also closed (d f *o = f * d o = O), so f * takes Z k ( N ) to Z k ( M ) . On the other hand, f * also takes B k ( N ) to B k ( M ) , since f * ( d q ) = d( f *q) . This shows that f * induces a map

zk ( N ) / B ~ ( N ) + z ~ ( M ) / B ~ ( M ) ,

also deno ted by f * : f*: H ~ ( N ) + H ~ ( M ) .

For example, consider the case k = 0. Ii N is connected, then H O ( N ) is just the coIIection of constant functions c : N + R. Then f " ( c ) = c o f is also a constani funciion. If M is connected, then f * : H O ( N ) + H ' ( M ) is just the identity map under tlie natural identificatioii of H O ( N ) and H O ( M ) with R . If M is disconnected, with components M,, a E A , then H O ( M ) is isomorphic to the direct sum

R where each W , % R ; U E A

ilie map f * takes c E W into the element of @ R . with a" component equal to c . V N is aIso disconilected, with components Ng, E B , then

takcs the elemeiit {cg) of @ g E B Rg to (41, where c; = cg when / ( M u ) c Ng.

A inore interesting case, and the only one we are pi-esently in a position to Iook at, is the map

f * : H n ( N ) + Hn (M)

when M and N are both compact connected oriented n-manifoIds. There is no natural way io make H n ( M ) isomorpl-iic to IR, so we reaIly want to compare

for o an 11-form on N. Clioose oiie wo wi th JN o 0 # O. Then there is some number a sucll tliai

r F

Siiice o H lM o is an isomorphism of H n ( M ) and R (and similarly for N) it folIows tliai for evay form o we Iiave

TIie ilumber a = deg f, w1iicIi depends onIy on f, is called tIle degree of J: If M and N are not compact, bu t f is propei- (the inverse image of aily compact set is compact), tlien we have a map

f * : H,"(N) + H,"(M)

and a number deg f, sucli that

h r al1 forms o on N with compact support. Until one sees the proof of the nexi iheorem, it is almost unbelievabIe that tlis number is always an inlegm.

12. THEOREM. Let f : M + N be a proper map between two connected oriented n-maiiifolds (M,P) and (N, u). Let q E N be a regular value of f. For each p E f -' (q), let

1 if f*p : Mp + N4 is orientation preserving signp f = ( (using the orientations pP for Mp and v4 for N4)

- 1 if is orientation reversing.

TIien d e g f = 1 sipp/ ( = 0 i f f - ' ( ~ ) = 0 ) .

~ € f - ' ( 4 >

PROOF. Notice first that regular values exist, by Sard's Theorem. Moreover, f -'(q) is finite, since i t is compact and consists of isolated points, so the sum above is a finite sum.

Let f -' (q) = ( p l , . . . , pk}. Choose coordiiiate systems (Ui, xi)' around pi such tliat all points in U j are regular values of S, and the Ui are disjoint. \Ve want to clioose a coordinate system ( V, y ) around q such that f -' ( V) = U1 U

. - U Uk. To do this, first choose a compact neighborliood W or q, and let

W' c M be the compact ser

w'= f - ' ( w ) - (u1 u . ' . u U k ) .

TIien f ( W') is a dosed set which does not contain q. We can therefore choose V C W - f ( W'). Th i s eiisures tliat f -' (V) C U1 U. -UUk. Finally, redefine Ui to be Ui n f -'(VI.

Now clioose w on N to be w = g dyi A A (13'" wliere g 2 O has compact support contained in V. TIieil support f *w c U1 u . - u Uk. SO

Sinc,e f is a diffeomorpliism li-om each Ui to V we have

S, f*w=S, w ir f is orieiitation preserving

=-S, w if f is orientation reversiiig.

Siiice f is orientatioii pi-esenriiig [or reversing] precisely wlien s ip , f = 1 lor -11 tllis proves the theorem. *t,

As an immediate application of tIie iheorem, we compute the degree of tlie "antipodal niap" A : S" + S" defined by A ( p ) = -p . ' We have already seeii that A is orientation presenring or reversing at aU points, depending on wliether rl is odd or even. Siiice A-' ( p ) consists orjust one point, we condude tha t

deg A = ( - ] ) " - l .

We can draw an interesting concIusion from this result, but we need to intso- duce ailotller imporiaiii concept first. Two fuiictions f, g : M + N between two Cm manifoIds are called (smoothly) homotopic ir there is a smooth function

tlle map H is calIed a (smooth) homotopy between f and g. Notice that M is smootlily contraciible to a poiiit po E M ifand only if the identity map of M is liomotopic to tlie coiistaiit map po. Recall tliat Tor every k-form o on M x [O, 11 we defiiied a (k - 1)-form ío on M such thai

ir*# - io*o -- d ( 1 o ) + 1( i lo ) .

M7e used tliis fact to sllow tliat al1 cIosed forms oii a smoot1~1y contraciible maií- ifold are exact. We can iiow prove a more general result.

1 3. THEOREM. If J; g : M + N are smoothly homotopic, then the maps

f*: H ' ( N ) + H ' ( M )

g* : H ' ( N ) + H ' ( M )

are equal, f * = g*.

PROOF. By assumprioii, ihere is a smootIi map H : M x [O, 11 + N with

f = : H o i O

g = H o i i .

Aiiy element of H' ( N ) is tlie equivaleiice class [o] o i some closed k-form o on N. Tlien

g*w - f * o = ( H o i i ) * o - ( H O iO)*o

=: i l * ( H * w ) - io*(H*w)

=: d(1H *o) + í ( d H * o )

= d ( í H * w ) + 0.

But tliis means that g * ( [ o ] ) = f * ( [ o ] ) . a:+

14. COROLLARY. Ir M and N are compact oriented n-manirolds and t11e maps f, g : M + N are homotopic, then deg f = deg g.

15. COROLLARY. T i 11 is even, then there does not exist a nowhere zero vector field on S".

PROOF \/Ve have aIready seen that the degree or the antipodal map A : S" + S" is ( - ] ) " - l . Since tlie identity map has degree 1 , A is iiot homotopic to the identity ror I I even. But if there is a nowhere zero vector field on S", then we can construct a I-iomotopy between A and the identity map as roIIows. For each p, there is a unique great semi-circle yp irom p to A ( p ) = - p whose iangeni vector at p is a muItipIe of X ( p ) . Define

For n odd we can expIicitly construct a nowhere zero vecior fieId on S". For p = (xi , . . . , xn+i ) E S" we define

this is perpendicular to p = ( x i , x2, . . . , x , + ~ ) , and therefoore in S",. (On S' this gives the standard picture.) The vector field on S" can then be used to give

a Iiomotopy between A and the identity map. For another applicaiion or Theorem 13, consider the retraction

r : R" - ( O ) + S"-' Y ( P ) = P / I P I . Ir i : S"-' + R" - ( O ) is tl-ie inclusion, tlien

r o i : S"-' + S"-' is the identity 1 or S"-'.

Tlle map

is, of course, not the identity, but i t is I-iomotopic to the identity; we can define ihe llomotopy H by

A 1-etrac tion with tllis property is called a deformation retraction. Whenever 1. is a deformation i-etractioil, the inaps ( r oi)* and (i o).)* are the identity. Thus, for tlle case of S"-' c R" - ( O ) , we have

and

r.* o i* = (i o r)* = identity of Hk(R" - {O))

i* o r.* = ( I . o i)* = identity of H k ( s " - ' ) .

So i* aiid i." ai-e inverses of eacll other. Thus

H k ( S " - ' ) ';. Hk(R" - ( O ) ) for al1 k.

Iii particular, we Iiave H"-'(R" - { O ) ) R . A generator of H"-' (R" - { O ) ) is tlle closed form r*aJ.

We are now going to compute H k (R" - ( O ) ) for al1 k . We need one further observa tion. The manifold

M x ( O ) c M x R'

is clearly a deformation retraction of M x R'. So H k ( M ) H k ( M x IR') for al1 I .

Now

~ ' I A - 'IB) - 0 o n A n B

and H' (A n 3 ) = H'([R3 - (O)] x R) H'(R3 - (O)) = O.

So ~ J A - ~ J B = dh for some O-form h on A n B. Unlike the previous case, we cannot simply consider VA - dh, since this is not defined on A . To circumveni this difficulty, note that there is a partition of unity ($A,$B) for the cover (A, 3 ) of R3 - (0):

$A+$B = 1

d $ ~ + d $ ~ = O

support $A c A

suppor t$~ C B.

Now, if $ ~ h o n A n B

denotes O o n A - ( A n B ) ,

and similarly for $AA, then

$ ~ h is a Cm form on A

$AA i s a C W f o r m o n 3 .

VA - d ( $ ~ h ) =: VA - $B dh - d $ ~ A

= 'IA + ($A - 1) dh + d$A A

= 'IA - dh + d($Ah)

= 'IB + d($Ah).

So we can define a Cm form on R" - (O) = A U 3 by letting it be VA - d ( $ ~ h ) on A, and q~ + d ( $ ~ h ) on 3. Clearly,

so w is exact. Tlie general inductive step is similar. Q

\/Ve end this chapter with one more calculation, which we wiU need in Chap- ier 11.

17. THEOREM. For O < k < n we have H~<(w") = O.

PROOE Tlie proof that H:(R") = O is left to the reader Let o be a k-form on Rn with compact support, O < k < n. We know that

o =: d q for some (k - 1)-form q on R". Let 3 be a closed ball containing support u. Then on A = R" - B we have d q = O. Since A is dfleomorpliic to

R" - ( O ) alid k - I < n - 1 we Iiave from Theorem 16 thai

q =: dh for some (k - 2)-form h on A .

Lei f : R" + [O, 11 be a CbD funciion wiili f = O in a neighborllood of 3 and f =: 1 on Rn - 23, where 2B denotes the ball of twice the radius of B. Then d ( f A) makes sense on al1 of R" and

ihe form q - d ( f A) clearly Iias compaci suppor t contained in 23.

PROBLEMS

1. The Riemann iniegral umstrs iheDmboux iniegrab Let f : [a, b] + W be bounded. For a partition P = ( l o < < 1,) of [a , b] , let n7i = 1 7 7 ~ ( f ) be the inf of f on [li-I, l i] and define Mi = Mi( f ) similarly. A choice for P is an n-tuple 6 = (ti,. . . , t,,) with ti E l i] . M'e define the "lower sum", "upper sum", and "Riemann sum" for a partition P and choice 6 by

CIear-ly L(f, P ) 5 S( J; P , 6 ) 5 U ( f , P ) . 14% cal1 f Darboux integrable if the sup of al1 L( f , P ) equals the inf of all U ( f , P ) ; this sup or inf is called the Darboux integral of f on [a , b] . 1We cal1 f RXemann integrable if

Iim S ( f , P , t ) exists; l lPI l+O

i11e limii is called tlie Riemann integral of f on [a, b] .

(a) We can ddine S ( f , P , 6 ) even if f is noi bouiided. Sliow however, thai lim S ( f , P , 6 ) cannot exist if f is unbounded.

I I P l + O (b) If f is coniinuous on [a, b] , then f is Riemann aiid Darboux integrable on [a, b] , and the two iategrals are equal. (Use uniform continuity of f on [a, b].) (c) If f is Riernaiin integrable on [a, b] , then f is Darboux integrable on [a, b] and tlie two integrals are equal. (d) Let n7 5 $ 5 M on [ a J b ] . Let P = ( S O < < S,) and Q = ( l o < < I,) Ile two pariitions of [a, b] . For e a ~ h i = 1,. . . , n, let

ei = lengtli of , l i]

- sum of Iengtlis of al1 [S,, S, 3 wliich are con tained in t i ] .

,- - --

[S,- , S,] 'S coi~iained in [li-l, li] slladed lei~gihs - add up to e!

Sliow ihat, if Mi denotes ihe sup of f on [rl-1, li], then

Tl~ere is a similar result for Iower sums. (e) Show that E;=l ei + O as 11 Pll + 0, and deduce Darboux's Theorem:

lim U( f, P ) = inf(U( f, Q ) : Q a partition of [a, b]) IP11+0

lim L ( f ,P ) = sup(L(f, Q) : Q a pariition of [a,b]). ll p 11- 0

(f) If f is Darboux iniegrable on [a, b], then f is Riemann integrable on [a, b]. (g) (OsgoodYs Theorem). Let .f and g be iniegrable on [a, b]. Show that foi choices 6 , t ' for P,

Hial: If Igl < M on [a, b ] , then I f ( tJi)g(Fi) - f (ti)g(tJi)l < Mlf (t'i) - f (tt)l. (h) Sliow thai 1, f dx + g d j ~ , defined as a limit of sums, equals

2. Compute Ic d0 = Jfo,ll C* do, where c(r) = (cos 2nr, sin2xr) on [O, 1 1 .

3. For 11 an inieger, and R > O, jet C R , ~ : [O, l ] + R2 - {O) be defined by

~ ~ , n ( t ) = (R cos 212711, R sin 2~711).

(a) S h w thai there is a singular 2-cube c : [O, 1l2 + R2 - {O) such that CR, ,n -

CR2,n = ac.

@) If c : [O, 11 + R2 - {O) is a i~y cuive witli ~ ( 0 ) = c(J), show ihat there is some n sucli that c - cl,, is a boundary in JR2 - (0). (c) Show tliat n is unique. Ji is called the windingnumber of c around O.

4. Let f : @ + @ be a polynomial, f(í) = ~ " + a ~ í " - ~ + . - - + a n , where n 2 l . Define c ~ , f : [O, l ] + G by c ~ , f = f o cR,1.

(a) Show that if R is large enough, ihen c ~ , f - CR,n is the boundary of a chain in @ - (O). Hinl: Note that C ~ , , , ~ ( [ ) - [cR,1 ( I ) ] " , and wriie

(b) Sliow tl~ai f (z) = O for some z E G ("Fundamental TI~eorem of AIgebra"). Hinl: If f ( 2 ) # O for al1 z with 121 5 R, ihen ~ R , J - co,f is a boundary,

5. Some approaches to iniegraiion use singular simplexes insiead of singular cubes. Alihough Stokes' Tlleorem becomes more complicaied, tl~ere are some advantages in using singular simplexes, as indicaied in the next Problem.

Lei An C R" be ihe sei of al1 x E Rn such ihat

A singular rl-simplex in M is a Cm function c : A , + M, and an 12-chain is a formal sum of singular 17-simplexes. As before, let I n : An + RQe the inclusion map. Define ai : + A , by

n-1 1 a0(x-) = ( [ 1 - zi=, xZ], x , . . . , X

1 i - t n - l )

a i ( x ) = (X , . . . ,A+ , O i , . . , x n l ) O < i < n,

and for singular rz-simplexes c, d e h e sic -- c o a i . Then we define

(a) Describe geometrically the images a i ( A n - 1 ) in A, . (b) Show that a2 = 0.

- (c) Sllow thai if o = f dxl A - - A dxi A . A dx" is an (n - 1)-form on W n , then

(Xmiiate the proof for cubes.) (d) Define Jc o for any k-chain c in M and k-form o on M , and prove thai

for any (k - 1)-form w. 6. Every x E Ak+1 can be writien as tx', for O 5 r 5 1, and x' E ao(Ak).

h4orcover, x' is unique excepi wlien r = O. For any singular k-simplex c : A k -t

define C : Ak+1 + W" by

We then define F for chains c in the obvious way.

(a) Show tliat 8c = O implies that c = 8F. (b) Let c : [O, 11 + JR2 be a closed curve. Show tliat c is ltol the boundary of any sum a of singular 2-cubes. Hinl: If 80 = a i c i , what can be said about Ci ai ? (c) Sliow ihai we do llave c = 80 + c' wliere c' is deSe?zm~le, tliat is, c' ([O, 11) is a poin t. (d) If c~ (O) = ~ ~ ( 0 ) and cI(l) = c2(1), show that ci - c2 is a boundary, using either simplexes or cubes.

7. Let o be a I-form on a manifold M . Suppose that Jc o = O for every closed curve c in M. Show that o is exact. Hin~: If we do have o = dS, tllen for any curve c we llave

E

8. A manifold M is called simply-connected if M is connected and if every smooth map f : S' + M is smoothly contractible to a point. [Actually, any space M (not necessarily a manifold) is called simply-connec ted if i t is connected and any continuous f : S' + M is (con tinuously) contractible io a poin t. It is not hard to show that for a manifold we may insert "smooth" at both places.]

(a) If M is smoothly conti-actible to a point, then M is simply-connected. @) S ' is not simply-connected. (c) S" is simply-connected for n > l . Hinl: Show that a smooth f : S1 + S" is not onto. (d) If M is simply-connected and p E M, then any smooth map f : S1 + M is smoothly contractible to p. (e) If M = U U Ir where U and V are simply-connected open subsets with U n V conilected, then M is simply-connec ted. pliis gives ano ther proof that S" is simply-connected for n > 1.) Hint: Given f : S' + M , partition S' into a fmite number of intervals each of which is taken into either U or V. (f) If M is simply-connected, ihen H 1 ( M ) = O. (See Problem 7.) 9. (a) Let U c R2 be a bounded open set such that W' - U is not connected. SIlow ihat U is not smoothly contractible to a point. (Converse of

Prooblem 7-24.) Hinl: If p is in a bounded component of R2 - U, show that tliere is a curve in U which "surrounds" p. @) A bounded connected open set U c IR2 is smoothly contractible to a point if and only if i t is simply-connected. (c) This is false for open subsets of R3.

1 O. Let o be an n-form on an oriented manifold M". Let @ and q be two partitions of uiiity by functions with compac t support, and suppose thai

(a) This implies tliai JM 9 w converges absolutely. (b) Show that

x J M m w = E t : J M q m w . #e@ #E@ *e*

and show the same result with o replaced by lo 1. (Note that for each 9, there are only finitely many xdiicl-i are non-zero on support 9.) (c) Sliow iliat Ceo* JM @ lo1 < m, and that

\Ve define tliis common sum to be j, u. (d) Let A , c (n,iz + 1) be closed sets. Let f : R + R be a CbQ function witIi iAII f = (-])"/ti and support f c U, A, . Find two partitions of unity @

and q sucli tliat &E, IR 9 f dx and '&E, IR i,il. f dx converge absolutely to different values.

1 1. Followiiig Problem 7-1 2, define geometric objects corresponding to odd 1-elab've teiisors of type (f) and weight w (w any real number). 12. (a) Let M be ((x, JJ) E R2 : I(x, y) 1 < I ), together witli a proper portion

of its boundary, aiid let w = A- dy. Show that

elen ihough bot1i sides inake sense, using PI-obleni 10. (No computatioiis iieeded-iioie tliai equaIity would hold if we had the entire boundary.) (11) Similarly, fiiid a countei-example to Stokes' Theorem when M = (0, 1) aiid w is a O-form whose support is not compact. (c) Exaniiiie a partition of uiiiiy for (O,?) by fuiiciions with compact support io see just wliy ilie proof of Stokes' Tlieorem breaks down iii this case.

13. Suppose M is a compact orientable 77-manifold (with no boundary), and 19 is an (n - 1)-form on M. Show that de is O at some point.

14. Let MI , M2 C R" be compact n-dimensional manifolds-with-boundary with M2 c Mi - aMl. Sl-iow that for any closed (n - 1)-form w on Mi,

15. Account foi- the factor I/lp 1" in Lemma 7 (we have r,(vp) = ( I / ~ p l ) ~ , ( ~ ) , but this only accounts for a factor of l/lpln-', since there are n - 1 vectors vi , , vn-1).

16. Use the formula for r*dxi (Problem 4-1) io compute **ot. (Note that

tl-ie map i o r : R" - (0) + R" - (O) is just r, considered as a map into R" - {O}.)

1 7. (a) Let M" and Nm be oriented manifolds, and let w and q be an n-form and an m-form with compact support, on M and N, respectively We will orient M x N by agreeing that vi, . . . , u,, wr, . . . , w, is positively oriented in ( M x Mp $ Ng if V I , . . . , Un and wi , . . . , Wm are positively oriented in M, and N,, respectively. If ni: M x N + M or N is projeciion on the ith factor, show that

where

g (p ) = JN - ) V . ~ I ( P , - 1 = q h(pY 01.

(c) Every (m + n)-form on M x N is h n1 * m A n2* TJ for some w and q .

18. (a) Let p ER"-(O}. Let wi,. . . ,wn-2 E IR? andletu E IR> be (hp), for some h E R. Show that

(b) Let M C IR" - (O) be a compact (n - 1)-manifold-with-boundary which is ilie uiiion of segrnents of rays through O. Show that jM r*o' = 0.

(c) Let M c IRn-(O] be a compact (n - 1)-manifold-wi th-boundary whicl-i in ter- sects every ray througli O at mosi once, and let C(M) = (hp : p E M, h 2 O}.

Sliow tl-iai

/M r*o' = / r*ol. C ( M ) f l S z

Tl-ie latiel. integral is il-ic nieasure of tlie solicl arigle subtended by M. For ibis reason we often denote r*ol by d e n .

19. Fol. all {-Y, y,-) E lR3 except tliose wiih A* = O, y = O, r E (-00, O], we deline $(x, y, z ) to be ilie angle be tween il-ie posiiive z-axis and the ray from O through (x, y, r ) .

(a) # ( x , y , z ) = arctan(Jx2 + y 2 / z ) (with appropriate conventions). (b) If v ( p ) = IpI, and 8 is considered as a function on W 3 , @ ( S , y , z ) = arctan y / x , then ( u , 8 , #) is a coordinate system on the set of all points ( x , y , z ) in IR3 except those with y = O, x E [O , cm) or with x - O , y = 0, z E ( - c m , O ] . (c) If u is a longitudinal unit tangent vector on the sphere S 2 ( r ) of radius r, then d#(u) = l . If w points along a meridian through p = ( x , y , z ) E SZ ( r ) ,

t hen

(d) If 8 and 4 are taken to mean the restrictions of 8 and # to [certain portions ofl S', then

o' = h d8 A d 4 ,

where h : S2 + IR is

h ( x , y , z ) = -m (tiie minus si91 comes fmrn the orientation).

(e) Conclude that a' = d ( - cos # d e ) .

(f) Let r2: R2- (O) + S' be the retraction, so that d9 = rz*i*o, for the form o on It2. Sliow that

r2*d@ = de.

If n : R3 + R2 is the projection, then the form d9 on [par t of] IR3 is just n*d@, foi- tlie form d9 on [part ofJ R2. Use this to show that

(g) Also prove tliis directly by using the result in part (c), and the fact thai r,(vp) = v,<,>/lpl for v tangent to S2(1pl). (h) Conclude that

(i) Siniilarly, express d0 , on R" - (O) in terms of do,,, on R"-' - (O).

20. Prove iliai a connected manifold is the union UI U U2 U U3 U . , where ille Ui are coordinate neighborhoods, with Ui í l Uj # 0, and the sequence is eventually outside of any compact set.

2 1. Let f ; M" + N" be a proper map beiween oriented n-manifolds such that f, : Mp + NI(,) is orientation preseniing whenever p is a regular point. Sllow that if N is connected, then either f is onto N, or else al1 points are critica1 points of f.

22. (a) Sliow tliat a polynomial map f : C + C, giveii by f (z) = z" +ai 2"-' + - . +a,, is proper (n 2 1).

(b) Let f l(z) = 71ín-I + (n - I ) a ~ z " - ~ + . +an-1. Show that we have f ' ( E ) = lim [ f (c + w ) - f (i>]/w , wliere w varies over complex numbers.

w+o (c) M7rite J'(x+iy) = u(x, y )+iv(x, y) for real-valued func tions u and v. Sllow that

a v au -(x, y) - i-(x, y). ay ay

Hinl: Choose w to be a real h , and then to be ih. (d) Conclude that

If '(x + iy ) l 2 = det D f (x, y),

where f ' is defined in part (b), whde Df is the linear transformation defined for any dflerentiable f : R2 + R2. {e) Using Problem 2 1, give ano ther proof of the Fundamental Theorem of Al- ge bra. (f) There is a still simpler argument, not using Problem 2 1 (which relies on many theorems of this cliapter). Show directly that if f : M + N is proper, then the number of points in f -' (a) is a locally constant function on the set of regular values of f. Sliow thai tliis set is connected for a polynomid f: C + C, and conclude that f takes on all vdues.

23. Let M'-' C R" be a compact oriented manifold. For p E Rn - M, choosi an ( u - 1 )-sphere E aiaund p such that aU points inside C are in R" - M. Let rp : R" - ( p ) + E be the obvious retraction. Define the winding number w (p) of M around p to be the degree of rp 1 M.

(a) Sllow tllat this defii~ition agrees with that in Problem 3. @) Show tllat tllis definition does not depend on the choice of C. (c) Sllow that w is constant in a neighborhood of p. Conclude that w is constant on each component of IRn - M. (d) Suppose M coiitains a portion A of an (ti - 1)-plane. Let p and q be points

close to tl-iis plane, but on opposite sides. Show that w(q) = w(p) z t l . (Show tliat rq 1 M is homotopic to a map which equals rp 1 M on M - A and which does not take any point of A onto the point x in the figure.) (e) Show tliat, in general, if M is orientable, then W n - M has at least 2 components. Tlie next few Problems show Iiow to provc tlie same result even if M is not orientable. More precise conclusions are drawn in Chapter 11.

24. Let M and N be compact n-manifolds, and let f , g : M + N be smoothly honiotopic, by a smootli liomotopy H : M x [O, 11 + N.

(a) Let q E N be a regular value of H. Let #/-'{y) denote the (fiiiite) numbei. of points in f -' (q). Show tliat

# f -' (q) = #g-i (q) (mod 2)

HUZI: H-' (y) is a compac t l -manifold-\vi tli-bouiidary The iiumber of poin h iii its boundary is clearly even. (Tliis is one place where we use tlie strongei- form of Sard's Theorem.) (b) Show, more geiicrally, tliat tliis result holds so long as y is a regular value of I~otli f and g.

25. For two maps f, g : M + N we will write f - g to iiidicate that f is smoothly Iiomotopic to g.

(a) If f g, theii thei-e is a smooth homotopy H': M x [O, 11 + N sucli tliat

' { p , ) = f . ) for I in a neighborliood of O,

H' { j ~ , 1 ) = g(p) for iri a iieigliborhood of l .

01) -. is an equivalence i-elatioli.

26. If f is smootlily Iioniotopic to g by a smooth homotopy H sucl-i ihat p w

H{P, I ) is a diffeomorpIiism for eacli I, we say that f is smoothly isotopic to g.

(a) Reing smootlily isotopic is an equivalence relation. (b) Let 4 : IW" + IW he a Cm function wliich is positive on tlle interior of the uiiit hall, aiid O elscw1iei.e. I'or p E S"-', let H: W x Rn + W" satisfy

(Lach solution is defined for al1 r, by Tlieor-em 5-6.) Show that each s w H{t, s) is a diffcoinoi-pliism, wliich is smootlily isotopic to the identity, and leaves al1 points outside the unit hall fixed.

171 iegm [ion 2 95

(c) Show tl-iat by choosing suitable p and t we can make H(t, O) be any point in the interior of the unit bdl. (d) If M is connected and p , q E M, then there is a diffeomor~hism f : M + M such that f (p) = q and f is smoothly isotopic to the identity. (e) Use part (d) to give an alternate ~ r o o f of Skp 3 of Theorem 9. (f) If M and N are compact n-manifolds, and f : M + N, then for regular values ql, qz E N we have

(where #fml(q) is defined in Problem 24). This number is cdled the mod 2 degree of f. (g) By replacing "degree" with "mod 2 degree" iii Problem 23, show that if M c IR" is a compac t {n - ])-manifold, then R" - M has at leas t 2 components.

27. Let (X ' ) be a CbD family of CbD vector fields on a compact manifold M. (To be moi-e precise, suppose X is a Cm vector field on M x [O, 11; then Xf(p) will denote T ~ * X ( ~ , ~ ) . ) From the addendum to Chapter 5, and the argument wl-iich was used in the ~ r o o f of Theorem 5-6, it follows that there is a CbD family (4, ) of diffeomorphisms of M [not necessarily a l-parameter group] , with & = identity, which is generated by (X') , i.e., for any CbD function f : M + IR we

{X'f )(p) = lim f (4 i+h (~ ) ) - f (4' (P)) h+O ii

For a family o, of k-forms on M we define the k-form

Ut+h - a' ¿I, -- lim h+O 11

(a) Show that for q{t) = 4,*of we haw

(b) Let o o and o1 be riowhere zero 11-forms o11 a compact oriented n-manifold M, and define

o' = ( 1 - t)wo + /o1.

Sliow that the family 4, of di&omorpliisn~s generated by (X') satisfies

~ I * W = wo for al1 t

if and only if

(c) Using Problem 7-18, show that this holds if and only if

(d) Suppose that JM 00 = JM 0 1 , SO that wo - o1 = d l for some l. Show that there is a dfleomorphjsm J : M t M such that wo = fi*ol.

28. Let f : M' + W n aiid g : N' + R" be Cm maps, where M and N are compact oriented manifolds, n = k + I + 1, and f ( M ) n g ( N ) = 0. Define

alrg: M x N + S"-' C W" - (O)

We define the linkingnumber of f and g to be

wliere M x N is oriented as in Problem 18.

(a) [ ( f , g ) = ( - l )k '+ie(g, f ). @) Let H : M x [O, J ] t W" and K : N x [ O , 11 + W" be smootli homotopies with

such that

( H ( P , I ) : p E M } n { K ( q , r ) : q E N ) = 0 for e v e r y ~ .

Show that

[ ( f , g ) = [tf 2).

(c) For f , g : S' + W 3 show that

where

( f l ) ' (u) ( f 2>'(u> ( f 3 ) ' ( ~ >

A(u , u ) = det ( g l )'(u) (g2) ' (v) (g3) ' (v>

s l ( v > - f l ( ~ ) g 2 ( v ) - f 2 @ ) g 3 ( v ) - f 3 ( u )

(tlie factor 1 / 4 x comes from the fact that ls, o' = 4n [Problem 9-14]). (d) Show that L( f , g ) = O if f and g both lie in the same plane (first do it for (x, y)-plane). Tlie next problem shows how to determine L ( f ? g ) without calculating. 29. (a) For (a ,b ,c ) E W 3 define

For a compact oriented 2-manifold-with-boundary M c and (a , b , c ) M , le t

E

a ( a , b , C ) = JM dQ(a,b,c).

Let (a , b , c) aiid (a', b', c') be points cIose to p E M , on opposite sides of M. Suppose ( a , b,c) is on tlie same side as a vector wp E W 3 p - M, for which the

triple w,, (vi ),, ( ~ 2 ) ~ is positively oriented in R~~ when (vi ),, (u2), is positivel y oriented in Mp. S how that

lim fi (a , b, c ) - S2 (a', b', c') = -471 ta,b,c)+ p

(a',b7,c')+

Hint: First sliow that if M = a N , then L? ( a , b , c ) = -471 for (a , b , c ) E N - M and S2 (a , b , c ) = O for ( a , b , c ) N .

(b) Let f: S' + W' be an imbedding such that f ( S 1 ) = aM for some compact oriented 2-manifold-~4th-boundary M . (An M with this property always exists. See Fort, Zpologv qf 3-A..lnn23[dr, pg. 138.) Let g : S' + lT3' and suppose

The figure on the left shows a non-orientable surface whose boundary is the "trefoil" knot,

l ~ i i t the surface on the right-including the hemisphere behind the plane of the paper- i r orientable.

that wben g(t ) = p E M we llave dg/dr $ M,. Let u+ be tlie number of inter- secrions wl-iere dg/dt poiiits iil the same direction as tlie vector wp of part (a), and the number of other intersections. Show that

(c) Show that

S1iow tliar 17 = for tlle pairs shown below.

30. (a) Let p , q E R" be distinct. Choose open sets A , 3 c 8" - (p , y) so tl-iat A and 3 are diffeomorphic to Rn - (O), and A fi 3 is diffeomorphic to IR". Using an argument similar to that in the proof of Theorem 16, show that

H'{R" - (p , q)) = O for O < k < 11 - 1, and that H"-' {R" - (p, q)) has dimension 2. @) Find the de Rham col-iomology vector spaces of R" - F where F C Rn ís a finite set.

31. We define the cupproduct u : H k ( M ) x H'{M) + Hk+'(M) by

(a) Show t11at u is well-defined, i.e., w A q is exact if w is exact and r~ is closed. (b) Sllow that u is bilinear.

k l (c) If ci E H ~ { M ) a n d p E H'{M), then c i u p = (-1) p u a . (d) If f : M + N, and ci E Hk{N), E H'{N), then

(e) Tlie cross-product x : H' {M) x H' {N) + Hk+' { M x N ) is defined by

Sliow tllat x is weI1-defined, aild tllat

(f) If A : M + M x M is the "diagonal map", given by A ( p ) = ( p , p ) , show that

L Y V ~ = At(a x p ) . 32, On the n-dimensional torus

n times

let d e i denote ni*d9, where n i : T" + S' is projection on the ith factor.

(a) Sliow that al1 dei' A A de'" represent different elements of Hk(T"), by finding submanifolds of T" over which they have different integrals. Hence dim H k (T" ) 2 6). Equality is proved in the Problems for Chapter 1 l . (b) Show that every map f : S" + T" has degree O. Hint: Use Problem 25.

CHAPTER 9 +

RIEMANNIAN METRICS

1 n previous chapters we have exploited nearly every construction associated with vector spaces, and thus with bundles, but tliere has been one notable

exception-\ve have never mentioned inner products. The time has now come to make use of this neglected tooI.

An inner product on a vector space V over a field F is a bilinear function from V x V to F, denoted by (v, w) i-, (v, w), which is symmetric,

arid non-degenerate: if v # 0, theil there is some w # O such that

For us, the field F will always be R. For each r with O 5 r 5 17, we can define an inner product ( , ), on lT3" by

this is non-degenerate because if a # 0, then

n 1 1 r+ 1 - C ( a i 1 2 > O. ((a , . . ., an) , (a , . . . , a r , -a , . , -an)Ir -

i=l

In particular, foi- i. = II we obtain the "usud inner product", ( , ) on R",

(a, b) = E aibi.

For this inrler product we have (a, a ) > O for aily a # O. In generd, a symmetric bilinear function ( , ) is called positive definite if

A positive definite bilinear function ( , ) is clearly i~on-degenerate, and consequently an iniier product.

30 1

Notice that an inner product ( , ) on V is an element of T ~ ( v ) , so if f: W + V is a lii~ear transformation, then f * ( , ) is a symmetric bilinear function on W. TIiis symmetric bilinear function may be degenerate even if f is one-one: e.g., if ( , ) is defined on R2 by

2 2 (a, b) = a ' b1 - a b. :

aiid f: IR + IR2 is f (a> = (a, a>-

However, f * ( , ) is clearlg iiondegeiierate if f is an isomoi.pl~ism ozzlo V. Also, if ( , ) is positive definite, then f * ( , ) is positive definite if and only if f is or~e-one.

For aily basis V I , . . . , v, of V, witli corresponding dual basis U*], . . . , U*,,, we can write

In this expressioii.

so symmetry of ( , ) implies tl-iat tl-ie matrix {gij) is symmetric,

Tlie matrix {gij) lias anorhcr important iilierpretation. Since an inner producr ( , ) is lii-iear in rlie seco~ld argument, wc can define a liiiear fuilcrional$, E V*, for eacli v E V, by

$v{w) = (u, w).

Sirlce ( , ) is linear in tl-ie fii-st argument, tlie inap v I-+ 4, is a liiiear transfor- ~nation fi-om V io V*. No~-i-degeiiei.acy of ( , ) implies that $, # O if U # 0. Tlius: if V is fii-iiie dimensional, an i1111er 111-oduct ( , ) gives irs al] isomor-phism CY : V + V*, witl-i

(v,w) = Q(v)(w).

Clearly, rl-ie matrix (gij) is jusr tbe matrix of CY : V + V* witl-i 1-espect to tl-ie I~ases {vi} [or 1' ai-id { v * ~ } for V *. Thus, 11o1-i-degeileracy of { , ) is cquiualei-ii to the cor-idition thar

Positive d~f i r - i i t~~- i~ss of ( , ) correspo~~ds to rhe more complicated coi-idition thar the n-iatrix (gij) be "positive definitc", meaning that

II

''>O gija a for al1 a l , . . . , an witl-i at least one a' # O. i = 1

Given aiiy posiliue deJilzib iiiner product ( , ) oii 1/ we define the associated

norm l l I l by

llvll = JT;;;) (the positive square root is to be taken).

In IR" we denote the norm corresponding to ( , ) simply by

Tlie principal pi-operties of 1 1 11 are tlie followiilg.

l . THEOREh4. For al1 u, w E V we have

(1) ll0vll = 101 + llull. (2) 1 ( u , w)l 5 11 v 1 1 11 w 1 1 , with equality if aild only if v and w are linearly

dependent (Schwarz inequality).

(3) Ilv + wll 5 llvll + 11 w 1 1 (Triangle iiiequality).

PRQQE (1) is trivial. (2) If v aiid w are linearly dependent, equality clearly holds. If ilot, then O # hv - w for al1 h E R, so

O < l l h ~ - w 1 1 ~ = ( A v - w , h v - w )

= h211v112 - 2 h ( v , w ) + llw112.

So the right side is a quadratic equation in h with no real solution, and its discrimiiiaiit must be iiegative. Tlius

4(v , w ) ~ - 411vl1211wl12 o.

Tlie fuiic.tioi.i 1 1 11 has certaii-i uiipleasai-it properqies-for example, the functioii 1 1 017 IR" is ilot dflereiitiable at O E IRn-wliich do riot arise for the function 11 1 1 2 . This lattei fui-ictioii is a "quadratic function" on V-in ter-ms of a basis ( v i ) for V it can be wi-itten as a "homogeiieous polynomial of degree 2" in the

More succii-ictly, n

An invariai-it definition of a quadratic furlction can be obtained (Problem 1) from the following obsentatioi-i.

2. THEOREhil (POLARIZATION IDENTITY). If 11 11 is the norm associated to an inner product ( , ) on V, then

PXOOF Compute.

Theorem 2 shows that t\vo iilnei- products which induce the same norm ar-c theiiiselves equal. Similarly, if f : V += V is 1101-m 131-esen~ing, that is, 11 f (v) 11 =

llvll fol. al1 v E 1/, then f is also iniler product presenfing, that is, (f (u), f (w)) =

(v, w) for al1 v, w E V. \Ve wil1 now see that, "up to isomorphism", diere is only one positive definite

ii-iner product-

3. THEOREhil. If ( , ) is a positive defiiiite iiiiier product on ai-i n-diinei-i- sional vector space V, then there is a basis vi,. . . , v, for V such that (vi, vi) = 6,. (Such a basis is called orthonormal with respect to ( , ).) Consequentl!; there is an ison~oi-pliism f : Rti + V such that

In other words, f*( 7 > = { 9 >.

PRO0.F Let Wr,. . . , wtl 13e ai-iy basis for V. \Ve obtain the desired basis by applviiig tlie "Granl-Scl-imidt orthonormalization proc.ess" to this basis:

Since wi # O, we can define

aiid clcady Ilui 11 = l . Suppose that we have constructed u], . . . , u k so that

(vi, v j > = 6, I s i , j s k

and span V I , . . . , vk = span W I , . . . , wk.

Then wk+] is linearly independent of V I , . . . , vk. Let

It is easy to see that

So we can define

and continue inductively. *3

A positive definite inner pi-oduct ( , ) on V is sometimes called a Euclidean 71zeh-i~ 011 V. Tilis is because we obtain a metric p on V by defining

T h e "triangle iilequality" (Theorem l(3)) shows that this is indeed a metric. 147e also caH llvll the length of v.

\die have onfy one more algebraic trick to play. Recalf that an inner product ( , ) on V prcwides an isomorphism a : V 4 V* with

Using the natural isomorphism i : V + V4*, defined by

1dre can ilow use /3 to define a bilinear functioii ( , )* on V* by

(A,p}* = /3(A)(p) = i a - l ( A ) ( ~ ) = P ( ~ - - ~ ( A ) ) .

Now, tlie symmetry of ( , ) can be expressed by tbe equation

Letting #(u) = A, u(w) = p,

this can be written 1 (A) = Au-l (A)),

which shows that ( , )* is also symmetric,

Consequently ( , )* is aii inner product on the dual space V* (in fact, the one which produces j3).

To see what this al1 means, choose a basis (ui) for V, jet (u*i) be the dual basis for V*, and 1et

n

Then

(gij) is the matrix of u : V + V* with respect to (vi) and (u*¿)

(gjj)-' is thc matrix of u-': V* + V witli respect to (u*i) and (q)

so

(gG)-' is the matrix of B : V* + V** witli respect to and ( ~ * * i ) .

Tlius, if we let g i j be the entries of the inverse matrix, (g") = (gjj)-l, so tliat

theii

= g " ~ i @ u j , ifwe considerui E V**.

Onc can clieck directly (Pi-oblem 9), without the invariant definition, tliat this equation defines ( , )* indepelidently of the choice of basis.

Notice that if ( , ) is positive definite, so that

then, letting a(v) = h, we have

so ( , )* is also positive definite. This can also be checked directly from the defiiiition in terms of a basis. In the positive defiilite case, the simplest way to describe ( , )* is as follows: TIle basis u*], . . . , v*, of V* is orthonormal with respect to ( , )* if aiid only if v i , . . . ,u, is orthoilormal ~ 4 t h respect to ( , ).

Similar tricks can be used (Problem 4) to produce an inner product on al1 tlie vector spaces T k ( v ) , T,(V) = Tk(v* ) , and a k ( v ) . Howwer, we are intcrested in oiily oile case, which we will i~o t desci-ibe in a completely invariant way. The vector space Qn(V) is 1 -dimensional, so to produce an inner product on it, we need only describe whidl two elements, o and -o, will have length 1. Let V I , . . . , v, aild W r , . . . , w, be two bases of V which are orthonormal with respect to ( , ). If we write

So the transpose matrix A' of A = (ujj) satisfies A - A' = 1, which implies that det A = f l . It follows fi-om Theoi-ein 7-5 that for aily w E S2"(V) we have

It clearly follows that

We have thus distinguished two elemen ts of R" (V); they are both of the form v * ~ A . - 'AV*, for (vi) an ortho11ormal basis of V. \IZ1e will cal1 these two elements

the elements of norm 1 in Rn(V). If we also have an orientation p, then we can furtlier distinguish the one which is posiiive when applied to any (vi,. . . , U,) Hfith [u1, . . . , Un] = p; \ve will cal1 it the positive element of norm 1 in nn(V) .

To express tlie elements of norm 1 in terms of an arbitrary basis wi, . . . , w., we cl-ioose an ortbonormal basis vl, . . . , U, and write

Problem 7-9 implies that

then

det(gjj) = det(At A) = (det Al2 .

In particular, det(gij) U o l w ~ ) ~ ~ posiiiu~. Consequently, the elements of norm 1 in R" (V) arc

\Ve noiv apply our new tool to \~ector buiidles. If c = a : E + B is a vector l~uiidle, we define a Riemannian metric on [ to be a function ( ) whicli assigns to eacli p E 3 a positive definite iiiner product { , ), on a- '(p), and which is coiltinuous iil tbe cense that for aily two continuous sections si , $2 : 3 + E, the functioil

( ~ 1 3 ~ 2 ) = p (SI(] . ' ) ,S~(P))~

is also coiltinurius. If 6 is a Cm vector builclle over a Cm manifold we can also speak of Cm Riemannian metrics.

[Another approach to the definition can be given. Let Euc(V) be the set of al1 positive definite inner products on V. Ifwe replace each n-'(p) by EUG(X-] (p)), and Jet

EucO) = u ~uc(rr-'(p)), PEB

then a Riemannian metric on can be defined to be a section of EM({) . Tbe only problem is that Euc(V) is not a vector space; the new object E~ic(c) that wc obtaiii is not a vector bulidle at all, but an iiistance of a more general structure, a fibre buiidle.]

4. THEOREM. Let { = n : E + M be a [Cm] k-plane bundle over a Cm maiiifold M. Tlieil tliere is a [Coo] Riemailnian me tric on c. PROOE Tliere is aii opeii locally fiiiite cover O of M by sets U for which there exists [Cm] trivializations

ru: =-'(U) + U x IR^. On U x IRk, tliere is aii olwious Riemannian metric,

((p, 4, (p,b))p = ( ~ , b ) .

For u, w E n-'(p), define

(u, w)pO = ( I U ( ~ ) ? ~u (w))p.

Then ( , ) u is a [Cm] Riemannian metric for CIU. Let (#u) be a partition of uiiity subordinate to 0. We define ( , ) by

(u, = E #u(p) (u, w)p u, w E x - ~ ( ~ ) . UEO

Then ( , ) is coiitiiiuous [Cm] and eacli ( , )p is a symmetric bilinear function oii n-' ( p ) . To sliow that it is positive definite, iiote that

eacli #u (p) (u, u): 2 O, aiid fol. some U strict inequality liolds. +:'

[The same argument shows that aily vector bundle over a paracompact space lias a Riemannian metric.]

Notice that the argunient in the filial step would not work if we had merely picked iioii-degeiierate iiiiier piaducts ( , ) 111 fact (Pi.oblem i), there is no ( , ) on TS' wliicli gives a symmetric I~ilinear fuiiction on each sZP which is iiot positive defini te 01. iiegative defiiiite bu t is still non-degenerate.

As an application of Theorem 4, we settle some questions which have ti11 now remained unanswered.

3. COROLLARY. If 6 = x : E + M is a k-plane bundle, then { c*. PROOF. Let ( , ) be a Riemannian metric for c. Then for each p E M, we have an isomorphism

U p : x-i (p) + [x-1 (p)]*

defined by

ol,(v)(w) = ( u , W p v, w E x-l (p).

Continuity of ( , ) iinplies that the union of al1 ap is a homeomorphism from E lo E' = U,,,[x-' (p)]*. O

6. COROLLARY. If { = x : E + M is a 1-plane bundle, then { is trivial if aiid only if is orieiitable.

PROOF The "only if" part is trivial. If t has an 01-ientation p aiid ( , ) is a Riemaiinian metric o11 M tlien there is a unique

with

p = 1, [s(p)I = Pp

Clearly s is a section; \ve then define an equivalente f: E + M x R by

ALTEMATIVE PROOF \Ve know (see the discussion after Theorem 7-9) that if c is orientable, theil there is a nowhere O sectioil of

so that t* is trivial. But { t*. +3

All these coi~sidei.atioiis take on special significai-ice wl-ien our bundle is the taiigeiit bui~dle T M of a C* manifold M . Iii this case, a C* Riemanniaii inetric ( , ) for T M , which gives a positive definite inner product ( , ), oi-i

each Mp, is called a Riemannian metric on M. If (x, U) is a coordinate system on M, then on U we can write our Riemaniliail metric ( , ) as

wherc the Cm fuiictioils gij satisfy gij = gji, since ( , ) is symmetric, and det(gli) > O since ( , ) is positive definite. A Riemannian metric ( , ) on M is, of coui.se, a covariai-it tensor of order 2. So for every Coo map f: N + M there is a covai-iant tensor f*( , ) on N, which is c1early symmetric; i t is a Riernai-iniai-i metric on N if and only if f is an immersion is one-one for a11 11 E N).

The Riemaililian metric ( , )*, wl~ich ( , ) induces on the dual bundle T4M, is a coi1travariant teiisor of order 2, and we can write it as

Our- discussion of innei- products irlduced on V* shows that for each p, the matrix (g"(p)) is the inverse of the matrix (gjj (p)); thus

Similarly, foi- each p E M the Riemanniail metric ( , ) on M determines two c1cments of S2"(Mp), the elemei-i ts of norm l . \Ve have seen that they caii 11e written

I Jdet(glj(p)) dxl(p) A . - . A dxn(p).

If M has ai1 or.ieiitation p , then pp aUows us to pick out the positive element of r1oi.m 1, and we ol~tain an i?-fosm on M; if x : U + R" is orzenfaiio~z pres~ving, tlien o11 U tliis form caii be written

Even if M is iiot oi.icntabIe, we obtaii-i a "voIun1e e1ement" on M, as defined in Chaptei- 8; in a coor-diiiate system (x, U) it can be written as

Tliis volume elen~eiit is deiioted by d V, eveil thougll it is usual1y not d of ai-iytliii-ig (even wl~en M is orientable and it can be considered to be an n-form),

and is calied tlie volume element determined by ( , ). We can then define the volume of M as

F

This certainly makes sense if M is compact, and in tlie non-compact case (see Problem 8-10) i t eitl~er converges to a defiilite iiumber, or becomes arbitrarily large over compact subsets of M , in which case we say that M has "infinite volume".

If M is ail 11-dirneii.sioiial manifold (-witli-boundary) in R", with tlie "usual Riemannian metric"

n

then gij = d i j , SO

d V = Idxl A - - A dxnI,

and "volume" becomes ordinary volume. Tliere is an eveii more importaiit construction associated with a Riemannian

metric on M , ~rhicli will occupy us for the I-est of tlie chapter. For every Cm cunte y : [a, b] + M, we have tangent vectors

and can t1ierefoi.e use ( , ) to define tlieir length

14Jc caii then define the length of y from a to b,

If y is mesely pulc~wrse sttzooilr, n~eaiiiiig tha t there is a parti tion a = ro < . . t,, = b of [u, b] sucli tliat y is smootli on eacli [ ~ i - i , li] (with possibly different

left- and right-hand derivatives at t i , . . . , tn-]), we can define the length of y by

IYhenever tliere is no possibility of misuiiderstaiiding we will denote L,b simply by L. A little ai-pmeni shows (Problem 15) tliat for piecewise smooth curves in IRn, with the usual Riemannian metric

dx' @ d x i 7

tliis defiiiition agrees ivith tlie definition of length as the least upper bound of tlle lengths of inscribed polygonal curves.

We can also define a function S : [a, b] + IR, the "arclength function of y" by

Co~~sequeiitly dy /dr has constant length 1 precisely when S ( t ) = t + constant, thus precisely when s(1) = E - a. Then

We can reparameterize y to be a curve on [O, b - u] by defiiiing

p ( r ) = y(t - a ) .

For the new curve y we have

iiew s ( l ) = L;(?) = = old s(i f u ) - old s (u)

= t .

If y satisfies S ( [ ) = r we say that y is parameterized by arcfength (and then often use S iiistead of r to denote the argument in the domain of y).

Classically, the norm 11 11 on M was denoted by ds . (This makes some sort of sense aren in modern notation; equation (s) says that for each cuwe y and corresponding s : [ a , b] + IR we have

on [ a , b].) Consequently, in classical books one usually sees the equation

Nowadays, this is sometimes interpreted as being the equivalent of the modern equation ( , ) = x y , j = l g i j d ~ ' @ d x j , but what i t always actually meant was

Tlie symbol dx'dx j appearing here is no l a classical subai tute for dx' O d x j - the value ( d ~ ' d s j ) ( ~ ) of dx'dxj at p should not be interpreted as a bilinear function at ali, but as the quadratic function

H dxi ( p ) ( u ) * d x j ( P ) (u) v E Mp ,

and we would use the same symbof today. The classical way of indicating dxi @ d x j was very strange: one wrote

E gij dx'dxj where d x and Sx are independent infinitesimals. i,j=l

(Classically, the Riemannian metric was not a function on tangent vectors, bu1 tlie inner product of two "infinitely smaU dispiacements" dx and Sx.)

Consider now a Riemannian metric ( , ) on a cotlneckd manifold M. If p, q E M are any two points, then there is at least one piecewise smootli cuwe y : [a , b] + M from p to q (tliere is even a smooth curve from p to q) . Define

d ( p , q ) = inf { L ( ~ ) : y a piecewise smooth curve from p to q ) .

It is clear that d ( p , q ) > O and d ( p , p ) = 0 . Moreover, if r E M is a third point, then for any E > O, we can choose piecewise smooth curves

y, : [u , b] +- M from p to q with L ( y i ) - d ( p , q ) < E

y2: [ b , ~ ] + M from q to r with L ( y 2 ) - d ( q , r ) < E.

If we define y : [a,c] + M to be y, on [a,b] and y2 on [b,c] , then y is a piecewise smooth cuwe fmm p to r and

L ( Y ) = L(Y,) + L(y2) < d ( p , q ) + 4 % 4 + 2 ~ .

Since tliis is true for al1 E > O, it follows that

d ( p , r ) 5 d ( p , 9 ) + d(q , 4. [ I f w did not allow piecewise smooth cuntes, there would be difficulties in fitting together y, and y2, but d ~rould still turn out to be the same (Problern 1 T).] The function d : M x M + R has aíl properties for a metric, except that it is not so clear that d ( p , q ) > O foi- p # q. Tliis is made clear in the following

7. THEOREM. The function d : M x M + R is a metric on M , and if p: M x M + IR is the original metric on M (wliich makes M a manifold), then ( M , d ) is homeomorphic to ( M , p).

PROOF. Both parts of the theorem are obviously consequences of the folíowing

7'. LEMMA. Let U be aii opeii neighborhood of the closed ball 3 = { p E R" : jpl 2 11, let ( , ), be the "Euclidean" or usual Riemannian metric on U,

n

aiid let ( , ) be any otlier Riemannian metric. Let 1 1 = 11 1 1 , and 11 I l be the corresponding norms. Then there are numbers m, M > O such that

aiid consequelitly for any cunte y : [a, b ] + 3 we have

??lLe(y) 5 L ( Y ) < M L , ( Y ) .

PROOE Define G : B x S"-< R by

G ( p , a ) = llap llp.

Then G is contiiiuous and posiiive. Since 3 x S"-' is compact there are numbers m, M > O such that

il7 < G < M on B x S"-'.

Now if p E B and O # bp E Rnp, let a E S"-' be a = b/lbi. Then

1?7lbl < lblG(p,a) < Mlbl;

siiice l b j G ( p , ~ ) = lb1 . l l ~ ~ l l ~ = IiObIlOpIIp = IIbIIp,

tliis gives the desired inequality (wliich clearly also holds for b = 0). O

Notice that tlie distance d ( p , q ) defined by our metric need liot be L ( y ) for any piecewise smooth curve from p to q. For example, the manifold M might be IR2 - {O), and y might be - p . Of course, if d ( p , q ) = L ( y ) for come y,

then y is clearly a shortest piecewise smooth curve from p to q (there miglit be more than one shortest curve, e.g., the two semi-cirdes between the points p and - p on S') .

ln order to investigate the question of shortest curves more tl~oroughly, we liave LO employ tecliiiiques fi-om the "calculus of variatioils". As an introduction to such teclii~iques, we consider first a simple problem of this sort. Suppose we are giveli a (suitably differentiable) function

Wc seek, amoiig al1 functions J.: [a , b ] + IR with f ( a ) = a' and f (b) = b' oiie

whicli will maximize (or minimize) the quantity

For examplc, if-

then we are looking for a function f on [u,b] which makes the curve t H

( E , f ( 1 ) ) between (a, u') and (b, b') of shortest length

As a second example, if

then we are trying to minimize the area of tlie surface obtaiiied by revolviiig tlie graph of f around the A--axis, which is given (Problem 12) by

To appl-oach this sort of problem we recall first tlie metliods used for solving t1.ie n~uch simpler problem of detei-ininiiig t l~e inaximum or niinimum of a function f : R + R. To solve tliis problem, \ve examine tlie critical points of f, i.e., those points x for wliicli f l (x) = O. A critica1 poilit is not iiecessarily a maximum or minimum, or even a local maximum or minimum, but critical points arc the only caiididates for maxima or mii.iin.ia if f is everywliere difTer- eiitiable. Similarly, for a functioii f : IR2 + IR we consider poiiits (x, y) E R2 for which

This is tlie same as saying that the cuntes

have derivative O at O. \Ve might try to get more information by considering the condition

o = (f 0 c)'(O)

for every cuiw c : (-E,&) + R2 with ~ ( 0 ) = (x, y), but it turns out that these conditions follow from (*), because of the chain rule.

To find maxima and minima for

we wish to proceed in an analogous way, by considering curves in Ihe sel ufa l l funclions f: [u, b] + R. This can be done by considering a "variation" of f, that is, a function

a : (-E, E) x [u, b] + R

such that

@(O, 1) = f U).

The functions 1 H a(u, I) are then a family of functions on ( - E , E) which pass through f for u = O. We will denote this function by &(u). Thus & is a function from (-E, E) to the set of functions f : [u, b] + R. If each ¿?(u) satisfies &(U) (u) = u', & (u)(b) = b', in other words if

for al1 u E (-E,&), then we cal1 a a variation of f keeping endpoints fixed For a variation a we now compute

S

k ( ? m m

"ci 1 % I

k l x m m

E 9

11 5 CQ

As is usual in classical notation, the arguments of functions are either put in indiscriminately or left out indiscriminately-in this case, not only are the arguments t and (1 , f ( r ) , f f ( t ) ) omitted (resulting in the disappearance of tht fuilction f for which we are solving), but the dependeiice of SJ on a is not indicated (which can make diings pretty confusing).

If f is to maximize or minimize J , then SJ(a) must be O for every variation cx of f keeping eiidpoints fixed. As in the case of 1-dimensional calculus, there is no reason to expect that the condition SJ(a) = O for al1 tr will imply that f is eveii a local maximum or miilimuin fos J , and h7e empliasize this by introduciiig a definition. \Ve caB f a critica1 point of J (01- an extremal for J) if SJ(a) = O for al1 variations a of f keeping endpoints fixed. The particular form (**) into which we have put SJ iiow allo~ts us to deduce an important condition.

8. THEOREM (EULER'S EQUATION). The C2 function f is a critica1 point of J if and only if f satisfies

JF ( r , , ( r ) ) = o . - 7 . f ' dr ay ax )

PROOI;: Clearly f must niake t l~e integral in (**) vanisli for e u q l

which vaiiishes at a and b. So the theorem is a consequence of the followiiig simple

8'. LERIMA. If a continuous function g : [u,b] + R satisfies

for every Cm fuiictioi~ q on [a,b] with ?](a) = q(b) = 0, tben g = 0.

PROOE Choose q to be #g where # is positive 011 (a , b ) and # ( a ) = # ( b ) = 0. +3

As an example, consider tlie case where F(r , s ,g ) = m. Tlie Eules cquatioil is

hence 0 = ( 1 + f r2)f" - ftj-'/ = ( 1 - f t + f12) f",

wliich implies that f" = O, so f is linear. Notice that we ~ ' o u l d have obtained the same result if we had considered t l~e

case F(t,x, y) = 1 + Y2, for then the Euler equatioii is simply

This is analogous to the situation in 1-dimensional cafculus, where the critica]

points of f i are the same as tliose of f, since

For tlie case of tlie surface of revolution, wliere F(l, x , Y ) = x J 1 + Y', [he Euler equation is

this Ieads to the equatioii

whic.11 we will a1so write in the cIassica1 form

To solve this, we use oiie of the Ko standard tricks (leaving justification of the details to t1ie reader). We let

Then

so our equation becomes

1 - log(l + p 2 ) = log y + constant 2

y = constant J;+pi

and thus (see Problem 20 for tlie definition and properties of tlie "liyperbolic cosine" function cosh and its inverse)

Replacing c by 1 / c , we write this as

The graph of ex + eYX

cosli x = 2

is sliown below; it is syn-imetric about the y-axis, decreasing for A- 5 0, and iricreasing for x 2 0.

So our surface must look like the one drawn below. It is, by the way, not trivial to decide whether there a7-e constan ts k and c which will make the graph of (*) pass through (u, a') and (b, b'). Problem 2 l investigates the special case where a' = b'.

It is easy to generalize these considerations to the case where f : [a, b] t Pn and

In this case \ve consider a: ( - E , & ) x [a, b] t Pn with G(0) = f, and compute that

Tlius, any critica1 point f of J must satisfy the n equations

We are ~iow going to apply these results to the problem of finding shortest paths in a manifold M. If y : [u, b] t M is a piecewise smooth cunte, with y ( a ) = p and y ( b ) = q , we define a variation of y to be a function

for come E > O, such that

(1) ~ ( 0 7 ~ ) = ~ ( 0 , (2) there is a partition a = ro < E] < . < EN = b of [a, b] so that a is Coo

on each strip (-E, E) X [~i - l , li].

\IZ1e cal1 a a vai-iatioii of y keeping endpoints fixed if

a (u ,a ) = p for al1 u E (-E, E).

@(u, b) = q

As before, we let G ( u ) be the path t H &(u, E). We would like to fiild which paths y satisfy

for a11 variations a keeping endpoints fixed. However, we will take a liint from oui. first exan-iple ai-id first fii-id the critica1 points for the "energy"

which has a much ilicer iiltegrand; aftenvards we will consider tlie relati011 between tlie two integrals.

\Ve can assumc that cach y 1 [ti-], Ei] 1ies i i i some coordiiiate system (x, U) (otheiurise we just refine the partition). If (u, E) is the standard coordinate system in (-E,&) x [a, b] we write

Then aa/aE(u,E) is the tangent vector at time E to tlie curve &(u) . If we adopt the abbreviations

then

If we usc tlie coor-diiiate system x to ideiitify U witli Rn, aiid consider t he g , as fuiictio~is o11 Rn, then we are considering

wliere 1

Il

F ( x , y ) = - 1 g G ( x ) . $ j j . 2 . .-

i l j - 1

arid

In arder to obtain a symmetrical looking result, we note that a little index juggling gives

From (***} we now obtain

d E (á.(u)l [di- I di]) 1

Remember that y is oniy piecewise Coo. Let

¡/Y + Y (ti-])

-(li ) = rigilt hand tangeilt vector of y at li clt ¡/Y -(ijr) = left lland tangent vector of y at ii. ¡/l

Notice that tlie fiilal sum iil the above formula is simplv

To abbi*eviate thc iii tegral somewhat we introduce the symbols

Tliese depend oii the coordinate system, but the integral

dZyr n dyi d y j x [ i j y l I ( ~ ( F ) 1 - 1 di ,

r=l i , j 2 1 dl di

which appears in our result, clearly cannot. Consequently, we will use the exact same expressiorl for each [ii-], ii], even though dXerent coordinate systems may actually be involved (and hence dXerent gij and y').

Now we just have to add up these results. Let

Then we obtain the following formula (where there is a convention being used in the integral).

9. THEOREM (FIRST VA RIATION FORh4ULA). For any variation a , we have

dzy' Il

I=i

(111 tlie case of a variation a leaving endpoints fixed, tlie sum can be written from 1 to N - 1.)

Tliis result is not very pretty, but there it is. It should be rloted that [ij, /] are noi the c.omponents of a tensor. Nevertheless, later on we will have an invariant iiiterpretation of the first variation forniula. For the time being we present, with apologies, this coordinate dependent appr-oach. From the first variation foi.inii1a it is, of course, simple to obtain conditions for critica1 points of E.

10. COROLLARY. If y : [ a , b ] + M is a Cm path, then y is a critical p i n t of E; if and only if for every coordinate system ( x , U ) we have

PR00E Suppose y is a critical point. Given t witli Y ( E ) E U, choose a partition of [ a , b ] with E E ( E j - , , l i ) for some i, and such that y 1 [ E i - ] , E j ] is in U. If a is a variation of y keeping endpoints fixed, then in the first variation formula \ve can assume that the part of the integral from E j - 1 to E j is written in terms of ( x , U ) . The final term in tlie formula vanishes since y is Cm. Now apply tlie me tliod of proof in Lemma 8', clioosing al1 aa l /au (0, 1 ) to be O, excep t one, wliich is O outside of ( E i - ] , t i ) , but a positive function times the term in brackets on (Ei-], 1 1 ) . *3

In order to put the equations of Corollary 10 in a standard form we introduce anothe r set of symbols

Our equations can now be written

We know from tlie standard theorem about systems of second order dairential equations (Problem 5-4), that for each p E M and each v E M,, there is a unique y : ( -E , E ) + M, for some E > O , such that y satisfies

Moreover, this y is Cm on (-E,&). This Iast fact shows that if y, : [O,&) + M and y2 : (-E, O] + M are Cm functions satisfying this equation, and if moreover

then y, and y2 togetfier give a Cm function on (-.$,E). Naturalfy, we coufd replace O by any other t . We now have the more precise result,

1 1. COROLLARJ7. A piecewise Coo path y : [a, b] + M is a critica1 point for E: if and only if y is actually Cm oii [o, b] and for every coordinate system (x, U ) satisfies . . .

for y ( 1 ) E U.

PROOE Let y be a critical point. Choosing the same u' as before (al1 u' are O outside of (ti-l, ti)), \ve see that y [ti ,] , ti] satisfies the equation, because tlie final tcrm in tlic first variation formula still vanishes, Now choose u so that

We already kiiow that the integra1 in tlie first variation formula vanishes. So we obtain

which irnplies that al1 AIi % are 0. By our previous remarks, this means that y

is actually Cm on al1 of [a, b]. 44

As t11e simplest possible case, consider tlie Eucljdean metric on Rn,

Here gij = 6,, so al1 agil/axk = 0, and r$ = O. Tlie critical poinis y for the energy function satisfy

d Z y k = o.

d12

Tlius y lies along a straiglit line, so y is a critica1 point for the length function as well. The situation is iiow quite dXerent from the first variational problem we considered, when we considered only curves of the form E H ( E , f ( E ) ) .

Any reparameterization of y is also a critical point for length, since length is independent ofparameterization (Problem 16). This shows that there are critical points for length which definitely aren't critical points for energy, since we have just seen tliat for y to be a critical point for energy, the component functions of y must be liiiear, and lience y niust be parameterized proportioiially to m- length. This situation always prevails.

12. THEOREM. If y : [a, b] + M is a critical point for E, then y is parameterized proportionally to arclength.

PROOE Obsen~e first, from tile definitions, tliai

Now we have

Replacing ag,/axl by tlie ~ ~ a l u e given above, tliis can be written as

Since y is a critical 11oiiic for E, I~otli terms in paientlieses are O (Coi-ollaiy 10). Tlius rlie leiigth lIdy/dilj is coiistant. <+

The formula

occurring in this proof will be used on several occasions later on. It will also be useful to know a formula for agij/axk. To derive onne, we first dXerentiate

to obtain

Thus we have

We can find the equations for critica1 points of the length function L in exactly the samc way as we treated the energy function. For the moment we consider only paths y : [u, b] + M with d y / d ~ # O everywhere. For the por- tioii y 1 [li-] , li] of y contained in a coordinate system (x, U), we have

dyf dyJ ( Y - 1 dl.

i, j=I

Coiisideriiig our coordinate system as Rn? we are now dealing with the case

NTe introduce the arclength function

S ( [ ) = G(Y). Then

After a little more calculation we finally obtain the equations for a critical point of L:

It is clear from this that critical points of E are also critical points of L (since they saiisfy d 2 s / d t 2 = O). Conversely, given a critical point y for L with d y / d t # O everywl-iere, the function

is a difGeomorp hism, and w e can consider tl-ie reparameterized cuive

y O S-' : [O, ~ : ( y ) ] + M.

This reparameterized curve is automatically also a critical point for L, so it inust satisfy tlie same differential equation. Since it is now parameterized by arclength, the lhird cerm vanishes, so y o S-' is a critical point for E.

Tl-iere is only one detail which remains unsettled. Conceivably a critical point for L might have a kink, but be Coo because it has a zero tangent vector

there, as in tlie figure below. In this case it ~tould not be possible to reparame-

tcrize y by arcleng-th. Problem 3'7 shows that this situation cannot arise. Henceforth we will cal1 a critical point of E a geodesic on M (for the Rie-

mannian metric ( , )). This name comes from the science of geodesy, which is concerned witli the measurement of the earth's surface, including surveying and the measurement of degrees of latitude and longitude. A geodesic on the earth's surface is a segrnent of a great circle, which is the shortest path between two points. Before we can say whether this is true for geodesics in general, wliich are so far merely known to be critical points for leng-th, we must initiate a local study of geodesics.

Tl-ie most elementary properties of geodesics depend only on facts about dxercntial cquations. Observe tliat the equations for a geodesic,

llave an important homogeneity property: if y is a geodesic, then r I+ y (c l ) is also clearly a geodesic. This feature of the cquatioil allows us to irnprove the result given by tlie basic existence aiid uniqueness theorems.

13. THEOREM. Let p E M. Then there is a neighborhood U of p and a number E > O such that for every q E U and every tangent vector v E Mq with lIvII < E there is a uniquc geodesic

satisfying d ~ u

Y,(O) = q, -j$OI = v.

PR00.F. The fundamental existencc and uniqueness theorem says that there is a neighborhood U of p and & ] , E * > O SO that for q E U and v E Mq with 1 1 ~ 1 1 < EI there is a unique geodesic

~ t i t h the required initial conditions. Clioose E < E] E*: Tlien if IvI < E and 11 [ < 2 we have

So we can define y, (E ) to be (~21). +f,

If v E M, is a vector for wl-iic1.i there is a geodesic

then \ve define tl-ie exponential of v to be

(Tl-ie reason for tliis termiiiology will be explaiiled in tl-ie next chapter.) The geodesic y can thus be described as

Since M, is an 11-diineiisioiial \lector space, iliere is a natural way to give ii a C* sii-ucture. If 13 c M, is tlle set of al1 \~eciors v E M, for whicli exp,(v) is defined, then the map

exp, : 13 + M

is Cm, since il-ie solutions of ilie differential equations for geodesics have a Cm flo~! 1dentif~ii-i~ the tangent space (M,), at v E M, with M, itself, \ve have ail ii-iduced map

(expq) v* : M, + MeXPq (U).

Iii. pariicular, we claim tliat the map

111 fact, io ol~rain a cuiw c ii-i rl-ie manifold M, with dc/d~(O) = v E M, = (Mq)o, we can let c(r) = tu . Theii exp, o c ( ~ ) = expq(tu), the geodesic witli iangent vector v at time O, so

(espq)o*(v) = - exp, (c(t)) = v.

PROOK Theorem 13 says that the vector O E Mp has a neighborhood V in the manifold TM such that exp is defined on V. Define the Cm function F: V + M x M by

F(u) = ( W , exp(u)).

Let (x, U) be a coordinate system around p. We will use the coordinate system

(f', . . . , i n , x l , - . .,Y),

described above, for lr-'(U). If rj: M x M -> M is projection on the ith

factor, then

is a coordinate system on U x U. Now, using the fact that

is the identity, it is not hard to see that at O E Mp we have

Consequently, F* is one-one at O E Mp, SO F maps some neighborhood V' of O diffeomorpliically onto some neighborhood of (p, p) E M x M. We may assume tllat V' consists of al1 vectors v E Mq with q in some neighborhood U' of p and [IvII < E. Cboose W to be a smaHer neighborhood of p for which F(V1) ii W x W. 43

Given a W as in [he theorem, and q E W, consider the geodesics through q of the form t I+ expq ( t v) for l[u 11 < E. These fill out Uq. The close analysis of geodesics depends on the following.

15. LEMMA (GAUSS' LEMMA). In U,, the geodesics through q are perpendicular to the hypersurfaces

{exp, (u) : 11 u11 = constant < E ) .

FIRST PR00E Le t u : IR + M, be a smoo th curve with 11 u (E) [I = a consta111 k < E for al1 E, and define

a(u, E) = exp, (u u([)) - l < u < l .

We are claiming that for every such a we have

A calculation precisely like tllat in tlle p-oof of Theorem 12 proves the foilowing equation, in which tlle arguments (u, E) and a(u, E) are omitted, for conveniente:

The first term on tlle right is O since each curve u H #(u, E) is a geodesic. Similarly, we obtain

wl~ich is just hice the second term on the right of (1). But aa/au(u, E) is just the tangent vector at time u to tlle geodesic u H exp,(u u(~) ) , where Ilu(~)Il = k; so ilaa/aull = k. Tlius the second term on the right of (2) is also O. So

( , isindependentofu.

But a(0, E) = expq (O) = q , SO aa/ar (O, r ) = O. It follows that

SECOA'D PROOF. Let u : R + Mq be any smooth cuive with 11 u([) 11 = a constant k < E for aii E, and define

B(u,E) = expq(l . v(u)) (note carefull~ the roles played by r and u).

Then fl is a variation of the geodesic y([) = expq(r u(O)), defined on [O, 11. By

tl-ie firsi variation formula, we have

tlie iniegral vanishing since y is a geodesic. But each curve j(u) has energy

16. COROLLARY. Lei c: [u,b] + Uq - { q ) be a piecewise smooth cuive,

for O < u(!) < E and /lu(E)lI = 1. Then

with equality if and only if u is monotonic and u is constant, so that c is a radial geodesic joiiiing IWO conceritric sphericai shells amund q.

PROOF If @(u, E) = exp, (u . v(E)), then C(E) = a (u(E), E) and

Since

with equality if and only if a ~ l / a i = O, and hence ~ ' ( r ) = O. Thus

~ ~ i t l i equality if and only if u is monotonic and v is constant. +:+

17. COROLLARY. Let W and E be as in Theorem 15, let y : [O, 11 + M be the geodcsic of leilgth < E joiiling q, q' E W, and let c : [O, 1 1 + M be any piecewise Coo path from q to q'. Then

with equality holding if and only if c is a reparameterization of y.

PROOE We can assume that q' = exp4(rv) E U, - {q} (otherwise break c up into smaller pieces). For S > 0, the path c must contain a s e p e n t which joins the splierical sliell of radius S to the spherical sliell of radius r, and lies between them. By Corollary 16, the Iength of this segment lias length 2 r - S. So the length of c is 2 r , and clearly c must be a reparameterization of y for equality to hold. +:+

\Ve thus see that suficimi~ smallpieces of geodesics are minimal paths for arclength. We can use Corollary 17 to determine the geodesics on a few simple surfaces, without any computations, if we first introduce a notion which will play a crucial role later. If ( M , ( , )) and (M' , ( , 1') are Cm manifolds with Riemannian metrics, then a one-one Cm function f : M + M' is called an isometry of M into M' if f*( , )' = ( , ) . For example, ~flection through a plane E* c Rn+' is an isometry I : S" + S". It is clear that if c : [O, 11 + M is a Cm cunte, then the length of c with respect to ( , ) is the length of f 0 c with respect to ( , )'; and if c is a geodesic, then f o c is likewise a geodesic.

For the isometry I : S" + S" mentjoned above, the fixed point set is the great circle C = S" n E*. Let p, q E C be two points with a unique geodesic C' of minimal length between them. Then I (C f ) is a geodesic of the same length as C f between I ( p ) = p and I ( q ) = q. So C' = I (C f ) , which implies thai C' c C, so that C ¡S a geodesic. Siiice there is a great circle through any point of S" in any given direction, these are al1 the geodesics.

Notice tliat a portion of a great circle ~ l i i c h is larger than a semi-circle is defiliitely not of minimal length, evaz among t iearb pa0i.r. Antipodal points on

tlie sphere have a continuum of geodesics of minimal length between them. Al1 o~lier pairs of points have a unique geodesic of minimal length between tliem b i i~ a11 infii~ite family of non-minimal geodesics, depending on how many times tl-ie geodesic goes around tlie spliei-e and in which direction it starts.

The geodesics on a right circular cylinder Z are the generating lines, the

ciirles cut by planes perpendicular to the generating lines, and the helices on 2. In fact, if L is a generating line of Z, tl-ien we can set up an isometry I : Z - L -t R2 by rolling Z onto R2. Tl-ie geodesics on Z are just the images

ui-ider 1-' of tl-ie straight lines in R2. Two points on Z have infinitely many geodesics be tween them.

We are now in a position to wind up our discussion of Riemannian mei- 1-ics on M by establisl-iiilg an important coililection between the Riemannian metric ( , ) and the metric d : M x M + R it determines,

d ( p , q ) = inf{L(y) : y a piecewise smooth cunte from p to q}

Notice that on both tlie spl-iere and tl-ie it-ifinite cylii-ider every geodesic y defined on an iiiterval [u, b] can be extended to a geodesic defined on al1 of R. This is false on a cylinder of bounded l-ieight, a bounded portion of Rn, or Rn - {O]. l n general, a manifold M wit1-i a Riemai-inian metric ( , ) is called geodesically complete if eveiy geodesic y : [u, b] + M can be extended to a geodesic from R to M.

18. THEOREM (HOPF-RINOW-DE RHAM). If ( , ) is a Riemannian metric on M, then M is geodesically complete if and only if M is complete in the metric d determined by ( , ). Moreover, any two points in a geodesically complete manifold can be joined by a geodesic of minimal length.

PROOF. Suppose M is geodesically complete. Given p, q E M with d(p, q) = r > O, choose Up as in Theorem 14. Let S c U, be the spherical shell of radius S < E. There is a point

on S such that d(po, q) 5 d(s, q) for al1 S E S . We claim that

this will show that the geodesic y(l) = expp(rv) is a geodesic of minimal length between p aild q. To prove this result, we will prove that

First of all, since every cunte from p to q must intersect S, we clearly have

So d(po, q) = i - - S. This proves that (**) holds for E = S. Now let 10 E [S, 1.1 be tl-ie least upper bound of al1 1 for which (**) holds. Then

(**) llolds for 10 also, by continuity. Suppose 10 < r. Let Sr be a spherical shell

of radius S' around y (10) and le t E S' be a point closest to q. Tlleil

d(y(lo),q) = mil l (d(~(~o) ,s) + d(s,q)) = 6' +d(po', cl), SES'

Hence d ( p , po') > d ( p q 9 ) - d ( p ~ ' 9 9) = lo + 6'.

Bui tlie path c obtaiiied by following y from p to ~ ( 1 0 ) and then the minimal geodesic from y (10) to por l i s length precisely 10 + 6'. So c is a path of minimal Icngtli, and must therefore be a geodesic, ~ f h i c h means that it coincides with y. Hence

y(I0 + 6') = po'.

Hence (***) gives d ( ~ ( 1 0 + 6 ' ) , q ) = r - ( ~ g + 6 ' ) ,

sliowing that (**) holds for 10 + S'. Tliis contradicts t he choice of 10, so it must be tliat 10 = i-. In other words, (**) Iiolds for 1 = r, which proves (*).

From tliis result, it follows easily that M is complete witli the metric d . In fact, if A c M has diameter D , and p E A , then the map expp: Mp + M maps the closed disc of radius D in Mp onto a compact set containing A. 111 otlier words, bounded subsets of M have compact closure. From this it is clear tliat Cauchy sequences converge.

Conversely, suppose M is complete as a metric space. Given any geodesic y : (u, b ) + M, choose E, + b. Clcarly ~ ( 1 , ) is a Caucliy sequence in M, so ii coiivesges io some point p E M. Using Theorem 14, it is not difficult to show that y can be cxtcndcd past b. Consequently, by a least upper bound argument, any geodesic can be extended to R. *:*

As a particular consequclice of Thcorem 18, note that tliere is always a minimal geodesic joiniiig any two points of a compact manifold.

ADDENDUM

TUBULAR NEJGHBORHOODS

Lei M" c N"+k be a submanifold of N, with i : M + N the inclusion map, so that for every p E M we have i*( M,) C N, . If ( , ) is a Riemannian metric for N, tlien we can define M,' C N, as

Lci E = U M,' and m: E + M t a k e ~ , ' t o p .

Ir. is not hard to see that v = -¿ir : E + M is a k-plane bundle over M, the normal bundle of M in N.

For example, the normal bundle v of S"-' E R" is the trivial 1-plane bundle, for v Iias a sectioii consistiiig of unit ouiwasd iiormal vectors. On the oiher haiid

if M is tlle Mobius strip and S' c M is a circle around tlie center, then it ir noi hard io see ihai tl-ie iiormal bundle v will be isomorpbic to the (non-tiivial) I~uridle M + S'. If we consider S' C M c P', then the normal bundle

of S' iii IP2 is exactb' tlie same as tlie iiormal 11uiidle of S' in M, so ii too ir non-trivial.

Our aim is to prove that for compact M the normal bundle of M in N is always equivalent to a bundle a : U + M for which U is an open neighborhood of M in N, and for which the O-section S : M + U is just the inclusion of M into U. In the case where N is the total space of a bundle over M, this open neighborhood can be taken to be the whole total space. But in general tlie neighborhood canilot be al1 of N. For example, as an appropriate neighborhood of S' c IR2 we can choose IR2 - {O}.

A bundle rr : U + M with U an open neigl-iborhood of M in N, for wliich the O-section s : M + U is the inclusion of M in U, is called a tubular neighborhood of M in N, Before proving the existente of tubular neighborhoods, we add some remarks and a Lemma.

If a : U + M is a tubular neighborhood, then c1early

rr o s = identity of M,

s o a is smoothly homotopic to the identity of U,

so a is a deformation retraction, and Hk(U) * H k ( ~ ) ; thus M has the same de Rllam col~omology as an open neighborhood. Moreover, if we choose a Riemailnian metric ( , ) for a : U +- M and define D = {e E U : (@,e) 5 1): tllen D is a submanifold-with-boundary of U, and the map 7110: D + M is also a deformation retraction. So M also has the same de Rharn col~omolog-)~ as a dosed neighborhood.

1 9. LEMMA. Let X be a compact metric space and Xo c X a closed subset. Let f : X + Y be a local homeomorphism such that f 1x0 is one-one. Then there is a neigl-iborhood U of Xo such that f IU is oae-one.

( (x ,y) E X x X : x # y and f(x) = $(y)).

Then C is closed, for if (x,, y,) is a sequence in C with x, + x and y, + y, then f (x) = lim J'(x,) = lim f (y,) = f (y), alid also x # y since f is locally one-one.

If g : C + R is g(x,y) = d(s , Xo) + d ( y , Xo), then g > O on C. Since C is compact, there is E > O such that g > 2~ on C. Then f is one-one on the E-neigl-iborhood of Xo. 43

20. THEOREM. Let M c N be a compact submanifold of N . Then M has a tubular neigliborhood T : U + M in N: which is equivalent to the normal bundle of M in N.

PROOF, Choose a Riemanniaii men-ic ( , ) for N , with tlie corresponding norm 11 11, and metric d : N x N + R . Lei

It follows easily from Theorem 13, and compactness of M, tl-iat exp is defined o11 E, for sufficieii tly small E > O. We claim that for sufliciently sm al1 E, the map exp is a diíTeomorpliism from E, onto U,. This will clearly prove the theorem.

Let V c E be the set of a non-critica1 points for exp. Then V 3i M (considered as a subset of E via thc O-section), and VI = V n El is compact; since exp is one-one on M c I/i, it follows from Lemma 19 that for sufficiendy small E

tlie map exp is a difTeomorphism on E,. It is clear also that exp(E,) c U,. To prove tliat exp is onto U,, choose any

q E U,, and a poirit p E M closest to q . If y : [O, 11 + N is the geodesic of 1engtl-i < E witli y(0) = p and y(1) = q , it is easy to see that y is perpendicular to M at p (compare the second proof of Gauss' Lemma). This means that q = exppdy/dt(0) where dy/d1(0) E E,. 43

O11c of tlie ii-iiei-esting features of Theorem 20 is that al1 the paraphernalia of Riemanriian metrics and geodesics are used in its proof, while they do not even appear in tlle statement. Theorem 20 will be rieeded only in Cl-iapter 11, wl-iere we will also need the following modification.

Rienzalzn ian Adel?.ics 347

2 1. THEOREh4. Let N be a manifold-with-boundary, with compact boundary aN. Theii aN has (arbitral-illl small) opeii [and dosed] neighboiqlioods for which there are deformation re tractjons onto aN.

PRO0.F Exactly the same as the proof ofTlieorem 20, using only inward poiiiting normal vectors. +3

PROBLEMS

1. Let V be a \lector space over a field F of characteristic # 2, and let h : V x V + F be srmmetric and bilinear.

(a) Define q : V + F by q(v) = h(v, u ) . Show that if e l , . . . , #n is a basis for V*, then

n

for SOme U i j . (b) Show that

(e) Suppose q : 1/ + F satisfies q(-u) = u, and that h(u, u) = q(u+ v) - q(u) - q(v) is bilinear. Show that

Conclude that q(0) = 0, and q(2u) - 4q(u). Tl-ien show that q(v) = h(u, u).

2. Let ( , ) be a Euclidean metric for V*. Suppose #i, Sri E V* satisfy #i A

m - m A #k = Sri A - A # O, and 1et W4 and W* be the subspaces of V* spanned by the ei and Sri.

(a) Show tl-iat w E W4 if and only if w A #i A . - . A #k = O. Conclude that w4 = w*. (b) Let u] , . . . , uk be aii or~lionoi-mal basis of WQ = Wy . If #i = & U j i U j ,

s11ow that the signed k-dimensioiia1 volume of the parallelepiped spanned by el , . . . , #k is det(aij). (Tlie sign is + if #i,. . . , Iias the same orientation as 01, . . . , Ok, and - otherwise.) (c) Usirlg Problein 7-9, sliow that tliis volume is the same for Srl ,. . . , @k.

(d) Conversely, if W4 = W*, and the si\gned volumes of the parallelepipeds are the same, sliow that #] A - A #k = A . - . A Srk.

If we ideiitify V witll V**, so tliat we llave a wedge product vi A- - - A vk of vectors vi E V, tlieil we liave a geometi-ic coiiditioii for equality with wl A . A wk. 111 L p n s sur la GWniArie des Q l i c e s de Rieinliiiii, É. Cartan uses this condition to d@n~ nk ( Y * ) as formal sums of equivaleiice classes of k vectors; he deduces geornen-ically i he coi~espondiiig conditions on the coordinates of vi, wi .

3. Let V be an it-dimensioi-ial vector space, and ( , ) an inner product on V wliich is iiot riecessarily positive defiiiite. A basis v i , . . . , u , for V is called orthonormal if (vi, v j ) = f 6,.

(a) If V # {O}, then there is a vector u E V with (u, u) # 0. (b) For W C V, let W' = (u E V : (v, W ) = O for al1 w E W}. Prove that dim W' 2 n - dim W. Hini: If {wi} is a basis for W, consider the linear functionals hí : V + R defined by h i ( ~ ) = (U, wi). (e) If ( , ) is non-degenerate on W, then V = W @ w', and ( , ) is also non-degenerate on w'. (d) V has an orthonormal basis. Thus, there is an isomorphism f : R" + V with f * ( , ) = ( , ), for some r (the inner product ( , ), is defined on page 301). (e) Tbe index of ( , ) is the largest dirnension of a subspace W c V such that ( , ) 1 W is negative definite. Show that the index is n - r , thus showing that r is unique ("Sylvester's Law of Inertia9').

4. Let ( , ) be a (possibly non-positive definite) inner product on V, and let vi,. . . , u, be an orthonormal basis (see Problem 3). Define an inner product ( , l k on nk (v) by requiring that

be an orthonormal basis, wit1-i

(a) Sliow tliat ( , l k is independent of the basis vi,. . . , uk. (Use Problem 7-16,) (b) S how thai

(c) If ( , ) has index i , theii

( u * ~ A . . A u*,, u*] A " . A U*,)" = (-lli.

(d) For tliose wlio k n w about @ and iIk. Usiiig the isomorphisms

(gkv)* and i Ik (v*) 4 ( i Ikv)* , define inner products on and iIkv by using tlie isomorphism V +- V* given by the ii-iner product on V. Show that tl-iese ii-ii-ier products agree witl-i the oi-ies defined above.

5. Recall tlie definition of u1 x x u,-1 in Problem 7-26.

(a) Sliow tliat (VI x - - - x v,-1, vi) =: O. (b) S ~ O M ' tli¿~t I u ] X * - X U,,-1 1 = l/deto, wliere gij = (ui, uj). H&L: Apply tlie result oi-i page 308 to a certain (11 - 1)-dimensional subspace of R".

6. Let c =: 71 : E += B be a vector bundle. An indefinite metric on 6 is a continuous choice of a non-positive definite inner product ( , )p on each 71-'(p). Show that the index of ( , )p is constant on each component of B.

7. This problem requires a li ttle knowledge of simple-connec tedness and covering spaces.

(a) There is no way of continuously c hoosing a 1 -dimensional subspace of S%, for each p E s2. (Consider the space consisting of the two unit vectors in each su bsp ace .) @) There is no Riemannian metric of index 1 on s 2 .

8. Let ( , ) and ( , )' be two Riemannian metrics on a vector bundle < = 71 : E + B. Let S be the set of e E E with (e,e) = 1, and define S' simiIar1y. Show that S is homeomorphic to S'. If { is a smooth bundle over a manifold M, show that S is dHeomorphic to S'.

9. Show by a computation tliat if tlie functions gij and gtij are related by

~ i t l l det(gjj) # 0, and the fiinctions gij, grij are defined by

then

Tliis, of course, is the classical way of defining the tensor [Iiaving the components] gV.

10. (a) Let ( , ) be a Riemannian metric on M, and A a teiisor of type (I), so tliai A@): Mp += Mp. Define a tensor B of type (0) by

If the expression for A in a coordinate system is

sliow that B = Bik dxi @ dxk, wliere

@) Similarly, define a tensor C of type (2) by

Show that if C has components ck/, then

The teiisors B and C are said to be obtained from A by "raising and lowering indices".

11. (a) Let Xl, . . . , X, be linearly independent vector fields on a manifold M with a Riemannian metric ( , ). Show that the Grarn-Schmidt process can be applied to the vector fields al1 at once, so that we obtain n everywhere orthonormal vec tor fields Y], . . . , Yn. (b) For the case of a non-positive definite metric, fii-id Y], . . . , Y" with (Yi, Yj) =

k 8 i j in a neighborhood of any point.

12. (a) If f : [u, b] + R is positive, show that tl-ie area of the surface obtained by revolving the graph of f around the x-axis is

lb 2 a f J G i .

(b) Compute tlie area of s2. 13. Let M c R" be an (11 - ])-dimensional submanifold with orientation p. The outward unit normal v(p) at p E M is defined to be that vector in Rnp of leilgth 1 such that v(p), ( u ] ) ~ , . . . , is positively oriented in IWP when (U] lp, . . . , )p is positively oriented in Mp.

(a) If M = ¿lN for a11 n-dimeilsional manifold-witl-i-boundary N C R", then v(p) is outward pointing in the sense of Chapter 8. (b) Let d V,.,-] Ix the volume element of M determined by the Riemannian metric it acquires as a submanifold of IR". S how that if we consider v (p) as an element of R", then

dVn-* (p) ((u1 Ip,. , (%-1 lp) = det j. un-1

Conclude tllat dVn-1 (p) is the restriction to Mp of

i=l (c) Note tliat VI x - x v,-1 = olv(p) for some <Y E W (by Problem 5). Sliow that for w E IR" we have

(w, v(p)) - (VI X - - - X un-1,v(p)) = (w,v1 x - - - x un-1).

Conclude that

vi(p) . d Vn-l (p) = resti-iction to Mp of - ( - ~ ) ' - ~ d x ~ ( ~ ) A ... ~ d x > ( p ) A . . . i \dxn(p) .

(d) Let M c IRn Ile a compact ri-dimensional maiiifold-with-boundary with v tl-ie outward unit normal on aM. Denote the volume element of M by dV,, and that of aM by Let X = xi uia/ax-' be a vector field on M. Prove th e Diuergence T/ieurem:

/M div X dVn = JdM (x, u ) dVn-l

(the function div X is dcfined in Problem 7-27) Hint: Consider the form w on M defined by

(e) Let M c lR3 be a compact 2-dimerisional manifold-with-boundary, with . . oi-ientation u, and outui7ard-unit normal v. Let T be the vector field on aM coilsisti~lg of positiidy orieir ted unit vectors. Dci.io te tlie volume element of M by dA, arid tliat of aM Ily ds. Let X be a vector field on M. Prove (the original) Stokes ' TIzeurem :

/M (V x X, u ) dA = (X, T ) ds

(V x X is defined in Prol~lem 7-27). J d M

14. (a) Let I ~ e t l ~ e i~olume of tlie uiiit ball in IRn. Show tliat

1

(b) If 1. = ll ( 1 - x2)("-')/' dx, show that

(c) Using Vi = 2, V2 = x, show that

n even

~ ( n + r ) / 2 ~ (n-1)/2 n odd.

1 - 3 - 5 - - - r i n"/2

(In terms of the r function, this can be written r(1+ n/2) (d) Let A.-I be the 01 - 1)-volume of S"-'. Using the method of proof in Corollary 8-8, but reversing the order of integration, show that

(e) Obtain tliis same result by applying the Divergence Tlieorem (Problem 13), with X ( p ) = pp.

15. (a) Let c : [O, 11 + IRn be a differentiable curve, where IRn has the usual Riemannian metric ( , ) = xi dxi @ dx'. Sliow that

(b) For tlie special case c : [O, 11 + IR2 $ven Ily c ( t ) = ( t , f ( t )), show that this length ,

i s tbe least upper bound of the lengths of inscribed polygonal curves.

Hint: If the inscribed polygonal cunre is determined by the points ( t i , ~ ( t i ) ) for

a partition O = to < - < E, = 1 of [O, 11, then we have

for some ti E [ti-1, ti].

(c) Prove tlle same result in the genera1 case. Hi7zi: Use the results of Prob- Iem &l, and uniform continuity of J- on a compact set.

It is iiatural lo suppose tliat the area of a surface is, similarly, the least upper bound of the areas of inscribed polygonal surfaces, but as H. Schwarz first obsened, tliis lcast upper bound is infinite for a bounded portion of a cylinder! To illustrate Schwarz's example 1 have plagiarized t l ~ e following picture from a book called Maazi~~u?nuuccitu.Ü A ~ m w ita M~ozoohj1a3urix, wntten by someone called M. C I I M I ~ ~ K .

Top view

To ir-icrrasc t l ~ e number of triangles, we maintain the hexagonal arrangement, t~i i i inove t l~e planes of tlie l-iexago~is closer together, so that the triai~gles are more r~early ir1 a plaiie parallel to the bases of the cylinder. 311 tliis way, we can increase the ilum ber of triaiigles iridefinitely, wl-iile t1-i e area of each approaches h1/2.

Tl-ie topic of suriace area for non-differen~iable sur-faces is a complex one, wliicb we will not go into here.

16. Let c : [O, 11 + M be a curve in a manifold M with a Riemannian metric ( , ). lf p : [O, 1 1 + [O, 1 1 is a diffeomorphism, show that

L(c) = L(c 0 p).

17. Show that the metric d on M may be defined using Cm, instead of piecewise Cm curves. (Show how to round offcorners of a piecewise Cm path so that tlie length increases by less than any @ven E > O; remember that the formula for length involves only first derivatives.)

18. (a) Let B c M be liomeomorphic to the ball ( p E Rn : Ipi 5 1 ) and let S c M be the subset corresponding to ( p E IRn : j p l = 1 ) . Show that M - S is disconnected, by showing that M - B and B - S are disjoint open subsets of M - S. (b) If p E B - S and q E M - B, show that d(p, q) 2 min d (p, q'). Use

g1eS

tliis fact and Lemma 7' to complete the proof of Theorem T . (In the theory of infinite dimensional manifolds, tl-iese details become quite important, for M - S does no, liave to be disconnected, and Theorem T is false.)

19. (a) By applying integration by parts to the equation on pages 318-319, show that

- S' -(t. f (t), f '(1)) dr] d i ; a ax

tbis result makes cense even if f is only C' . (b) Du Bois R~mond i hiima. If a con tinuous furlction g on [a, b] satisfies

for al1 Cm functions q on [a,b] with q(a) = q(b) = O, then g is a constant. Hi711: The constant c must be

We clearly llave

/ob "(t)[g(t) - -rl di = O,

so we need to find a suitable q with q3(t) = g(t) - c. (c) Coiiclude tliat if the C' func~ion f is a critica1 point of J, then f still saósfies the Euler equations (wliicli are not oprioioA meaningful if f is not c2).

20. The hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions sinh, cosh, and tanh are defined by

ex - ,-x ex + eeX sinh x s inhx = 3 cosh x =

3 ' t anhx = -.

2 - cosh x

(a) Graph sinh, cosh, and tanh. (b) Show that

2 cosh2 - sinh = 1

tanh2+l/cosh2 = 1

sinh(x + y) = sinh x cosh y + coshx sinh y

cosh (x + y) = cosh x cosh y + sinh x sinh y

sinh' = cosh

cosh' = sinh.

(c) For those who know about complex power series:

sin ix s inhx = -, cosh x = cos ix .

i

(d) The iiiverse fuiictions of sinh and tanh are denoted by sinh-' and tanh-', respectively, while cosh-' denotes the inverse of cosh 1 [O, m). Show that

21. Consider tl-ie problem of findii-ig a surface of revolution joining two circles of radius 1, situated, for conveniei-ice, at a and -a. We are looking for a function

Rie~nann ian Mehics

of the form x

f (x) = ccosh - C

where c is supposed to satis+

u ~ c o s h - = 1 (e > O).

C

(a) There is a unique yo > O wit11 tanh yo = 1 /yo. Examine the sign of 1 / y - tanh y for y > 0. (b) Examine the sign of coshy - Ysinhy for y > 0. (c) Let

u A , (c) = c cosh - c > O.

C

Show that A , has a minimum at alyo, find the value of A , there, and sketch the graph. (d) There exists c with c coshu/c = 1 if and only if u 5 yo/ cosh yo. If a =

yo/ cosh yo, tlien tl~ere is a unique such c, namely c = u/yo = 1 / cosh yo. If u < yo/ coshyo, then there are two such c, with c, < a/yo < C?. It turns out that tl-ie surface for cz has smaller area.

-yo/ cosh yo -a

(e) Using Problem 20(d), show that

Yo - cosh yo

-m.

[Tliese plieriomena can be pictured more easily if we use the notion of an envelope-c.f. Volume 111, pp. 176K The envelope of the l-parameter famill. of cunes

1- f, (x) = c cosh -

C

is determiried Ily solving the equations

af, (x) X x x O = - = cosh - - - sinh -.

M'e obtain X x - = f 3'0, y = ccosh- = ccosh yo, C C

so tlie envclope conskts of the straight lines

cosh yo y = + X .

Yo

Tlie uriique member. of tlie family througli (yo/ cosh yo, 1 ) is tangent to tlie erivelope at tliat ~oi r i t , I;br a < yo/ cosh 170, tlie grapli of f,, is tangent to

tire envelope at poirits P, Q E (-a,a), but the grapli of h2 is tangeiit to tlic cnvelope at points outside [-u, a]. Tl-ie poirit Q is called conjugate to P along tlie exti-emal f,, , and it is slrown in thc calculus of variations tliat tlie existericc of this conjugate point implies tl~at the portion of f,, from P to (o, 1) does no1

@ve a local minimum for \ f m. [Compare with the discussion of J

conjugate points of a geodesic in Volume IV, Chapter 8, and note the remark on pg. V396.)]

22. All of our illustratioiis of calculus of variations problems involved an F wliich does not involve t , so tliat the Euler equations are actually

(a) Show that for any f and 1;: R2 + IR we have

d - dt ( F - y':) = f f [ - -LE] aF a x dt a y ,

and conclude tl-iat the extremals for our problem satisfy

(b) Apply tliis to F(A-, y) = x m to obtain directly the equation d y / d x =

JCJ* - I which we evelitually obtained in our solution to the problem.

23. (a) Let x and x' be two coordinate systems, witli corresponding gij and gfil for the expression of a Riemannian metric. Show that

(11) For the c.or.responding [ij, k] and [@, y]', show that

so tliat [ij, k] are not tlie components of a tensor. (c) Also show tl-iat

24. Sliow that any Cm so-uctur-e on R is difTeomorphic to the usual Cm structure. (Consider tlie arcleligth function on a geodesíc for some Riemannian metric on R.)

25. Let ( , ) = x i d x i @ dxi be the usual Riemannian metric on R", and let Eitj gildui @ d u j be another metric, where u', . . . , u" again denotes the standard coordinate system on Rn. Suppose we are told that there is a difKeo- morphism f : Rn -t R" such that E , gij du' @ dul = f * ( , ) . How can we go about finding f ?

(a) Let a f /aui = ei : Rn -t R.. If we consider ei as a \rector field on Rn, show that f,(a/auf) = ei.

(b) Show that gij = (ei, e j ) .

(c) To solve for f it is, iii theory at least, sufficient to solve for the ei, and to solve for tliese we want to find dflerential equations

satisfied by t l~e ri's Show that we must have

Il def

Bi),k = E gkr~; , - - r=l 1=1

(d) Show thai

(e) By cyclically permuting i, j , k, deduce tbat

, so that A; = r&. I n Lqons sur In Géo7nétrú des Espaces de Riemnmr, E. Cartan uses tliis approacli to motivate the introduction of the r$. ( f ) Deduce the result A;] = fh directly from our equations for a geodesic. (Note tliat tlie cuntes obtained by setting al1 but one f' constant are geodesia, siiice they correspond to lines parallel to the xi-axis.)

26. If (Vf, ( , )') and (V", ( , )") are two vector spaces with inner products, wedefine ( , ) on V = V f @ V" by

(a) Show that ( , ) is an inner product. (b) Given Riemannian metrics on M and N, it follows that there is a natural way to put a Riemannian met ic on M x N. Describe the geodesics on M x N for this metric.

27. (a) Let y : [a,b] + M be a geodesic, and let p ; [a,B] + [a, b] be a diffeomorphism. Sl-iow that c = y o p satisfies

(b) Conversely, if c satisfies this equation, then y is a geodesic. (c) If c satjsfies

d2ck Il

dci dcJ dck - dt2 + E r$ ( ~ ( t ) ) ~ ~ = x p ( t ) for p : IR+ IR,

i, j==1

then c is a reparameterization of a geodesic. (Tlie equation p"(t) = p'(t)p(t) can be solved explicitly: p(t) = J' e M ( S ) ds, where M1(s) = p(s) .)

28. Let c be a curve in M with d t l d t # O eveqwhere, and consider the hy- persurf aces

{exp,(,)u : Il u ll = constant, where u E M,(,) with (u, dc/dt) = 0).

Show tliat for u E M,(r) with (u, dcldt) = O, the geodesic u H exp,(*)u . u is perpendicular to these liypersurfaces. (Gauss' Lemma is the "special case" where c is constant.)

29. Let y: [u, b] + M be a geodesic with ~ ( u ) = p, and suppose that expp is a difTeomorpliism on a neighborhood 13 c Mp of (tyt(0) : O 5 t 5 1). Show that y is a cunre of minimal length between p and q = y(b), among al1 curves in exp(0). (Gauss' Lemma still works on exp(O).)

30. If ( , ) is a Riemannian metric on M and d : M x M + IR is tlie corresponding metric, then a curve y : [u, b] + M with d(y (a), y (b)) = L (y) is a geodesic.

31. Scflwal-z's ulequalilif for continuous functions states that

witli equality if and only if J' and g are linearly dependent (over IR).

(a) Prwe Schwarz's inequality by imitating the proof of Theorem l(2). (b) For any curve y show tliat

with equality if and orlly if y is parameterized proportionally to arclength. (c) Let y : [u,b] + M be a geodesic with L : (~) = d(y(o), y(b)). If c(u) = y(u) and c(b) = y(b), show that

Conclude that E(y) < E (e) unless c is also a geodesic with

111 particular, sufficiently small pieces of a geodesic miiiimize energy.

32. Let p be a poilit of a niaiiifold M with a Rieniannian metric ( , ) . Clioosc a basis vi,. . . , v, of Mp, SO tliat we have a "rectangular" coordinate system o11 Mp given by xi aivi I+ (ui, . . . ,un); let x be tlie coordinate system ~o exp-l. defined in a neighborhood U of p.

(a) Sliow tliat in tliis coordi~iate system we llave r$ (p) = O. Hini: Recall the equations for a geodesic, aiid note tliat a geodesic y ilirough p is just exp composed with a straight line through O in M,, so that each y k is linear.

(b) Let r : U + W be r(q) = d@,q), so that r o y = Ck(yk)2 . Show that

(c) Note that

Using part (a), conclude that if I l y'(0) 11 is sufficiently small, then

so that d (r o y)2/dr is strictly increasing in a neighborhood of O. (d) Let 3, = { U E M p : llull 5 E) and SE = { U E Mp : llull = E). Show that tlie followirig is true for al1 sufficiently small E > O: if y is a geodesic such that y(0) E exp(SE) and such that yt(0) is tangent to exp(S,), then there is 6 > O (dependiiig on y) such that y(t) # exp(BE) for O # 1 E (-S,S). Hznt: If y'(0) is

tar~gent to exp(S,), then d(r o y)/dr = 0. (e) Let q and q' be two points witl-i r (q), r(q') < E and let y be the unique geodesic of Iength < 2~ joining them. Show that for sufficiently small E the maximum of r o y occurs at either q or q'. (f) A set U c M is geodesically convex if every pair q, q' E U has a unique geodesic of n~inimum length between them, and this geodesic lies completely in U. Show that exp ( ( u E Mp : 1 1 u 1 1 < E)) is geodesically convex for sufficiently small E > 0. (g) Let J.: U + W" be a diffeomor-pl-iism of a neighborhood U of O E R" into R". Sl-iow that for sufficiently small E, the inlage of the open E-ball is convex.

33. (a) There is an everywhere daerentiable curve c ( t ) = ( t , f ( 1 ) ) in IR2 such that

leng-th of c 1 [O, h] lim h-O lc ( h ) - ~ ( 0 ) 1 # 1 -

Hint: Make c look something like the following picture.

1

length fmm (f ,O) to (170)

length fmm

. . . ,

(b) Consider the situation in Corollary 15, except that c ( t ) = q if and only if t = U , and suppose u r ( t ) > O for r near O. If c is c', then v ( t ) approaches a limit as t + O (even though v(0) is undefined). Show that if c is c', then there is some K > O such that for al1 r near O we have

Hnt: In Mq we clearly have

Since expq is locally a difTeomorp11ism there are O < K I < K2 such that

for al1 tangent vectors v at points near q. (c) Conclude that

2 lh J u ' ( t ) 2 + I $ ( U ( t ) , 1 ) 11 di lirn L(c I [O, I11) - = lim - 1 . h-O d ( p , ~ ( h ) ) h-0 u ( h )

(d) If e is c', show that L(c) is the least upper bound of inscribed ~iecewise geodesic curves.

34. (a) Using the methods of Problem 33, show that if c is the straight line joining u, w E Mp, then

L (exp o e) lim = 1.

v,w+o L,(c)

(b) Siinilarly, if yviW is the unique geodesic jojnjng exp(v) and exp(w), and Yv. w = exp o cu,,, then

lim L ( Y ~ , w) = 1. v,w+o L(cuiW)

(c) Conclude that d (exp u, exp w )

ljm = 1. v,w+O flu-wlf

35. Let f : M + N be an jsometry. Show that f is an isometry of the metric space structures determined on M and N by their respective Riemannian

36. Let M be a manifold with Riemannian metric ( , ) and corresponding metric d. Let f : M + M be a map of M onto itself which preserves the metric d.

(a) If y is a geodesic, then f o y is a geodesic. (b) Define f t : Mp + MAp) as follows: For y a geodesic with y(0) = p, let

Show that f f f '(X) f f = f f X 11, and that f'(c X) = c f '(X). (c) Given X, Y E Mp, use Problem 34 to sliow that

- - IIxI12 + IfU12 - lim [d(exptX,expr~)I2 flxll - IlYlI r-0 IlrX lf . 111 y 11

Conclude tliat (X, Y), = (f '(X), f '(Y))J(,), and then that f'(X + Y) = f '(X) + f ' (Y1. (d) Par t (c) sho~vs tfiat f ' : M, + Ml(,) is a diffeomorphism. Use this to show that f is itself a diíTeomorphism, and hence an isometiy.

37. (a) For u , w E Rn ulth w # O, show that

lim IIv+lwII-IIvlI - -- ( v , ~ ) 1-0 1 llvll

The same result tlien Iiolds in aiiy vector space with a Euclidean metric ( , ). Hin~: If u : Rn + R is the norm, tlien tlie limit iS Du(v)(w). Alternately, one can use the equatiori (u, u ) = ijuli . iivii - cose wliere 8 is the arigle between u and v . (b) Coiiclude tliat if w js liriearly independent of u, then

lim iIv+fwll - llvll - lltwll &

(e) Let y : [O, 11 + M be a piecewise CI critical poirit for lerigth, aiid suppose that yl(to+) # yl(lo-) for some 10 E (O, 1). Clioose r l < 10 and consider the varjatioii cr for wliicli 6 ( 2 1 ) is obtairied by following y up to 11 , then tlie unique geodesic fi-om y(fI) to y(lo + u), and firially the rest of y. S110~7 that if 11 is

close eiiough to 10 then d L (iu (a)) /du 1 I,=o # O, a contradic tiori. Tlius, critica1 paths for leng-th carinot h a ~ e khks.

38. Consider a cylinder Z c IR3 of radius r . Find tlie metric d induced by tlie Riemannian metric it acquires as a subset of p. 39. Consider a cone C (without the vertex), and let L be a generating lbe. Unfoldiiig C - L onto lR2 produces a map f: C - L + lR2 whicli is a local

isometi-); but which is usuaUy not oiie-orie. Iinrestigate tlie geodesics on a cone (tlie number of geodesics between two points depends 011 the angle of the cone, arid soine geodesics may come back to their initial point).

40. Let g : S* + Pn be tlie map g(p) = [ p ] = (p, - p ] .

(a) Sliow tliat there is a uiiique Riemaririian metric (( , )) on P" such tliat g*(( , )) is tlie usual Riemanriian metric on S" (tlic one that makes the iriclusion of S" into lRnil an isometry). (11) Sliow tl-iat every geodesic y : IR + Pn is closed (tliat is, tliere is a riumber a sucb that y(t +a) = y(t) for al1 E), and tliat every two geodesics isltersect exactly once. (c) Sllow tbat tliere are isornetries of P" onto itself taking arly tangent vector at orie point to any tangent vector at any other point.

Tliese recults sbow that P' provides a model for "elliptical" non-Euclideaii georiieti-)! Tlie sum of the angles in aiiy triangle is > n.

41. Tlie Poincark upper halFp1ane 3f2 is the maiiifold ( ( x , 13 E lR2 : y > 0) m:itli the Riemannian metric

(a) Compute that

k al1 otl-ier rij = 0.

(b) Let C be a semi-circle in X 2 with center at (O, e) and radius R. Considering it as a curve t t-+ (1, y(t)), show that

(c) Using Problem 27, show that all the geodesics in X 2 are the (suitably parameterized) semi-circles with center on the x-axis, together with the straight lines parallel to the y-axis. ,

(d) Show that these geodesics have infinite length in either direction, so that the upper haif-plane is complete. (e) Show that if y is a geodesic and p 4 y, then there are infinitely many geodesics through p which do not intersect y. (f) For those who know a little about conformai mapping (compare with Prob- lem IV7-6). Consider the upper haif-plane as a subset of the complex numbers C. Show that the maps

are isometries, and that we can take any tangent vector at one point to any taiigent vector at any other point by some $,. Conclude that if length AB = length ~ ' 3 ' and length AC = length A'C' and the angle between the tangent vectors of p and y at A equais the angle between the tangent vectors of p' and y' at A', then length BC = length B'C1 and the angles at B and 3' and at C and C' are equal ("side-angle-side"). These results show that the Poincaré

upper lialf-plane is a model for Lobachevskian non-Euclidean geometry. The sum of the angles in any triangle is < x.

42. Let M be a Riemannian manifold sucli tliat every two points of M can be joined by a unique geodesic of minimai length. Does it necessarily follow that the Riemannian manifold M is complete?

43. Let M be a manifold with a Riemanliian metric ( , ), and choose a fixed point p r M. Suppose that every geodesic y ; [a, b ] + M with initial vaiue y (a) = p can be extended to ail of R. Show that the Riemalinian manifold M is geodesically complete.

44, Let p be a point in a complete non-comlacl Riemanilian manifold M. Prove that there is a geodesic y : [O , m) + M with the initiai value y(0) = p , having the property that y is a minimai geodesic between al-iy two of its points.

45. LRt M alid N be geodesically complete Riemannian manifolds, and give M x N the Riemannian metric described in Problem 26. Show that the Rie- mannian manifold M x N is aiso complete.

46. This problem presupposes knowledge of covering spaces. Let g : M + N be a covering space, where N is a Cm manifold. Then there is a unique Cm structure o1-i M wliicli makes g an immersion. If ( , ) is a Riemannian metric oli N, tlien g*( , ) is a Riemannian metric on M , and ( M , g * ( , )) is complete if and only if (N, ( , )) is complete.

47. (a) If M" C is a submanifold of N, show that the normal bundle v is indeed a k-plane bundle. (b) Usiiig tlie i-iotioii of Whitney sum @ introduced iil Problem 3-52, show that

48. (a) Show that tlie normal bundles v i , vz of M" c defined for two different Riemannian metrics are equivaient. (b) If t = rr : E + M is a smooth k-plalie bundle over M*, show that the normal bundle of M c E is equivaient to 5 .

49. (a) Given an exact sequence of bundle maps

as in Problem 3-28, wllere tlie bundles are over a smooth manifold M [or, more generally over a paracompact space], show that E2 2 El $ E3. (b) If t = r r : E + M is a smooth bundle, conclude that T E R. rr*(t) $

x * ( T M ) .

50. (a) Let M be a non-orientable manifold. According to Problem 3-22 there is S' c M so that ( T M ) I S ' is not oricntable (the Problem deals with the case where (TM)IS' is aiways trivial, but the same conclusions will hold if each (TM)IS' is orientable; in fact, it is 11ot hard to show that a bundle over S' is trivial if and only if it is orientable). Using Problem 47, show that the normal bundle v of S? M is not orientable, (b) Use Prol~lem 3-29 to condude that there is a neighborhood of some S' c M which is not orientable. (Thus, any non-orientable manifold contains a "fairly smail" non-oi.ientable open submanifold.)

CHAPTER 10

LIE GROUPS

T liis cliapter uses, and illuminates, many of the results and concepts of tlle pixeding chapter-s. It will also play an important role in later Volumes,

where we are concerned ~ 4 t h geometric problems, because in the study of these pi-oblems the groups of automorphisms of various structures play a central role, aiid tliese groups can be studied by the methods now at our disposal.

A topological group is a space G which also lias a group structure (the product of u , b E G beiiig denoted by ab) such that the maps

(a, b ) M ab from G x G to G

u H a from G to G

al-e coiitinuous. It cleai-ly suffices to assume instead that the single map (a , b ) w

nb-' is continuous. 147~ will mainly be iiiterested in a very special kind of topological group. A Lie group is a group G which is also a manifold with a Cm structure such that

are Cm functioiis. It clearly suffices to assume tliat the map (x, y) w xy-' is Cm. As a mattei. of fact (Problem 1), it eveli suflices to assume that tlie map (x, 1)) M xy is Cm.

Tlle simples1 example of a Lie group is R", with the operation + . The circle S' is also a Lie group. Oiie way to put a group structure on S' is to consider it as tlie quotient group R/Z, wliere Z C R denotes the subgroup of iiitegers. The f~iiictions A+ M cos2nx and x M sin 2nx are Cm functions on R/Z, and at each poilit al least one of them is a coordinate system, Thus the map

( A - , y ) W .Y - }' W x)'-' m m m

R x R + R + S ' = R / Z . wkich can be expressed in coordiliates as one of thc two maps

(x, 1 1 ) M cos 2x(x - 11) = cos 2nx cos 2n y + sin 27rx sin 2ny

(x, y) M sin 2n(x - y) = sin 27rx cos 27r y - cos 27rx sin 2xy,

is Cm: consequently the rnap (A-, y) w xy-' from S' x S' to S' is also Cm.

If G and H are Lie groups, then G x H, with the product Cm structure: and the direct product group structure, is easily seen to be a Lie group. In particular. the torus S' x S' is a Lie group. The torus may also be desuibed as the quotient group

the pairs (a. b) and (a', b') represent the same element of S' x S' if and only if a' - a E Z alid b' - b E Z.

Many important Lie groups are mati-is groups. The general linear group GLOI, R) is the group of al1 non-singular real 17 x 17 matrices, considered as a subset of R"'. Since the function det : R"' + R is continuous (it is a polynomial map), the set GL(i1, R) = deth'(R - (O)) is open, and hence can be given tlie Cm struciuir wliich makes i t an open submanifold of R"'. Multiplication of mati-ices is Cm, siiice tlie entries of A 3 are polynomials in the entries of A and B. Sinoothness of tlie inverse map follows similarly fi-om Cramer's Rule:

( ~ - ' ) ~ i = det Aij/det A ,

where A'J is the matrix obtained from A by deleting row i and column j. Oiie of the most importaiit esamples of a Lie gi-oup is the orthogonal group

O(II), consistiiig of al1 A E GL(r1, R) witll A A' = 1, wliere A' is tlie transpose of A. Tlis condirion is equivaleilt to tlie coiidition tliat the rows [alid columns] of A are 01-tlioiiormal, which is equivalent ro tlle coiidition that, with respect ro the usual basis of R", the inatrix A i-epresents a linear ti-ansformation wliich is aii "isomerry", Le., is norm presenling, and thus inner product presenling. Problem 2-33 presen ts a proof tliat O (17) is a closed submanifold of GL(i1, R), of dimension i l I r? - 1)/2. To sliow tliar O(I?) is a Lie group we must sliow rliat the map (x, y) i-, xy-' which is Cm on GL(i1, R), is also Cm as a map from O(]?) x O(i7) ro O(II). By Propositioii 2-1 1, i t suffices ro sliow rllat ir is coiitinuous; but this is true because the inclusion of O(i7) + GL(n, R) is a homeomorphism (sii~ce O ( I ~ ) is a sii11niaiiifold of GL(n , R)). Later in the cliapter we will havc anotlier way of proving that O(i?) is a Lie group, aiid in particular, a maiiifold.

Tlie argument in the previous paragraph shows, generally, tliat if H c G is a subgroup of G aild also a submanifold of G, rheii H is a Lie group. (Tliis gives aiiotlier prooftliat S' is a Lie groiip, for S' c R2 caii be coiisidered as tlle gioup

motions, Le., isometries of R". A little arpmenr shows (Problem 5) tliat even. element of E(]?) can be writren uniquely as A T where A c O(n) , and r is a translation,

T ( X ) = T a ( X ) = X + U .

]Ve can give E(]?) the Cd¢ st~.iicture which makes it diffeomorphic to O(n) x R*. Now E(n) is iiot tlie direcr pi-oduct O(i7) x IR" as a gi-oup, since translations aiid ortl~ogoiial trallsformations do not generally commute. In facr,

SO

Ara A-' = r ~ ( a ) , Ar, = %(o} A .

Consequen tly:

wliich shows that E(i7) is a Lic gmup. Clcarly tlie compoiient of e r E(r7) is the subgroup of Al Ar with A E SO(i?).

For any Lie group G, if a E G we define [he left and right translations. L,: G + Gaild R,: G + G,bv

L,(b) = ub

Ru ( b ) = bu.

Notice tliat L, and R, are bot1.i diffeomol-pliisms, witll iiiverses La-, and R,-1. resliect ively. Corisequently, tlle maps

are isomorphisms. A vector field X on G is called left invariant if

La,X = X for al1 a E G.

Recall tliis means thai

It is easy to see tliat tbis is rrue if we merely liavc

La, X, = Xa for al1 a E G.

Co~lscquently, giveli X, E G,. tl1e1-e is a uliique left invarianr \rector field X on G wllich lias rlle value Xp at e .

2. PROPOSITION. E1ler-y left invariant vector field X on a L e group G is Cm.

PROOE It suffices to prove tbat X is Cm iii a neigliborhood of e, since the diffeomorphism L, theti takes X to the Cx \lector field L., X around a (Prob- lem 5-1). Ler (x, U ) be a coordinate system around e. Choose a neighbor- I~ood V of e so rliat a! b E V implies ab-' E U. Tlien for a E V we have

Since the map (a, b) H ab is Cm on V x V we can write

lbr some Cm fuiictioti f' on x(V) x x(V). Tllerl

xxi(a) = xe (A-' o L.)

wliich shows tliat x x i is Cm. This implies that X is Cm. O

n a(xi o L,)

= Ccj j= 1

ax j

3. COROLLARJ'. A Lie group G aiways has a trivial tangent bundle (and is consequently orietitable).

n

where X, = ci a

j = 1 e

PROOE Cl~oose a basis Xle, . . . , X,, for G,. Lct Xl,. . . , Xn be the left invari- alit vector fields wirll these values ar e . Tlien XI , . . . , Xn are clearly everywliere litiearly il~dependetit, so we can define an equivalente

A left invarianr vector field X is just one that is La-related to itself for all a. Consequently, Proposition 6-3 shows that [X, Y] is left invariant if X and Y are. Henceforrh we will use X, Y, etc.: to denote elements of G,, and X, Y, erc., to denote the left invariant vector fields with R(e) = X, y(e) = Y, etc. We can then define an operation [ , ] on Ge by

The \lector space G,, together with this [ , ] operation, is called rhe Lie algebra of G, and will be denoted by k(G) . (Sometimes the Lie algebra of G is defined insread [o be the set of left invariant \rector fields.) We wiIl also use the more customary notation Q (a German Fraktur g) for J ( G ) . This notation requires some conventions for particular groups; we write

QI(II,IR) fortheLiealgebraofGL(n,R)

o(n) for the Lie algebra of O(n).

In general, a Lie algebra is a finite dimensional vecror space V, with a bilinear operation [ , ] satisfying

[X, X] = o [[X, Y], Z ] + [[Y, Z], X] + [[Z, X], Y] = O 'yacobi identity"

for al1 X, Y, Z E V. Since the [ , ] operation is assumed alternaring, it is also skew-syrnmetric,

[X, Y] = -[Y, X]. Consequently, we cal1 a Lie algebra abelian or commutative if [X, Y] = O for all X, Y.

Tl-ie Lie algebra of IR" is isomorphic as a vector space to IR". Clearly L(IRn) is abelian, since t11e vector fields a/axi are left invariant and [a/dxi, d/axj] = 0. The Lie algebra 6 ( S 1 ) of S' is 1-dimensional, and consequently must be abelian. If are Lie algebras with bracket operations [ , 1; for i = 1,2, then we can define an operation [ , ] on the direct sum V = 6 @ V2 (= VI x V2 as a set) by

[(Xi, X2), (Y19 y211 = ([Xl, Ylli, [X2, y212).

It is easy to check that this makes V into a Lie algebra, and that L(G x H) is isomorpl~ic to J ( G ) x L ( H ) with this bracket o~eration. Consequently, the Lie algebra L(S1 x - - x S' ) is also abelian.

The structure of gi(n,IR) is more complicated. Since GL(n,R) is an opcil sulnnanifold of IR"', the tangent space of GL(n,R) at the identity I can br

identified with Wn2. If \*e use the standard cooi-dinates xij on Wn', then an i7 x n (possibly singular) matrix M = (Mij) can be identified with

Let fi be tbe left invariant \rector field on GL(n, W) corres~onding to M. 1% compute the function M x k i on GL(n,B) as follows. For every A E GL(ri,W),

NOMI the function xk' o LA: GL(li, W) + GL(r1,R) is the linear function

witli (constant) partial derivatives

So if N is another n x 17 matrix, we have

Eom this we see that

[M, ElI = ~ ( M N - N M ) ~ ' a l ; k,'

axki

thus, if we identify 91 ( n , W) with Wn2, the bracket operation is just

[M,N] = MN - NM.

Notice that in any ring, if we define [a, b ] = a b - b a , then [ , ] satisfies the Jacobi iden ti h.

Since O(ii) is a submanifold of G L ( n , W ) we can consider O ( n ) l as a subspace of G L ( n , R ) ] , and tlius identify o ( n ) with a certain subspace of Wn2. This subspace may be determined as follows. If A : ( - E , E ) + O ( n ) is a curve with A(0) = 1, and we denote ( A ( t ) ) , by A j j ( i ) , then

which shows that A i j f ( 0 ) = - A ~ ~ ' ( o ) .

Thus O ( i l ) ~ C w"' can contain only matrices M wliich are skew-symmetric,

Tliis subspace has dimeiisioii ~ ( 1 1 - 1)/2, wliich is cxacrly the dimensioii of O ( n ) , so O ( U ) ~ must coilsist exactly of skew-symmetiic matrices. If we did not hiow [he dimeilsioil of O(iz ) , we could use tl-ie foltowing line of reasonii-ig. For each i , j with i < j , wc can define a cunre A : W + O(n) by

~ 4 t h sin t and - sin t at (i, j ) and ( j , i ) , 1 'S on tlie diagonal except at (i, i) and ( j , j), and O's elsewhere. Then the set of all ~ ' ( 0 ) span the skew-symmetric matrices. Hence O ( I I ) ~ must consist exactly of skew-symmetric matrices, and O(n) must have dimension n(n - 1)/2.

\Ve do not need any new calculations to determine the bracket operation in o(11). In fact, considel. a Lie subgroup H of any Lie group G, and let i : H + G be the inclusion. Since i, : H, + G, is an isomorp hism into, we can identify f& witli a subspace of G,. Any X E H, can be extended to a left invariant vector field X" on H and a leR invariant vector field on G. For each a E H C G, we have left translations

La : H + H , L,: G + G

and L a o i = i o La.

rY LI

i, X (a) = i, La, X = L,,(i* X) = X (u). rY

Iti, other woids, X" and f are i-related. Co~isequentl~ if Y E &, then [X, Y] LI LI

aiid [X, Y] are i-related, which means that

Tlius, H, c Ge = Q is a subalgebra of Q, thar is, H, is a subspace of Q wliich is closed uiider the [ , ] operation; moreover, He with this induced [ , ] operation is jusr 1) = L(H) .

This coi-respoiidetice between Lie subgt-oups of G and subalgebras of 9 turns out to work in the othei- direction also.

4. THEOREh4. Let G be a Lie group, and 11 a subalgebra of 9. Then there is a utiique connected Lie subgroup H of G whose Lie algebra is $.

PROOF. For u E G, let A, be tlie subspace of Gu consisting of al1 ?(o) for X E 11. The fact that 11 is a subalgebra of g implies that A is an integrable distribution. Let H be rhe maximal integral manifold of A containing e. If b E G, then clearly Lb*(Aa) = Abur so Lb* leaves the distribution A invariant. It follows immediately that Lb permutes the various maximal integral manifolds of A among rliemselves. Iti particular, if b E H, tlien Lb-[ takes H to the tnaximal iiitegrai manifold containitig Lb-, (6) = e, SO Lb-] (H) = H. This iinplies rhar H is a subgi-oup of G. To prove tliat it is a Lie subgroup we just iieed to show tliat (u, b) I-+ ub-' is Cm. Now tliis map is clearly Cm as a map into G. Using Theorem 6-7, i r follows that it is Coo as a map into H.

The pt-oof of uniquetiess is left to the reader. *3

There is a very dificult theorem of Ado which states that every Lie algebra is isomorphic to a subaigebra of GL(N7 R) for some N. It then follows from Theorem 4 that e v q Lie aigebra ti tiomo@ic to /he he aigebra ofsome Liegroup. Later on we will be able to obrain a "local" version of tliis result. We will soon see to what extent the Lie aigebra of G determines G.

1% continue the study of Lie groups along the same iSoute used in the stud!. of groups. Having considered subgroups of Lie groups (and subaigebras of tlieir Lie algebras), we next consider, more generaily, homomorphisms between Lie groups. If #: G + H is a Cm homomorphism, then #,,: Ge + He. For anv a E G we clearly have

- so if X E Ge, and 5 = #.,X is the left invariant wctor field on H with value #,, X at e, t hen

Thus X" and 5 are #-i-elated. Consequently, tlie map #,, : Q + 1) is a Lie algebra homomorphism, tliat is,

Usually, we will denote #,, simply by #, : g + I). For example, suppose tliat G = H = R. There are an enormous iiumbei-

of homomorphisms #: R + R, because R is a wctor space of uncountable dimciision wer Q, and ewry linear transformation is a group homomorphism. But if # is Cm, then the conditioi-i

implies thar

which ineans that #(t) = ct for some c (= #'(O)). It is iiot hard to see tliat eveii a coi-itii-iuous # must be of rhis form (one first shows that # is of this form on tlie

rational numbers). ]Ve can identify L(R) with R. Clearly the map #, : IR + R is just rnultiplication by c.

Now suppose that G = R, but H = S' = R/Z. A neighborhood of the identity e E S' can be identified with a neighborhood of 0 E R, giving rise to an identification of l ( S 1 ) with R. The continuous homomorphisms #: R + S' are clearly of the form

once again, #, : R + R is multiplication by c. Notice that the oiily continuous homomorphism # : S' + R is the O map

(since (O) is the only compact subgroup of R). Consequently, a Lie aigebra ho- momorp hism Q + 11 may not come from any Cw homomorphism # : G + H. However, we do have a local result.

5. THEOREM. Let G and H be Lie groups, ai-id a: Q + b a L e aigebra homomorphism. Then there is a neighborhood U of a E G and a Cm map #: U + H such that

#(ab) = # ( n ) # ( b ) when a, b,ab E U,

and sucli that for every X E 9 we have

Moreover, if tliere are two Cw liomomorphisms #, S r : G + H with #,, = $*e = a, and G is connected, then # = @ .

PROOE Let f (German fiaktur k) be the subset f c Q x b of al1 (X, @(X)), for X E Q. Since is a homomorphism, f is a subalgebra of Q x b = L(G x H). By Tlieorem 4, there is a unique connected Lie subgroup R of G x H whose Lie algebra is f. If xl : G x H + G is projection on tlie first factor, and w = x11 K , rhen w : K + G is a Cm homomorphism. For X E Q we have

so w, : K(e,el + G, is aii isomorphsm. Consequeiitly, there is an opcn neigh- l~orhood V of ( e , e ) E K such that w takes V diffeomorphicaily onto an open neigliborliood U of e E G. If x2: G x H + H is ~rojection on the second factor, we can define

# = x2 O u-' on U.

Tlle first condition on # is obvious. As for the second, if X E Q, then

Given #,S, : G + H , define tlie one-one map 8 ; G + G x H by

The image Gr of 8 is a L e subgroup of G x H and for X E g we clearly have

so L(Gr) = f . Thus G' = K, which iinplies that $ ( u ) = #(a) for al1 a E G. Q

6. COROLLARY. If two L e groups G and H have isomorphic L e algebras. then they are locally isomorpliic.

PXOOE Given an isomorpliism 0 : g + I), let # be the map $ven by Tlieo- rem 5. Since #,, = is aii isoinoi-phism, # is a dfleomorphism in a neighborhood of r E G. e3

Xenlark: For tllose wl-io know about siniply-coilnected spaces it is fairly easv (Problem 8) to conclude that two simply-coriiiected Lie groups with isomoi-phic L e algebras are actually isomorpliic, aild that al1 connected Lie groups with a given Lie algebra are covered by [he same simply-connected Lie gi-oup.

7. COROLLARY. A connected Lie group G with an abelian Lie algebra is itself abeliaii.

PROOE By Corollary 6, G is locally isomorpbic to IR", so a b = bu for a , b in a neighborhood of e. It follows that G is abeliaii, sii-ice (Problem 4) a i i ~

neighborhood of e geiierates G. +$

8. COROLLARY. For every X E G,, thei-e is a uiiique Cm I-iomomorphism # : R + G such tl-iat

FIRSTPROOE Define 0 : R + 6 ( G ) by

@(a) = (xx.

Clearly 0 is a Lie algebra homomorphism. By Tlieorem 5, on some neigliboi- liood (-E,&) of O E R there is a map #: ( - E , & ) + G with

# ( S + 0 = # ( ~ ) # ( l ) lsl, l t l , 1s + 1 1 E

and

To extend # to R we write every t with It 1 E uniquely as

1 = k ( ~ / 2 ) + Y k an integer, Ir 1 < &/2

and define

# ( ~ / 2 ) m # ( ~ / 2 ) . # ( r ) [ # ( ~ / 2 ) appears k times] k 3 0 # ( i ) =

# ( - ~ / 2 ) # ( - ~ / 2 ) - #(Y) [ # ( - & / S ) appears -k times] k < 0.

Uiiiqueiiess also follows from Theorem 5.

SECOJVD (DIRECV PROOE If .f : G + IR is C m , aiid # : R + G is a C m homomorpl-iism, then

= lim S(# ( t )#( ld) - f ( $ 0 1) h+O h

Thus # iriust be an integral cunre of X", whicb proves uniqueness. Con~rersely~ if # : R + G is aii integral curve of X, tben

is aii iiitegi-al curve of X wliich passes through # ( S ) at time i = O. The same is clearly true for

i E- #(S + t ) :

so # is a hoinomoi.pliisrn. MTe kno\rr that integral cunres of X" exist locally; they can lx extended to al1 of R usiiig [he method of the first proof. *3

A homomorphism # : IR + G is called a l-parameter subgroup of G. We thus cee that there is a unique 1-parameter subgroup # of G with given tangent wrtnr d#/di (0) E G,. l e have already examined the 1-parameter subgroups ui R. More interesting things liappen when we take G to be IR - {O), with multiplicatioil as tlie group operation. Then all Cm homomorphisms # : IR + IR - {O), with

#(S + t ) = #(s)#(I),

must satisfy

The solutions of tliis equation are

Notice that IR - (O) is just GL(1, I R ) . A l Cm l~omomorphisms # : IR + GL(n, I R ) must satisfy tlie analogous differential equation

where iiow denotes matrix multiplication. The solutions of these equations can be written formally in the sarne way

where exponeiiriation of matrices is defined by

This follows from t he facts in PI-oblem 5-6, some of which will be briefly reca- pitulated here.

If A = (ujj) and IAl = max lau 1, then clearly

+ * . m + ( 1 1 1 A I ) ~ + * + O a s N + m .

( N + K)!

so tlie series for exp(A) converges (the (i, j)'h entry of the partial sums converge), and convergence is absolute and uniform in any bounded set. Moreover (see Problem 5-6), if AB = BA, then

exp(A + B) = (exp A)(exp B).

Hence, if # (1) is defined by (m), then

#'(I) = lim exp(t#'(O) + 17#'(0)) - exp(t#'(O) 1 h+O h

= lim [exp(l7#'(0)> - 11 h+O h exp(t#'(O)

= lim 1 ! 2 ! h-to h exp (1 #'(O) 1

so # does satisfy (*). For any L e gi-oup G, we now define the "exponential map"

exp: g + G

as follows. Given X E 9, let #: R + G be the unique Cm homomorphism witfi d#/dt(O) = X. Then

exp(X1 = #(] l .

We clearly have

9. PROPOSITION. The map exp: Ge + G is Cm (note that G, R" has a iiatural Cm structure), and O is a regular poirit, so that exp takes a neighborhood of O E Ge diffcoinorpliically oiito a rieigfiborhood of e E G. If S,: G + H is any Cm homomorphism, tlien

exp o S,* = S, O exp. expl lexp

S, G-H

PROOE The tangent space (G, x G)tx,,) of the Cm manifold G, x G at the point (X,u) can be identified with G, @ G,. We define a vector field Y on G, x G by

4 \

Y,,,., = o @ ?(a).

(S')L x Si -

Tlien Y ]las a flow cx : R x (G, x G) + G, x G, which we know is Cm. Since

exp X = projection on G of a(], O @ X),

it follows tliat exp is Cm. If \ve ideiitifv a \lector u E (G,)o witli G,, tlien tlie curve ~ ( i ) = t v in G, has

tangent \lector v at O. So

So exp+o is tIle identity, aild lle~ice one-oiie. Tlierefore exp is a diffeomorphism i i i a neigl~borhood of O.

Given $: G + H, and X E G,, let #: R + G be a Iiomomorpliism with

Con~equeiitly~ exp($*X> = S, O # ( i ) = $(exp X). e3

10. COROLLARY. Every o~ie-one Cm Iiomomorpliisrn #: G + H is ail immersion (so #(G) is a L e subgroup of H).

PROOE If = O for some iioii-zeio X E Q, then also #*,(X) = 0. 13ut rlien

= exp #+e(lX) = #(exp(lX)),

contradictiiig tlie fact tllat # is oiie-one. *=r

11. COROLLARY. Every coiltinuous I-iomomorphism #: R + G is Cm.

PROOF, Let U be a star-sllaped open neighborhood of O E G , on which exp is one-one. For any a E e x p ( ~ U ) , if a = exp(X/2) for X E U, then

a = exp(X/2) = [exp( x/4)12, exp X/4 E exp(; U) .

So a has a square rooi iii exp(; U). Morewer, if a = b2 for b E exp(4u), then b = exp(Y/2) for Y E U, so

exp(X/2) = a = b2 = [exp(y/2)12 = exp Y.

Since X/2, Y E U it follows that X/2 = Y, so X/4 = Y/2. This shows that every a E exp(; U) has a unique square root in the set exp(; U).

Now dioose E > O SO that #(í) E exp(; U) for 11 1 E. Let #(E) = exp X, X E exp(f U). Since

[#(&/2)12 = #(E) = [exp X/212, i t follows from ihe abwe that # ( ~ / 2 ) = exp(X/2). By induction we have

#(&/Y) = e ~ p ( X / 2 ~ ) .

Helice

#(m/2" E) = #(~/2")" = [exp(X/2")lm = exp(n7/2" X).

12. COROLLARXi. Every continuous Iiomomorphism # : G + H is Cm

PROOE Clioose a basis Xl, . . . , X, for G,. The map t i-, #(exp tXi) is a coiitiiluous l~on~omorphism of R to H, so tl-iere is Yi E He such thai

#(exp iXi) = expiYj.

Thus,

(*> # ((exp 11 Xi ) - . - (exp i, x,~)) = (exp ti YI ) - . (exp hYn). Now tlle map S,: Rn + G given by

is Cm and clearly

so is a diKeomorpliism of a tieighboi-liood U of O E R" onto a neighborllood V of e E G. Then on V,

# = (# O S,) O S,? aiid (*) sliows tllat # o S, is Cm. So # is Cm at e, and tllus everywhere. 9

1 3. COROLLARXr. If G and G' are Lie groups wllich are isomorphic as topological groups, then they are isomorphic as Lie groups, that is, there is a d f i o - morphism between them wllicll is also a group isomorphism.

PROW Apply Corollary 11 to the continuous isomorpliism and its inverse. +:+

Tlle propert jes of tlie particular exponential map

exp : V' (= 1 , R ) + GL(n, R)

may 11ow be used to show tllat O(n) is a Lie group. It is easy to see thal

e x p ( ~ ' ) = (exp M)'.

h/loreo\~er, sirlce exp( M + N) = (exp M) (exp N) when MN = NM, we have

(exp M)(exp -M) = I .

(exp M)(exp M)' = 1;

Le., exp M E O(n). Co~lversely, any A E O(!]) sufficiently close to I can be written A = exp M for come M. Let A' = expN. Then I = A A' = (cxp M)(exp N), so es11 N = (exp M)""' = exp(- M). For sufficiently small M and N tliis implies tliat N = -M. So exp M' = A' = exp(- M); lieilcc M' = -M. It follows tliar a ncighborhood of I in O(n) is an n(n - 1)/2 dimensional su bmailifold of GL(i1, R). Since O(iz) is a subgroup, O (11) is itself a submanifold of GL(ir , R).

Just as in GL(n, R), tbe equation exp(X + Y ) = exp X exp Y liolds wlie~ievci [X, Y] = O (PI-ol~lein 13). 111 general, [X, Y] measures, up to first order, the cstent to which tliis equatioii fails to hold. In tbe following Theorem, aiid i11

its piaof, t~ iiidicate diat a functioii c. : R + G , has tlie property that c ( t ) / t is l)ouiided for small t, we wiD denote it 11y O(t3). Thus O(f3) will denote differeni fi~~ictions at differeiit times.

14. THEOREM. If G is a Lie group and X, Y E G,, theii

f ' (1) exp t~ exp / Y = exp{f(x + Y ) + i [ ~ , YI + 0 ( t 3 ) J

(2) exp(-tX) exp(-t Y) exp tX rxp t Y = exp{f 2 [ ~ , Y] + O(t 3)i (3) exp tX exp t Y exp(-tX) = exp(ty + r2[x, Y] + ~ ( t ~ ) } .

(i) Ff(a) = Fa ( f ) = L.. x(/) = X(J o L ~ ) = l o f a exp UX) .

Similarly,

m

Irf(a) = - f(a m exp uY).

For fixed S, let #(i) = f(expsX expiY).

Then

d (iii) #'( i)=-f(expsXexpiY)=

di f l.=, f(exp sx exp I Y exp u Y)

Applying (iii) io yf insiead of / gives

m m

(iv> #"(/) = [Y (Yf )](exp sX exp /Y).

Now Taylor's Theorem says thar

Suppose that f (e) = O. Then we have

Similarly, for any F.

- Substituiingin (v) for F = f , F = y~ and F = Y ( Y f ) gives

(vi) f'(expsXexpiY)=s(yf)(e)+i(Yf)(e)

Now for small i we can wriie

exp iX exp t Y = exp Z ( i )

for some Coo function Z witll values i1.i G,. Applying Taylor's formula to Z gives

Z ( r ) = ízI + i2z2 + 0(t3),

for some Zi , Z 2 E G,. If f (') = O , then clearly f ( A ( t ) + 0 ( t ')) = f ( A ( 1 ) ) + O({ ' ) , SO by (vi) we have

Siiice we can tnke tfle .f?s to be cooi-dinate fuilctioi~s, comparison of (vii) and (viii) gives

w11 i cli givw 1

thus proving (1). Equatioi~ (2) follows immeciiately from (1).

To prwe (3), again cl-ioose f with f ( r ) = O. Tl~en similar caiculations give

(ix) f (exp t X exp t Y exp(-t X))

then we also have

Comparing (ix) and (x) gives the desired result. e3

Notice tliat for-mula (2) is a special case of Tl~eorem 5-16 (compare aiso with Problems 5-1 6 and 5-13).

T l ~ e work involved in prmii-ig Tl-ieorern 14 is justified by its role ii-i the following beautiful theorem.

15. THEOREh4. If G is a Lie group and H c G is a closed subset which is also a subgroup (algebraically), the1-i H is a Lie subgroup of G. More precisely tl~ere is a Cm sti.ucturc on H , triith die relntiue topoiogy, tllat rnakes it a Lie subgroup of G.

PROOE We attempt to recoiistruct the Lie a1gebi.a of H as follows. Let B c Ge be the set of al1 X E G, such tliat expiX E H for al1 t .

Assulioll I . Let Xi r Ge with Xi + X and let t j + O witl~ each ti # O. Suppose exp tiXi E H for a11 i. Then X E 11.

Pro!$ We can assurne ti > 0, since exp(-tiXi) = (exp li Xi)-' E H. For t > 0: le[

1 ki (1) = largest integer 5 -.

ti

ki(r) exp(ki (t)ti Xj) = [exp(ii ~ j ) ] E H,

ki(t)tiXi + t X .

Thus exp tX r H , since H is closed and exp is continuous. We clearly also haw expiX E H for i c O, so X E 1). QE.D.

]Ve now clairn that 1) c G, is a vector subspace. Clearly X E 1) irnplie~ sX r 1) for al1 s E R . If X , Y E 11, we can write by (1) of Theorern 14

exp tX exp t Y = exp{t(X + Y ) + iZ(i)i

wliere Z(i) + O as i + O. C1loose positive tj + O aiid let Xi = X + Y + Z(ii). Then Assertion 1 irnplies that X + Y r b. Alternatively, we can write, for fixed i.

taking 1irnits as n + ca gives exp t (X + Y ) E H. [Sirnilarly, using (2) of Tlieorern 14 we see tl~at [X, Y] E b, so that 1) is a

subalgebra, but we will ilot even use this fact.] Yow let U be an open neigliborllood of O r Ge on whicla exp is a d8'Teornoi.-

pl~isrn. Then exp(b n U) is a submanifold of G. It c.lear1y suffices to sllow thai if U is srnall enough, tlleil

H n exp(U) = exp(1) n U).

Clloose a subspace 1)' C G, cornplernei~tary to 11, so tliat G, = 1) $ 11'.

Assutioil 2. Tl-ie rnap # : G, + G defined by

#(X + X') = exp X exp Xf X E Ij, X ' E 1)'

is a dXeornorpllisin i i i sorne neigl-iborliood of O.

Rooj: Cl~oose a basis X1, . . . , Xk , . . . , X, of G, witli XI , . . . , Xk a basjs for [l. Tlieii # is given by

Siiice tlie rnap .x:=, a iX i H ( a l ,. , . ,on) is a diffeomorphisrn of G, onto W". it suftices to show thai

@(al , . . . ,a,) = exp

is a diffeomorphism in a neigliborhood of O E IR". This is clear, since

Assc~~olr 3. Tlicre is a i~eighborllood V' of O in 11' such that exp X' $ H if 0 # X' E V ' .

Plo$ Clioose aii iiiner product on lj' aiid let K c 11' be the compact set of al1 X' E I)' with 1 5 IX'I 5 2. If the assertion were false, there would be Xi' E b' ~vitli Xj' + O and esp Xi' E H. Clioose integers 1 7 i wcitli

Choosirig a suheqiience if necessary, we can assirrne Xi' -+ X' E K. Since

it follows fi-orn A . T . T ~ . ~ z o ~ ~ 1 that X' E 11, a contradiction. Q1E.D.

\Ve cari now cornpletc tlie proof of the tlieorem. Choose a neighborhood U = W x W' of G, or-i which exp is a d~eornorphisrn, with

W a neighborhood of O E 11

W' a neighborhood of O E 11'

siicl~ that W' is contained in V ' of Assn-tion 3, and # of Asscriioli 2 is a daeornor- pliism on W x W'. Clear1)-

To pr-we the reverse iriclusion, let a E H n exp(U). Then

n = exp X esp X' X r W, X' r W'.

Since n, exp X E H we obtain exp X' E H, so O = X' , and a E exp(b fl U). +$

Up to iiow? we liave concentrated on tlie left invariant vector fields, but rnan? properties of Lie groups are better expressed in terrns of forrns. A forrn w is called Iefi invariant if L,*w = o for al1 a E G. This rneans that

Clearly? a left invariant k-forrn o is deterrnined by its value o ( r ) E SZ' (G,). Heiice: if o', . . . , o" are left iiivariant 1-forrns such that w1 (E), . . . , wn(e) span G,*. theri e\-eI?I left invariaut k-forrn is

foi- certain conslanh al. 1 f o' (e), . . . , o" ( r ) is t1ie dual basis to Xi , . . . , Xn E G,. t11er-i any Coo vector field X can be written

X = f ' T j for Cm functions j.'.

T l ~ e i ~ w i ( x ) = f ' ,

so o' is Cm. It folloius that aiiy left invariant forrn is Cm. If o is leíi invariant, tllen for a E G we have

L,*dw = d( LO*#) = d o ,

so clw is also left in~~ariailt. The for.rnula on page 215 irnplies that for a lefi invarian t 1 -forrn w and left in\-ariant \rector fields y and Y we llave

- - d w ( X , Y ) = X"(w(p) ) - r(o(2)) - o ( [ X , Y ] )

N -

= - o ( [ X , Y ] ) .

tlie bracket being the operation in Q. Thc ir~ter-plg. betweeii left ii-i\.ai.iant arid 1-ight in\7ariarit \lector fields is tlie

sul~jiject of 1'1-oblern 1 l. Her-e \ve coiisider the case of for-nis.

16. PROPOSITION. Let *: G + G be @(a) =a- ' .

(1) A for-rn w is left invariant if and only if $*w is rjglit invariant.

(2) 1f E Q ~ ( G ~ ) , then $*oe = (-])'oe.

(3) If w is left and right invariant, then do = 0.

(4) If G is abelian, then 0 is abelian (converse of Corollary 7).

If o is left invariant, tlien

so @*o is riglit invariant. The converse is sirnilar:

(2) It cleasly sufices to prove this for k = 1. So it is enough to show that SI,,(X) = -X for X E Ge. Now X is the tangent vector at i = O of the curve 1 i-, exprX. So **e X is tlle tangent vector at r = O of 1 i-, (expix)- ' = exp(-f X); tl-iis tangent \lector is just - X.

(3) If w is a left and right invariant k-forrn, then

Sir-ice @*o ar-id o are 110th left invariant, we have

T l ~ e forni do is also lefi a i d nght invariant, so

But k $-'(do) = d(**w) = d((- 1 lko) = (- 1) dw.

(4) If G is akliail , then al1 left invaiialt 1 -fol-rns w are also rigl-it invariant. So do = O for al1 left invariant 1-foi*rns. It follows fiorn (x) tllat [X, Y] = O for al1 x , Y E n.

A ! ~ e ~ ? ? a k p r o d o f (4). B y Tlieorern 14, if G is abelian, then for X, Y E G, \ve h ave

H ence X, Y] + o(t3)p2 = ;[Y, x ] + 0 ( 1 ' ) / 1 ~ . 2 [

Letting r + O, we obtain [X, Y] = [Y, X]. $+

Since do is left invariant for any left invariant o, it follows tliat for a basis o ' , . . . , o" of iiivariaiit 1-forms we can express eacli dwk in terms of the o' A d . First clioose Xi , . . . , X, E G, dual to o' ( e ) , . . . , & ( e ) . There are constants C$ such that

ti

clearly we dso liaw n

Tlie iiumbers C$ are called tlie constants ofstrucnire of G (witli respect to h e basis Xl , . . . , X, of g). fi-om skew-sylniiietry of [ , ] aad tIicJacohi idcntity we obtain

Frorn (*) on page 394 we obtain

Ii iiiriis out that (2) is esactly wliat we obtaiii korn the relation d 2 0 k = O. Coii- dition (2) is tlius an ir-itegrability coiidition. lo fact, we can prwe (Probleni 30) that if C$ ale coiistaiits satisfyiiig (1) aiid (2), tlien we can find everywlicrr liiiea1.ly indcpciideiit 1-forms o' , . . . , o" iii a neigliborhood of O E W" sucli tliat

Morewer, the existente of such o' jrnplies (Prohlern 29) that we can define a rnultiplication (a, b) I-+ ab in a neighborf~ood of O which is a group as far as it can be and which has the o' as left invariant 1-forrns. Frorn this latter fact and (a suitable local version of) Theorern 5 we could irnrnediately deduce the following Theorern, for which we supply an independent proof.

17. THEOREM. Let G be a Lie group with a basis of left invariant 1-forrns o', . . . , o" aiid constants of structure c;. Let M" be a differentiable manifold aiid let O', . . . ,O" be everywhere linearly independent 1-forrns on M satisfying

Tlien for every p E M there is a neigllborhood U and a diffeornorphisrn f : U + G such that

0' = f*J.

PROOF. k t T C ~ : M x G + M and xz : M x G + G be the projections. Let

Then

d(ek - ók) = - E C; ( [e i A oj] - [ói A ó j ] )

i c j

By Proposition 7-14., M x G is foliated by 11-dimensional rnanifolds whose tangent spaces at each point are anniliilated by al1 8* - o*. Choose a E G and let r he tlie foliurn through (p , a). Now e ' , . . . ,en, ó', . . . , ón are linearly independent everywliere; so on r(p,alr which is tlie set ofvectors in ( M x G)(p,a) where ek - ók = O, tlie sets o ' , . . . ,8" and o',. . . , ón are each linearly inde- peiident. Hence T C ~ : r + M and x2: r + G are eacli diffeornorphis~ns i i i sorne neigliborhood of (p,a) . Tl~is rneans that r contains the graph of a diffeornorpliisrn f fi-orn a neigliborliood U of p to a iieighborhood of a.

Let f: U + M x G be the rnap

It is also possible to sap by how rnucll any two such rilaps differ.:

18. THEOREM. Let M be a coi~nected manifold, let G be a Lie group, and let . j ; , . j i : M + G be two Cm niaps sucll tliat

for al1 lefi in\.ai-iant 1-for-rns w . Tl~eii J j and J" differ by a left translation, tlial is, ther-e is a (uriique) a E G sucl~ tllat

PEDISTliLclAr PROOF. C m 1. M = IR and tlze 1x10 ntaps y,, y2: IR + G soiisfi y1 (0) = y, (0). ]Ve rnust show tliat y, = y2. For every left iiivariant 1-forrn w

we llavc.

1 r follows tliar

If wc r-egai-d y, as givei~? and wi-ite this equation out iri a coordinate system. t l - i i~i it bccoiiies an or-diiiaiy difl'cr-critial eqiiation for y2 (of t l~e type considered

in the Addendurn to Cl~apter 5), so it lias a unique solution with tlie iiiitial coridition y2(0) = y, (O). But this solution js clearly y2 = y,.

Cme 2. M = IR, but 11le nzajis y,, y2 are a r b i l r a ~ . Choose a E G so that

If o is a left invariant 1-forrn, then

(L. 0 Y,)*(#) = y,*(La*w) = y,*(o) = y,'(o).

Since La o y, (O) = y2(0), it follows frorn Cme 1 that La o yl = y2.

C a s ~ 3. Ge~ia.01 cme. Let po E M. Choose a E G so tliat

For aiiy p E M there is a Cm curve c : IR + M i v i t l ~ c(0) = po and c(1) = p. Let yj = fi o C. Then

By C a e 2, we llave y2 (t ) = a . y, ( t ) for al1 f .

i i i particular for t = 1, so $2(])) = a - (P).

E L E G A A T P R O O F : Lec : G x G + G be projection on the ith factor. Clioose a l~asis o ' , . . . , o'' for tlle left iinrariant 1-forrns. For (a, b) E G x G, let

Tlieil A is aii integr-able distribution on G x G. In fact, if A(G) c G x G is tlie diagoiial subgi-oup {(a, a ) : a E G), tlieii tlie rnaxirnal integral rnanifolds of A are t l~e lefi cosets of A(G). Now define h : M + G x G b?.

By assurnptioii,

Since M is connected, it follows that Iz(M) is contained in sorne left coser of A(G). 1 n otliei- words, tl1ei-e are a , b E G witll

19. COROLLARY. If G is a connected Lie group and f : G + G is a CE map preserving left invariant forrns, then f = L, for a unique a E G .

\Yliile left ini-ai-iant 1-forrns play a fundamental role in thc study of G, t h c

left iilvariant 12-for-rns arvz also very irnportant. Clearly, al1 left invariant 11-form: ai-c a constant rnultiple of anv non-zero one. If o" is a left invariant n-forrn. then o" deter.rnii1es al1 orieiitation on G, aiid if f : G + R is a Cm fui-iction ivitll coimpac t support, we can define

Siiice a'' is usually kept fixed in any discussioi-i, this is often abbreviated to

The latter iiotation has adiyantages iii cer-tain cases. For exarnple, left invariai~cc of o" irnnlies that

/G m

f ( n ) da = jG j'(ba) da.

in other words,

S, fa" = S, ga', wherr glo) = . f (b«);

1 note that Lb is an orientation presening difkornoi-phisrn, so

~cllich proves the rorrnula]. \Ve can, of c.ourse, also consider right iiivariai~t n-f'oi.rns. Tliese gcncr-ally turn out to be quite different frorn t l ~ e left invariant 12-ioi-ms (see the csarnple in Pr-oblern 25). But in one case they coincide.

20. PROPOSITION. If G is conipact niid connected and w is a left irivariai~t 12-hl-in, then o is also rigllt invariant.

1'1~00.E Suppose o # O. For each a E G, t l ~ e lorrn Ra*o is left invariant. so tliere is a unique real nuinber f (a ) with

Siiice R,* o RbA = (Rab)*, we have

So f ( ~ ) C IR is acoinpact connected subgroup of R-(O). Hence f ( G ) = { 1). +$

]/Ve can abo consider Riernannian rnetrics on G. In the case of a cornpact group G there is ahvays a Riernannian rnetric on G which is both left and right invariant. In fact, if ( , ) is any Riernannian rnetric we can choose a bi-invariant 17-forrn an and define a bi-invariant (( , )) on G

We are finally ready to account for sorne terrninology frorn Chapter 9.

2 1. PROPOSITION. Let G be a Lie group with a bi-invariant rnetric.

(1) For any a E G, the rnap I, : G + G giwn by Ia(b) = ab- 'a is an isornetq. wllich reverses geodesics through a , i.e., if y is a geodesic and y (O) = a , then Iu (Y (t)) = Y (-0. (2) The geodesics y with y(0) = e are precisely the l-pararneter subgroups of G, i.e., the rnaps f J-+ exp(tX) for sorne X E g.

PROOE (1) Since I,(b) = b-',

the rnap l e , : G, + Ge is just rnultiplication by - I (see tlle proof of Proposi- tion l6(2)), so it is an isornetry on G,. Since

for any a E G, the rnap J , , : G, + Gu-1 is also arl isornetry. Clearly Ie reverses geodesics through e .

Since la = R,-l :

it is clear that la is an ison-ietry reversing geodesic through a .

(2) Let y : IR + G be a geodesic with y(0) = r . For fixed t, let

T l ie i~ y is a geodesic and Y (O) = ~ ( 1 ) . So

I t follows by induction that

Y ( n l ) = Y (0" for any integer n.

If 1' == 12'1 and 1'' == nfft for integers n f and n", then

so y is a hornornorphisrn on 0. By continuity, y is a 1-pararneter subgroup. T l ~ e s e are the only geodesics, since there are 1-pararneter subgroups with

any tangent vector at t = 0, and geodesics tbrough e are deterrnined by their- tarigent vectors at t = 0. *:*

]Ve conclude this chapter by iiitroducing sorne neat forrnalisrn which aHows us to write the expression for dok in an invariant way that does not use tlic constants of structure of G. If V is a d-dimensional vector space, we define a V-valued k-form on M to be a furiction o such that each o ( p ) is an alternating

maP

k times

d If u], . . . , ud is a basis for V, tlieri tliere are ordinary k-forrns o ' , . . . ,o sucli tliat for- A'], . . . , Xk E Mp we have

we will write sirnply

For- anv V-valued k-forrn w we define a V-valued (k + 1)-forrn do by

a simple c.alculation slio~vs that t l~is definition does not depend on tl-ie choice of basis u', . . . vd for V.

Líe G'7.aups 403

Sirnjlarly, suppose p : U x V + W is a bilinear rnap, where U and V have bases u l , . . . ,u , and u ] , . . . , ud, respectively. If w is a U-valued k-forrn

and q is a V-valued 1-forrn

is a MI-ualued (k + 1)-forrn; a calculation shows that this does not depend on the choice of bases u] , . . . ,u, or v i , . . . , ud. M'e will denote this W-valued (k + ¡)- forrn by p(w A q).

Tliese concepts have a natural place in the study of a Lie group G. Althougli there is iio natural way to choose a basis of left invariant 1-forrns on G, there U a natur-al q-valued 1-forrn on G, narnely the forrn o defined by

U$ng tlie bilinear rnap [ , 1: CJ x + g, \ve have, for any R-valued k-form q a11d any n-valued ¡-forrn h on G, a new g-valued (k + 1)-forrn [q A A] on G.

Now suppose tbat X i , . . . , X, E Ge = g is a basis, and that o', . . . , wn is a dual basis of leíi invai-iant 1 -forrns. T h e forrn o defined by (*) can clearly be writteii

n

Tlien

O n the other hand, n

Cornparing (1) and (2): we obtain the equations of structure of G:

The equations of structure of a Lie group ~ 4 1 1 play an irnportant i-ole in Voiume 111. For the pi-esent we rnerely wish to point out that the terrns d o and [o A o] appearing in this equation can also be defined in an invariant way. Foi- tlle terrn d o wejust rnodify tlle formula in Theorern 7-13: If U is a vector field on G and f is a g-va1ued fuiiction on G, then (Problern 20) we can define a g-valued function U ( f ) on G. O n the other hand, # ( U ) is a g-valued function on G. For vector fields U and V we can then define

Recall that the value at a E G of the right side depends only on the values U, and Va of U and V a t a . I fwe clioose U = R, V = Y for sorne X 7 Y E G r , then

rY rY

d o ( a ) ( y Q 7 Fa) = O - O - o ( a ) ( [ X , Y] , ) - - - - = - w ( e ) ( [ X , Y ] , ) since [ X , Y ] is left invaimiani

E - o ( e ) ( [ X , Y ] ) by definition of [ , ] in G ,

= - [ X , Y ] by definition of o.

= - [ o ( a ) ( y a ) , o ( a ) ( y a ) ]

It foHows that foi. any lector fields U and V we have

Pi-oMem 20 gives an iinrariant definition of & A V ) and shows tliat tliis equation is equivalent to tlie equations of structure.

MIARNING: In sorne books the equation which we have just deduced appears as d w ( U , V) = - L [ ~ ( U ) , 2 o(~)]. T h e appearance of the factor f Iiere Iias noi1rin.g to d o with the f in the other forrn of the structure equations. I t comes about because sorne books d o not use tIie factor ( k + l ) ! / k ! l ! in the definition of A. This rnakes their h A q equal to f of ours for 1-forrns h and v. T h e n the defiiiition of d ( x dx') as d o i A d x i rnakes their d o equal to f of ours for 1-forrns o.

PROBLEMS

1 . Let G be a group which is also a Cm manifold, and suppose that ( x , y) J-+ ñ-1. is CE.

(a) Find 1 - ' w h e n f : G x G + G x G i s J ( x , y ) = ( x , x ) l ) . (b) Show that ( e , e ) is a regular point of f. (c) Conclude that G is a Lie group.

2. Ler G be a topological group, and H c G a subgroup. Show that tlir closure of H is also a subgroup.

3. Let G be a topological group and H c G a subgroup.

(a) If H is open, then so is every coset gH. (b) If H is open, then H is closed.

4. Let G be a connected topological group, and U a neighborhood of e E G. Let U" denote al1 products al a,, for ai E U.

(a) Show that Un+' is a neighborl~ood of Un. (b) Conclude that U, Un = G. (Use Problem 3.) (c) If G is locaily compact and connected, then G is a-compact.

5. Let f : Rn + Rn be distance presenring, with f (O) = 0.

(a) Show that J takes straight lirles to straight lines. (b) Sliow that f takes planes to planes. (c) Show that f is a linear transforrnation, and hence an elernent of O ( n ) . (d) Show that any elernent of E(I z ) can be written A z for A E O ( n ) and z a translation.

6. Show tliat tbe tangent bundle T G o f a Lie group G can always be made into a Lie group.

7. \Ve have cornputed that for M E gl(n, IR) we have

(a) Show that this means that

(It is ñctually clear n that M defined in this way is left invariant, for LA* = LA since LA is linear.) (b) Fi~ind the i.ight invariant lector field with value M at J .

8. Let G and H be topological groups and #; U + H a rnap on a corinected open neighbrhood U of e E G such that #(ab) = #(a)#(b) when n, b,nb E U.

(a) For each c E G, consider pairs (1/, S,), where V c G is an open neigliboi-- hood of c with V V-' c U: and where S,; V + H satisfies $(o) S,(b)-] = #(ab-' ) for n, b E V. Define (Vi, S,]) y (V2, S,2) if S,] = S,2 on sorne srnaller neigliborhood of c. Show that the set ofall 7 equi\lalence classes, for al1 c E G , caii be rnade into a covering space of G. (b) Conclude that if G is sirnply-connected, t l~en # cari be extended uniquel\- to a liornornorphism of G into H.

9. In Theorem 5, show that # and S, are equal elten if t h q are defined only o11 a neigliborliood U of e E G, provided that U is connected.

10. Sliow that Corollary 7 is false if G is not assurned connected.

11. If G is a gi.oup, \ve define the opposite group G0 to be tlie sarne set 134th tlie rnultiplication + defined by a+ b = b .a. If g is a Lie algebra, ~ 4 t h operation [ , 1, we define th e opposite Lie aIgebra f1° to be tlie sarne set with t l~e opei-atiori [X, YI0 = -[X, Y].

(a) G0 is a group, and if S,: G + G is a I-+ a-': then S, is an isornoi.phism from G to GO. (b) no is a Lie algebra, and X I-+ -X is an isornorphisrn of g onto gO. (c) 6 ( G 0 ) is isornorphic to [6(G)1° = gO. (d) Let [ , ] be the operation on G, obtained by using right invariant \lector- fields instead of left invariant ones. Then (n, [ , ]) is isornorphic to 6 (G0) , and hence to gO. (e) Use this EO give aanother proof that g is al~elian wllen G is abelian.

12. (a) Show that

e X p ( O -a :)=( - coso sin a cosa sino

(11) Use the matrices A and B below to show tllat exp(A + B) is not generally equal to (exp A)(exp B).

13. Let X, Y E G, with [X, Y] = 0.

(a) Use Lemrna 5-13 to show that (exp sX)(exp t Y) = (exp t Y)(expsX). (b) More generally, use Theorem 5 to show that exp is a hornornorpllism on the subspace of G, spanned by X and Y. In articular, exp(X + Y ) = (exp X)(exp Y).

14. Problem 13 implies that exp i (X + Y) = (exp iX)(expt Y) if [X, Y] = O. A more general result liolds. Let X and Y be vector fields on a Cm manifold M with corresponding local 1-parameter families of local diReomorphisms {#r):

Suppose that [X, Y] = 0, and let vi = #[ o = $[ o #[.

(b) Using Corollary 5-1 2, show that

In other words, (vr) is generated by X + Y.

15. (a) If M is a diagonal matrix ~ ? i t h complex entries, show that

det exp M = e trace M

(b) Show that the same equatioii holds for al1 diagonalizable M with complex entries. (c.) Conclude that it holcis for al1 M witli complex eritries. (The diagonalizable matrices are deiise; compare Problem 7-1 5 .) (d) Using Pi-opositioil 9, sliow tliat for tlie homomorphism det: GL(n,R) + R - {O), the map ciet,: ~l(12, R) + J ( R - (0)) = R is just M J-+ trace M. (e) Usc tliis fact to give a fancy proof tliat trace M N = trace NM. (Look at trace(MN - NM) = trace[M, N].) (0 Prove the result iii part (d) ciii-ectly: ivitl~out usii~g (c). (Sincc det, and tracc are hornomorphisrns, it suffices to look at matrices with only one non-zero entry.) (g) Now usc tliis i-esult and Pr-oposition 9 to give a fancy proof of (c).

16. (a) Let U be a iieighborliood of the ideiitity (1 , O) of S' (considered as a

subsct of R2). Show that no matler liow small U is, there are elements a E U which have square roots outside U in acidition to their square root in U. (b) Show that foi. each n 2 1, there is a neighborhood U of e E G such thal eitery elemeiii in U has a unique nth root in U. (c) For G = S' . show tl~at tliere is no neighborl~ood U which has this propcrt); for al1 n.

17. (a) Let (x, V ) be a cooi-diiiate systein arouiid e E G with xi ( e ) = O. Let

for Cm functions J'. Show that

(b) If a, p : (-E, E) + G are difherentiable, show that

(c) Aso deduce this result fiorn Theorern l4(I). (Not even the full strength of (1) is needed; it suffices to kiio\v that exp tX exp t Y = exp{t (X + Y) + O(t)). Tlie argurnent of pan (a) is essentially equivalent to the initial parc of the deduction

of (1)s )

18. Let G be a Lie group, and let H C G be a subgr.oup of G (algebraically), such that every a E H can be joined to r by a Cm path lying in H. Let 6 c G, be tlie set of tangent vectors to al1 Cm paths Jying in H.

(a) Show tliat 11 is a subalgebra of G,. (Use Theorern 14.) (b) Let K C G be tlie connected Lie subgroup of G with L e algebra 6. Show that H c K. Hint: Join any a E H to r by a Cm curve c, and show that the tangent \rectors of c lie in tlie distribution constructed in the proofofT1ieorern 4. (c) Let t i , . . . , ck be curves in H with {cit(0)) a basis for b. By considering tlie rnap / ( t i , ..., t k ) = c~ ( i1 ) . . . ck ( i k ) , show that K c H. Thus, H is a Lie subgroup of G. It is even true that H c G is a Lie subgroup if H is path connected (by not necessady Cm paths); see Yarnabe, On an arcwire cotrnected sllbg~oup ofa L e group, Osaka Math. J. 2 (1950), 13-14. (d) If H c G is a subgroup and an irnrnersed subrnanifold, then H is a Lie su bgio up.

19. For a E G, consider the rnap b M aba-l = L,R,-'(b). The rnap

is denoted by Ad(a); usually Ad(a)(X) is denoted sirnply by Ad(a)X.

(a) Ad(nb) = Ad(n)oAd(b). Tlius we llave a 1iornoinorphisrn Ad : G + Aut(g), where Au/(n), the autornorphisrn group of g, is the set of al1 non-singular linear transforrnatioi~s of the vector space g onto itself (tllus, isornorphic to GL(n, R) i f n has dirnension n). Tlie map Ad is called the adjoint representation. (b) Show tl~ai

exp(Ad(a)X) = n(exp ~)a-' .

Hini: T1iis follows irnrnediatcly frorn one of our propositions.

(c) For A E GL(n, R) and M E gl(n, R) show that

(11 suffices to show this for M in a neighborhood of O.) (d) Show that

Ad(expíX)Y = Y + í [X, Y] + 0( i2 ) .

(e) Since Ad: G + g, we have the map

tangent space of A u t ( ~ ) at the Ad*e : (= Ge) + identity map i of Q to itself.

Tl~is tangent space is isornorphic to End(~) , where End(9) is the vector space of al1 linear transformations of g into itself: If c is a curve in Aut(~) with c(O) = l,, tllcn to regard c'(0) as an elcment of Aut(~) , we let it operate on Y E Q by

(Compare with the case 9 = Rn, A u t ( ~ ) = GL(n, IR), End(g) = 12 x n matrices.) Use (d) to show that

Ad*e(X)(Y) = [X, Y]. N N

(A proof may also be given using the fact that [X, Y] = L F ~ . ) The map Y J-+ [X, Y] is denoted by ad X E End(g). (f) Conclude that

Ad(exp X) = exp(ad X) == l o + ad X + (ad x ) ~ + - e . .

2!

(g) Let G be a connected Lie group and H c G a Lie subgroup. Show that H is a iiormal subgroup of G if and only if I) = L ( H ) is an ideal of Q = L(G), tllat is, if and only jf [X, Y] E lj for al1 X E n, Y E I).

20. (a) LRt f : M + V, where V is a finite dimensional vector space, wjth basis v i , . . . ,ud. For Xp E M p , define Xp(f) E V by

where f = xd I=I j' vi for f i : M + R. Show tliat this definition is indepeii- dent of the cl~oice of basis vi , . . . , vd for V.

(b) If o is a V-valued k-forrn, sliow that d o rnay be defined invariantly by the formula in Theorern 7-13 (using the definition in part (a)). (c) For p: U x V + W. show that p(w A q) rnay be defined invariantly by

Conclude, in particular, that

(d) Deduce tlle structure equations frorn (b) and (c).

21. (a) If o is a U -valued k-forrn and q is a V-valued I-forrn, and p: U x V + W, then

d ( p ( o A $1) = p ( d o A 9) + (- lIk p ( o A dq).

(b) For a pvalued k-foi-rn o and 1-forrn q we have

[o A q] = (-l)kl+'[qA o].

(c) Moreover, if A is a 9-valued n7-forrn, then

22. Let G c GL(II, IR) be a Lje subgroup. The inclusion rnap G + GL(n, IR) + IRn2 will be denoted l y P (for "point"). Then d P is an IRn2-valued 1-forrn (it corresponds to the ideiltity rnap of the tangent space of G jnto itself). We can also consider d P as a rnatrix of 1 -forrns; it is just tlie rnatrix (dxij), where each dxij is restricted to the tangent bundle of G. \Ve also have the ~ " ~ - v a l u e d 1 -forrn (or rnatrix of 1 -forrns) P-' dP, wliere - denotes rnatrix rnultiplication, and P-' denotes the rnap A I-+ A-' on G.

(a) P-' - d P = p(P-] A dP), where p: 8"' x 8" + 8"' is rnatrix rnultiplication. (b) LA*dP = A m dP. (Use f *d = df *.) (c) P-' d P is left invariant; and (dP) P-' is right invariant. (d) P-] d P is the natural Q-valued 1-forrn o on G. (It suffices to check that P-' d P = w at 1.) (e) Using d P = P o , show that O = d P - o + P - d o , where the rnatrix of 2-foi-rns P . d o is cornputed by forrnally rnultiplying h e matrices of 1-forrns d P and o. Deduce tllar

d o + o . o = O .

If w is the matrix of 1-forrns o = (oU), this says that

Check that tliese equations are equjvalent to the equations of structure (use the forrn do(X, Y) = -[o(X),o(Y)].)

23. Let G c GL(2, R) consjst of al1 matrices (O l) wih n # O. ~ o r conw-

iiieiice, denote the coordinates xl l and x12 on GL(2, IR) by x and 1..

(a) Show that for the natural g-valued forrn o on G we have

so tliat dx/x and dy/s are IeA iilvarjant 1-forrns on G, and a left invariant 2-form is (dx A dy)/x2. (b) Find tlie structure constants for these forms. (c) Show that

and find tlle right iiivariant 2-forrns.

24. (a) Show that tlie natural gI(n, R)-valued 1-forrn o on GL(n, R) is giverl by

. . 1 olJ = ddx"'

det (xaB) k=i

(11) Sliow that botli tlie leR and nght invariant ,12-forrns are rnultiples of

1 (dxJ1 A . . . / \dxn ' ) A . . . A (dxl" A . . . ~ d ~ " " ) . (det (xa8))"

25. The special linear group SL(,z, R) c GL(n, R) is the set of all matrices of deterrninant 1.

(a) Using Problern 15: sliow that its Lie a1gebi.a .r;l(~z, R) consists of al1 matrices with trace = 0.

(b) For the case of SL(2,R), sbow that

P ~ P = ( v d x - y d u vdy - y dv - u d x + x d u - u d y + x d v

where we use x, y, u, v for x", x12, x2' , x ~ ~ . Check tliat the trace is O by djfferentiating the equation xv - yu = 1. (c) Show that a left invariant 3-forrn is

26. For M , N E o ( n ) = L(O(n)) = { M : M = -M'), define

t (N, M ) = - trace M N .

(a) ( , ) is a positive definite inner p~-oduct on u(n) . (b) If A E O(n) , tliei~

(Ad(A) is defined in Problern 19.) (c) Tlie left inva-iant rnetric on O(!?) witli value ( , ) at is also right invarian t.

27. (a) If G is a cornpact Lie group, then exp: g + G is onto. Hint: Use Proposition 21. (b) Let A E SL(2,R). Recall that A satisfies its cl-iaracteristic polyilornjal, so A 2 - (trace A)A + I = O. Conclude that trace 1 -2. (c) Show that the following elernent of SL(2, R) is not A 2 for aiiy A . Conclude that it is ilot iil the irnage of exp.

(d) SL(2, R) does not llave a bi-invariant rnetric.

28. Let x be a cooi-diilate systern around r in a Lie group G, let ~ r i : G x G + G be the projectioiis, a i ~ d let (y , z) be tlie coordiiiate systern arouiid (e, r ) given by y ' = x ' o i r i , z i =x io i r i . Define#': G x G + R by

and let Xi be tlie left iilvariant vector field on G with

(a) Show that

(b) Using &Lb = Lab, s l i~w that

Deduce that xi (b)(xi 0 La) = @: (ab).

and then tliat - .

Letting $ = (3;) be tlic inverse rnatnx of @ = (y?:), we can write

This equation (or arly of nurnerous tl-ijng equivalent to it) is known as Lde'sfirsi fundame~tal Uieorem. Tlie associativity of G is jrnplicitly contained in it, since we used the fact that L, Lb = Lab. (c) Prwe tlie conuene uf G e S jrsi funda~nenhl Ilreurem, which states the followiiig. Let # = (#', . . . ,#") be a differentiable fuilction in a neighborhood of O E IR2" [with standard coordinate system . . , v",zl,. . . , sn] such that

#(a,o) = a for a E IR".

Suppose thei-e are differciitiable functions @; in a iieighborhood of O E IRn [with standard coordinate system rl,. . . , xn] sucli tliat

$j (O) = sj

a#' n Toi. (a, b) in a neighborhood (*) m("7b)=C@:(4(a ,b) ) .$; (b)

i= l of O E Ik2".

Tlien (a , b ) M #(a , b ) is a local Lie group structure on a neighborhood ofO E IRn (it is associative arid has inverses for points close enough to O, which serves as the ider-itity); the corresponding left invariant \rector fields are

[To prove associativity, note that

arid tlien sliow tliat #(a , # ( b , z)) satisfies the sarne equation.]

29. Lid secolrd fundonzenid tlteo~e~n states tl-iat tlie left invariant \rector fields Xi of a Lie group G satisfy

n

for certain constoak ~ 6 - i n otlier words, tlie bracket of two left invariant vector fields is left invariant. The airn of this problern is to prove tl-ie ~o?¿uer.e of&3 second fulzdo)nento2 tlteo~em, w1iicl-i states tl-ic following: A Lie alge bra of vector fields on a rieigliborhood of O E I R n , which is of dirnension iz over IR and cor-itains a basis for RUo, is tlie set of left invariant \lector fields for sorne local Lie group structure on a rieighborhood of O E IRn.

(a) Choose X I , . . . , X, in the Lie algebra so that Xi (O) = d/dxl l o arid set

tlien tlie oi are tlie dual rorrns, and coiisequently

(b) Let q : IR' x IRn + IR^ be tlie projections. Then

Consequeri tly? tlie ideal 9enerated by tlie forrns d (r ' ox2) - (11.: ox2) xl*oi is rhe same as the ideal 9 eeiierated by the forms rn*oj -xI'oj. Using the faci iliat tlie cjk are coirstants: SIIOMI that d(9) c 9. Hence W" x W" is foliated b\- 11-dimensional manifolds oii which tlie forms d(x i o x2) - (@/ O x2) xl*oi

a H j1ar-i ish . (c) ConcIude, as in tlie prooiof Theorem 17, that for fixed a, there is a iunct~on @, : W" + W" satisfying @,(O) = a and

n

a n

( b ) = E @f ( @ o ( b ) ) - i q b ) . ~ A - J i= l

Now ser #(a, b) = @,(b): arid use tlie converse of Lie's first fuiidamental tlieorem .

30. Lie's tliirdjiudnn?cr~i~l 11zeoren~ states that tlie C;k satisfy equations (1) aiid (2) on page 396, ;.e., tliat tlie left invariant vector fields form a Lie algebra undes [ , 1. The aim of tliis problem is to prove tlie cu:o,zverse o f h ' s tízird fundamental :Acooi.ern. \vIricli states that any 17-dimerisional Lie algebra is the Lie a1gebz.a Por- sorne l'ocal Lie group in a ~ieigliborhood of O E IRX.

Let C; be constants satisfyiiig equations (1) aiid (2) on page 396. We would like to find vector fields XI, . . . , X , on a neigl~borliood of O E IR" such rhat [X i . X,] = E;=, C[ Xk . Equivalently, we want to fiiid forms o' witli

Tlierl tlie 1-esult will follow from tlie conver-se of Lie's secorid fundamental tlieoreni.

(a) Lec h4 be functions on W x IR" such that

k h, (O, X) = 0.

Tliese ai-e equatioris "deperiding on the paranieters x" (see Problern 5-5 (b)). Notr tliat l$(r, O) = 6:r' so tliai h f ( i,O) = 6J. Let ok be tlie 1-form on W x WN defiried b!.

and wrile dok = hk + (di A a k ) .

where hk and ak do not involve d t . Show that

(b) Show that

(c) Let

Show that

+ terrns not involving d f .

Using

and equation (2) on page 396, show that

+ terrns not involving di

FinaUy deduce that

dek = dt A - E C; xjg1 + terrns not involving dt . j , i

(d) MIe can wnie

wliere gk. (O. .Y) = O (Mrli!;?). Using (c), show ihai f J

Cor-iclude that 8' = 0. (e) M;e 11ow ha\:(-

CHAPTER 11

EXCURSION IN THE REALM OF ALGEBRAIC TOPOLOGY

T liis cliapter exl~lores further properties of tlie de Rharn cohornolog-y \lector. spaces of a riianifold. Our rnain results will be restaternents, in terrns of thr.

de Rharn colrornology of fundamental properties of the ordinary cohornolog? whicli is studied 111 algebraic topologv. Because we deal only with rnanifolds. rnar-iv of the proof3 I~ecorne significantlv easier. On tlie other hand, we will be usir-ig sorne of tlie riiairi tools of algebraic topology rl-ius retaining rnuch of the flavor of rlia t subject. Along tl-ie wav \ye \vil1 deduce al1 sorts of in teresting consequences? irrcludirig a tl~eor-ern abou t tlie possil3ility of irnbeddirig ir-rnanifolds in ~ ? l + l

Lct M be a nianiiold with M = U U 1/ ior open sets U . V C M. Before exarniiiing tlie coliornology of M we will sirnply look at tlie vec.tor s p c e ck ( M ) of k-forrns orr M. Ler

l ~ e tlie inclusions. T l ~ c n we have two linear rnaps cu and /?.

defiried b!

Her-e iw*(o) is just the restr-iction of o to U'. etc. Clearly /? o a = & ln other w ~ ~ ' d ~ I iriiag~' a C ker /?. h4oreover; tlic converse liolds: ker /? C iniage a. Foq if P(hr ,h2) = 0: [he11 ir = A2 on U n V. so we can define o on M to be h I on U and h2 on 1/. arld tllen a(w) = (AI, h2). The equation irnage ar = ker /? is expressed by sayirig tliat tlie above dia~i-arn is exact at tlie rniddle vector space. \Ve can esrend thic diasrarn by putting tlie vector- space coritaining orily O at thc

e~ids; rl~e ai.r.o\trs at eitlier crid of the following sequerlce are tlie only possilil~- linear rnaps.

1. LEh/lh/lA. The sequericc

is exacr at al1 places.

I'ROOF. lt is clear tliat a is one-one. Tliis is equi\~alerit to exactness at ck ( M ) . sirice tlie irnage of the firsi rnap is (O) c ck ( M ) . Sirnilarly, exactness at C k (U 11) is equi\,alerit to #l be ir^^ onro. To projle tliar /? is onto, let (#u, #VI be a partiiiori ol'uriity suliordiiiate to ( U , V ) . Tlien o E c k ( U n V ) is

wl~er-e #vo denotes tlie Iorrn equal to #vo ori U n 1/, arld equal to O o11 U - ( U n V I . +>

By p u t t i r ~ ~ ir1 rlie rnaps (1. ~ - c - cari expand our diagram as follows:

so rliat rlie I-OM'S arme all exacr. lt is easy to check rhat rliis diagram cornrnuies. tliat is, any rwo cornpositions Irorn orie vector space to anotlier are equal:

Our first rnain theor~rn deperids onlv on tlie siinple algebraic structure in- herent in this d ia~~-a rn . To isolate this purelv al9ebr-aic structure, we rnakr* ilie following definitions. A cornplex C is a sequence of wctor spaces C" k = 0,1,2,. . . , togerller with a sequence of linear rnaps

satisfyilig dk+' o d A = 0: or biiefly, d2 = O. A rnap a: CI + C2 briweeii cornplexes is a sequerice of linear rnaps

sucli that tlie followirig diagrarn cornrnu tes for al1 k .

Tlie rnost irnportaiit exarnples of cornplexes air obtained by clioosin~ C" C' (M) foi. soine nianifold M, with dk tlie opei-atoi- d oii k-forins. Anotlier exarnple, irnplicir in our discussioii, is rlic direct sum C = Cl C2 of tmio coinplexes? defiiied 11).

l'or any cornplex C IW caii define tlie cohomology vector spaces of C by

k ker clX (') = image dk-1

Katuially, if C = (c'(M):, tlien H'(c) i s just H ~ ( M ) . lf a: Ci + C2 is a rnap betwren cornplexes. tl~eri we have a rnap. also denored by a.

To deíiiie a wc iiotr diat e17ery clerneiit of HQC,) is deterrniiied by sornr .Y E C? i'4tlt dd ik ( ,~ ) = O. Cornrniitativity of [he above diagi-arn s1io1z.s tliat

k k d2 (a (x)) = a di"(s) = O, SO ak (x) detei-rniiies aii elernent of H ~ ( c ~ ) ) , ~1hicl-i ~ $ ~ e define. ro b r @([he class deterinined bv A*). This rnap is ~vell-defined.

for if \ve cllange x to x + dik- ' (y) for sorne y E Clk-', then olk ((x) is clianped lo

ak(x + dlk-i(B)) = a k ( x ) + ak(dik- ' (y)) k k - I ( f fk - l = a ( x ) + d2 (Y) ) .

wliich deiel-iniiies tlle sanie elerneiit of H k (4). \Ylien Clk = C k ( ~ ) , ~2~ = @((N). and a: C k ( M ) + C ~ N ) is f * for f: N + M, then this map is jusi I * : H ~ ( M ) + H ~ ( N ) .

how suppose that \\le havc an eiact seqlrence of cornplexes

ivl~icli i.eally ineaiis a vasi coinmutative diagrarn in ~+lliich all rows are exacr.

Wia i does iliis iniply aboui tlic rnaps a : HB(C1 ) + H P ( c ~ ) alid B : H k (c?) 3 Hk(C3)? Tlir iiiccsi tliiny iliat could liappeii would be for ihe followiiig diagrarn lo bc exact:

a B O + Hk(C1) + Hk(C'2) + HL(C'3) + O .

l i s S o e . l:or csanildc, if U and V are oiei-lsppiiig poiíions of s2 IOI wliich ilier-e is a deforrnariori i.erractiori of U fl V inio S', then we llave aii exacf sequen cc

but nol an exact sequencc

h-e\lertheless, sornetliing very nice is true:

ff B 2. THEOREh4. lf O + CI + C2 + C3 + O is a short exact scqucncc o i cornplexes, tlieii tliere are linear maps

so that the follo\ving infinitely long sequence is cxact (eveqwhere):

PROOE Throughout tlie proof, diagrarn (*) should be kept at hand. Let x E

C3'< with d3'<(x) = O . By esactness of tlie rniddle ro\v of (*), there is y E c:" ~ 4 t h pk(y) = X. Tlieii

So d2k(y) E ker pk++' = iiiiageolk+'; tlius d2k(~7) = ak+' ( i ) for sorne (uniquej r E CI k + ' . h4oreo~~ei:

Since <uk++' is one-oiie! tliis irnplies that dik+'(s) = O , so z determines an ele- nieilr of Hk+ ' (Cl ): this eleiaent is defined to be 6k of the elernent of H k ( ~ 3 )

dcrcrrnined by s. ln order to prove tliai 6k is well-defined, ure iiiust check tliat the result does

iiot dcpend on thc choice of x E c~~ represeiiting the elernent of Hk(C3). So \ve Iiave lo show that we obtain O E Hk+'(C1) if7 we start 14th an elernent of tlie í'orrn d3kd1 (x') lor x' E G ~ - ' . 1n this case, let x' = pkdi (y'). Then

k-l -Y = d3 (,U') = dgkdl/?k-l(J~i) = /lkd2k-1(y1),

so we clioose d2k-1(Jl') as J.. This rneans that dik(?) = 0, and hence r = 0.

I t is also rrecessanl to check tliat our defiiiitiori is independent of tlie choice of y with /3k((i;) = x; tliá k left to the reader

Tlie proof tliat tlie sequerice ic exact consists of 6 sirnilar diagrarn chases. 14;~ will suppl:~ the proof tliat ker ol c imase 6. Let x E C'ik satiSfy di k ( ~ ) = 0, and suppose that ol"(x) E c~~ represclits O E kJk(c2). Thk meaiis that = dzkdi O') lor sorne 1) E c ~ ~ - ~ . NOM-

So ~ " ~ ( j t ) represeiiis air elernent of kJkdi(C3). h4oreover. tlie definition of S irnrnediately sliows that the irnay of tliis elernent under 6 is precisely tlie class represented by A-. $4

Ir is a wortliwliile exercise to clicck that the rnain step in tlie proof of Tlieo- rein 8-1 6 is preciselv tlie proof rliat ker ol c irnage 6, togetlier with tlie first pai-t of thc proof tl-iat 6 i s dl-defiiied. Al1 of Theoi.ern 8-1 6 caii he derived dir.ectl\. fi-on-i the folloi.i.iii~ corollanr of Lernrna 1 and Theorern 2.

3. THEOREh4 (THE MAX'ER-VJETORIS SEQUENCE). lf M = U U 1/: ~rliere U ar-id V are operi. tlien \ve Iiave an exact sequence (eventually endiny in 0's):

As several of tlie Problerns sliow. tlie colioniology of ncarly everythiiig can I J ~ computed b?. a suital~le applicaliori of thc Mayer-Vietoris sequence. As a simple cxamplc. 1 . i ~ corisider tlie torus T = S' x Sr, aiid tlie opeii sets U arid V illustrated 11elot.v. Since iliere is a deforination retractioii of U and 1'

orito cirrlcs. arid a defor-mation 1-etr-actiori of U n 1/ onto 2 circles. tlie h4aver- Vietoris sequeiicc i:

Tlle niap H ' (U n V) + H2(T) is iiot O (it is onto H2(T)), so its kernel is 1-dimensional. Thus the image of the rnap H1(U) H ' ( v ) + H1(U f l V) is I -dimensional. So the kernel of 11tG rnap is 1 -dimensional. and consequentiv tlie rnap H ' (T) + H' (U) $ H' ( V ) lias a l -dimensional irnage. Similar reasoning sIio~vs tliat this rnap also has a l-dimensional kernel. 1 t follows thar dirn H' (T) = 2. The reasoning used liere can fortunarely be systematized.

4. PROPOSlTl ON. lf tlie sequeilcc

is exact, tlieii

PROOF. BY inductiori on k. For k = 1 we llave the sequencc-

Exactness nieans rliar (O) C VI is the keriiel of tlie rnap Vi + O, which implier that VI = 0.

Assume tlie tlieorern for k - 1. Siiice tlle rnap V2 + V3 has kernel @(VI), it iiiduccs a rnap V2/ac(V1) + V3. h/loreover, tl~is rnap is one-one. So we have an exact sequcnce of k - J \lector spacei

O = d h V2/~(V1) -dirn V3 + . . . = - dirn VI + din? 5J2 - dim V3 + - . . .

whicli proves tlie [heoren? for k . +$.

Rarher than cor-i-ipure tI-ie col~omologv of otl-ier rnanifolds: we \vil1 use tl iv Mawr-\ktoris sequence to relate the dimensions of fik (M) to an entirel?. dirrerent set of iiumbers. arising fiom a "tiíanplation" of M, a new structurc which we will now define.

The standard 11-simplex An is defined as the ser

{ I A, = x E IR"" : O 5 x' 5 1 and x;f: x1 = 1 I '

(111 Pi.oblem 8-5. A, is dcfined to be a diflei-er-it. altbough horneomorpliic, ser.) Tlir siibset of A, obraiiied by settiiig 17 - k of the coordinates x' equal to O is Iion~comorphic ro A k . aiid is called a k-face of A,. lf A C M is a dif- leornorphic in-iage OS sorne A,, tlien tlie iii-iage of a k-face of A, is called a k-face of A . Bow 11v a triangulation of a c-oinpact ir-manifold M we mean a fii-iite collection of diffeornorpbic ima~es of A, wliich cover M arid whicl~ satisfv the fo1lowii-i~ coiiditioi-i:

lf a"! ri a';. # fl. tlieii for some k the iiltersection uni ri anj is a k-lacc of both ani ar-id a", .

1 t is a dficult ~heorerri rlrat evenl CK rnailifold has a triai-iplarion; for a proof see Muiikres. E/crmnncn/o~~~ ~ i j i r m ; i a / %/~o/ogy. or IYhitney, Gonrelnk lniqrol ion

T l l e o ~ . Assurning: that our manifold M has a tr-iangulation {uni) we will cal1 each uni an 77-simplex of the trianplation; any k-face of any un; will be caIIed a k-simplex of the triangulation, MJe Iet ark be tbe nurnber of these k-sirnplexes.

Kow let U be the disjoin union of open balls, one within each 71-sirnplex uni: and let Vndl be the cornplernent oftlle set consisting of the centers of these balls, so that V,-1 is a nei~hborliood of the union of al1 (n - 1)-sirnplexes of M. Tlien

M = U U l / , d l where U f~ V,-l lias the sarne cobornology as a disjoint union of an copies of S"-]. Consider first [he case where 11 > 2. The Mayer-jiietoi-is sequence breaks into pieces:

(2 ) For 1 < k i 17 - 1 .

Applyins Proposition 4 to these pieces vieldr;

dim Hk(Vn-1) = dim hJk (M) O ~ k ~ r z - 2

dim ~ " - ' ( l / , - ~ ) = dim H"-'(M) - dim H n ( M ) + a.. For the case n = 2 \ve easily obtain rl-ie same result ~ i t h o u t splitting up tlie sequence. \4;e now ii~troduce the Euler characteristic x(M) of M, defiiíed b!l

X(M) = dim H'(M) - dim H'(M) + d i m ~ ~ ( M ) - . . . + (-1)" dirn H"(M).

This makes sense for ano manifold in ~lli ich al1 hJk(M) are finite dimensional; we aiiticipate Iiere a later result that H k ( M ) is finite dimensional wlieiiever M is compact. Tlíe above equarions tl~en imply thar

~ ( " ~ - 1 ) = x(-l)k dim H ~ ( V , ~ - , ) k=o

= dim h J k ( ~ )

+ (-1)"-'[dim Hn- ' (M) - dim H n ( M ) + u,]

5. THEOREM. For any triangularion of a compact manifold M we llavc

I'ROOF. Iii tlle mailifold V,- we dehile a ilew opeil set U wllicll coiisists of a disjoint unioií of sets diffeornosphic ro IR"? oi-ie for ead-i (n - 1 )-face, joining the balls of the old U.

componeiits of iiew U (n 1= 3)

componenrs of new U (17 - 3)

M'c \vil1 let 1',,-2 be t11e cornplerne~~r oral-cs: in t11e ilew U , joining ihe ceniers of the balls in the old U .

Vn-2 js ~ l i r complemenr of

1;,-2 js the complemen t of

An argurneilt precisely like thar wIlic11 proves the equation

Similady \ve iilrroduce Vt1-3, . . . , VO; the last of r11ese is a disjoinr uilion of a0

sers eacll of ~~hic11 is srnoothly conrractil~le to a point. Hence x ( V o ) = cwo, w11iIe in al1 orl~er cases we have

Cornbining these equations, we havc

6. COROLLAR17 (DESCARTES- EULER). If a convex polyll edron has V vei-rices, E tdges, and F Paces, r l ~ e i ~

lf we turn frorn HQO H: we encounter a very dfiereni situation. If U c M is open, a iorrn o wirli cornpacr supporr C M rnay 1101 restrict to a forrn wirli compact supporr c U: rhe inclusion rnap of U inro M is nor proper. O n r 1 1 ~

other hand, if o is a form wirh compacr suppori c U: tlien o can be exteiided ro M by lerting ir be O ourside U; we will deiiore rhis extended íorrn 131-

If c!(M) dciioies tlie vcctor space of k-fornls wiili compact suppori on M, ivc

can define a iie~v sequciicc..

is exact.

PROOF Ii is clear iliar jw' @ - jv' is one-one; iii faci, each rnap ju' and jr~' is oile-one.

To pr-ove tliat i v ' + i y r is onio, ler w l ~ c a k-fhrrn wirh cornpact supgort on M, aiid let (#u,#v: hc a partirion of unity foi. rhe cover (U, V I . Tl ie~í

is clearly ilie in iap of ( # o o , # v ~ ) E C;(U) @ C k ( v ) . c

Ir is clear that iiiiage (jw' @ - jv') c ker(io' + iV8). To prove rlie coiiversc, suppose rliat

(Al, A l ) E C: ( U ) @ c:( V ) satisfies iU ' ( l i ) + iv'(A2) = 0.

This ilicaiis tliat ii = - k 2 . Siilcc supporr A l c U aiid suppori A;! C U. tliic slinws tl~ar siipposi iL1 c U n V aíid siipporr h2 C U n V. So (A1 ,Al) is tl1r i i i ia~e of A l E C/(U n V ) +3

8. THEOREhI (MAYER-VIETORIS FOR COMPACT SUPPORTS). If ihe manifold M = U U V for U , V open in M I rlieii there is a long exact sequence

PROOE Apply Tlieorern 2 io tlie sllort exaci sequence of cornplexes given by t l~e Lernrna. +3

Tllis sequence is rnuch h arder to work ~Yltll I han the M ayer-Vietoris sequence. For exarnple, suppose vire waiit to fiiid H,k for W" -(O]? wliich is dfleornorphic to

x B. Ifwr write S" = UUV in the usual way. so iliat U n V is diffeornorpliic ro S"-' x IR: rllen S" x W = (U x W) U (If x W), whei8e (U x B) n ( V x B) is diffeornorphic to SN-I x B2. The oiily i*ay to use iiiduction is io find H/ for al1 S" x R", stariiiig xzit l i S' x IRm. The details will be left to the reader; \ve ~411 rnerely recosd one furrlier i-esult! for latei- use. aiid tlieil proceed ro yet aiior 11es application of Tl~eorern 2.

9. COROLLARXt, If M = U U V for U , V opeii i i i M! tlien rkei-e is a dual loiig exact sequeiicc

PROOE M7e jusr have to show t11 at if tlie sequeiice of linear rnapc

is exaci at W2. theii SO is tlie sequeilce of dual rnaps aiid space5

For aiiy h E W3* we llave

SO ff* o p * = 0. h'ow suppose A E W2* satisfies a * ( h ) = O. Tlien h o ar = O. We clairn tl~at

tliere is 1: MI? + IR ~ i i l i h = #j*(h) , i.e., A = #j o h. Giveii a w E W3 wliich is

of tlie forrn /?(wl) : we define

i ( w ) = A(/?') .

Tliis rnakes sense, for if / ? (w l ) = # I ( w " ) , then w - w" = a(,-) for some r ? so A{w) - A{wM) = Acv(s) = O. Tliis defines h oii #?(W2) c W3. Now clioos< W c lV3 wiili W3 = /?(Wi) @ W. and define h 10 be O on U[. +

M;e I ~ O M ? co~isider a i-arlier differenr situation. Let N C M be a comnpaci sub- rnaiiifold of M. Then M - N is also a rnaiiifold. Mre r 11 erefore llave the sequencc

wliere c is "exteiision". Tliis scquence is tioi exact ai C$ (M): the kernel of i' coiitaiiis al1 o E c , ~ ( M ) ivhicli are O on N , wliile rlie iiriage of e contains al1 o E C: ( M ) ~sliicli are O in a iieigliborhood of hr.

To circumirent rhis clifficulty, ive will llave to use a technical devicc. M+ appeal first ro a resulr fi-om rhe Addeiidum to Chaprer 9. There is a compacr iiei~hboi~hood V of N and a map x : V + N such rhat V is a manifold-wit11- boundar).; aíid if j : N + V is tlie inclusion, then rr o j is the idenrity of n'. ~vliile j o rr is srnoor hly homotopic ro tlie idenriry of V. ! e now coiistruct scquence o i sucli neighborhoods V = V I 5i V2 5i V3 5i with ni Vi = A'-

KOW ronsider ~ W O forms ~i E ck(Vi), W j E ck ( V j ) . \Ve will cal1 wi atid w j

agirinalemii if rliere is I > i , j sucli tliar

It is clear that we can make rlie set of al1 equivalence classes into a vector sparc. $"N)? the "gerrns of k-forins in a iieighborliood of M". Moreoirer, it is easy [o define d : P ~ ( N ) + g k + ' ( N ) , so tliat we obtain a complex $. Finally. wi define a map of cornplexes

i i i tlle o l ~ i ~ i o ~ s ulajr: w I+ tlle equivalence class o í aily o1 Vi .

1 0. LEAtMA. The sequence

e i* O + C;(M - N ) -t C ; ( M ) -t $ l k ( N ) + O

IS exact.

PROOF. Clearly e is one-one. If o E c:(M - N ) . then o = O in some neighborliood U of N . Since N is

compact and nl Vi = N , there is some i sucli tliat Vi c U, arid consequeritl!. o = O on l/i. Tliis iiieans tliat i * e ( o ) = O. Co~i~fersely, suppose h E c ) ( M ) satisfies i* (A) = O. By defiiiition of $lk ( N ) , this means tliat h 1 Vi = O for some i . Hence hl M - N lias conipact support c M - N , and h = e(hl M - N ) .

Fiiially, any eleme~it of $lk (N) is represented by a form q on sorne V,. Let .f : M + [O, 11 be a Cm function wliicl~ is 1 on I/i+l, having support f c interior l/i. Then f q E c!( M ) , and f q represents tlie same element of $ l k ( N ) as 17; conseque~itly tliis element is i* ( f q ) . 4 3

11. LEMMA. Tlie coliomology vector spaces ~ ' ( 8 ) of the complex { $ k ( N ) ) are isomor~hic to H k ( N ) for al1 k .

PROOF, Tliis follows easily from the fact tliai j* : H' (vi) +- I f k ( N ) is aii isomorpliism for eacli lfi. Details are left to tlie reader. 43

12. THEOREM (THE EXACT SEQUENCE 01: A PAIR). If N c M is a coinpact submaiiifold of M , then there is an exact sequence

6 k + 1 ( ~ - N ) , . - . . . , H:(M - N ) + H ; ( M ) - H ~ ( N ) -+ H,

PROOF Apply Theor-em 2 to tlie exact sequence of complexcs given by Lemma 10, and theii use Lemnia 11. *t.

In tlie proof of tliis tlieorem, the de Rliam coliomology of the manifold-witli- l ~ou r ida r~ 1/1 entered oiily as an intermediar? (aiid we could have re~laced tlie Vi by tlieir iiitei-iors). But in the next tlieorem, whjch we will need later, it is the object of primary interest.

13. THEOREM. Let M be a manifold-witli-bouiidaiy, with compact boundary aM. Tlien there is aii exact sequence

PROOF, Just like the proof of Theorem 12, using tubular nejghborhoods I/i of aM in M. *t*

As a simple application of Tlieorem 13, we can rederive H;(Wn) from a kiio\vledge of Hk(Sn- ' ) , by clioosiiig M to be the closed ball 3 in R", witli H: (3) = H A ( B ) = O for k # O. The reader may use Tlieorem 12 to computi

H:(s" x IRm), by considering tlie pair (Sn x Rm, { p ) x IRm). Then Tlieo- rem 13 may be used to compute tlie cohomology of S" x Sm-' = a(Sn x closed ball iii Rm). For oui- iiext applicatioii we will seek bigger game.

Let M c R"+' be a rompart 11-dimensional submanifold of IRn+' (a compact "hypers~irface" of IRn+'). Using Tlieoi-em 8-17, the sequence of the pair (IRn+', M ) @ves

It follo\vs tliai

($1 iiumber of compoiients of IRn+' - M = dim H n ( M ) + 1.

But we also know (Problem 8-25) tliai

(**> iiumber of compoiielits of IR"+' - M 2 2.

14. THEOREM. If M c Rn+' is a compact hypersurface, then M is orientable, aiid Rn+' - M has esactly 2 coinpoi-ients. h/loreover, M is the bounday of each componeni.

PROOF. From ($1 and (**) we obtain

dim H n ( M ) + 1 1.2.

Sii-ice dim U n ( M ) is eitlier O or 1, we conclude tliat dim U n (M) = 1 , so M is 01-ieiitable; then (*) sliows tliai IR"+' - M has exactly two compoiienrs. Tlie proof in Problem 8-25 shows tliat every point of M is arbitrarily close to points iii diflerent componeiits of IRn+' - M, so every point of M is in the boundary of eacb of the m components. 4.

15. COROLLARJ7 (GENERALIZED [Cm] JORDAN CURVE THEOREM). If M c Rn+' is a submaiiifold liomeomorpliic to S", then Rn+' - M has two coinpoilents, and M is tlle boundary of eacli.

16. COROLLARY. Ncitlier the projective plaiie nor tlle KJein bottle can be jmbedded in R3.

Our iiext inain i-esult will combine some of tlie tlieorems we ali-eady have. However, tl-iere are a nuiill~er of teclinjcalities iiwolved, wliicli we will llave ro dispose of first.

Coiisidei- a boui-ided opei-i set U c R" wllicli is star-sllaped with respect to O. Tlien U can be described a s

U = ( t s : A- E S"-' and O 5 t < p(x)j

for- a ceriain fuiiction p : S"-' + R. \Ve will cal1 p the radial function of U.

If p is Cm, tlien we can prove that U is diffcomoi-pliic to the opeii ball B of radius 1 in R". The basic idea of the poof is to take tx- E B to p(x)t A- E U. Tliis produces difficulties at O, so a modification is necessary.

17. LEMMA. If the radial function p of a star-shaped open set U C Rn is Cm? tlieii U is difTeomorpliic to tl-ie open ball B of radius 1 in R".

PROOF. \Ve can aassume: ~ i t l iou t loss of generality, tliat p > 1 on S"-'. k t f : [O, 11 + [O, 11 be a Cm function with

f = O i1-i a neighborhood of O

f ' > - o f I 1 ) = l .

Define h : B + U b\:

Clearly h is a one-oiie n-iap of B oiito U. It is tl-ie identity in a neigl~borhood of O, so it is Cm. \vitli a non-zero Jacobian, at O. At any other point the same coiiclusioil follo\vs from tlie fact tliat t H t + Ip(x-) - 1 ) f ( t ) is a Cm functio~i witll strictly positive derivative. 4 3

111 general, rlie fiiiiction p i-ieed i-iot be Cm; it miglit not even be coiitinuous.

However: tlle discontinuities of p can be of a certain form oi~l!:

18. LEhdhlA. At each point s E S"-'. tlie radial f~inctioii p of a stai--shaped opeii set U c R" is "lower semi-contiiluous": for every E > O tliere is a neighborl~ood W of s i r i S"-' sucli tl-iat p (y ) > pIx ) - E for al1 E M'.

PROOF. Choose rx E U witl-i p ( x ) - t < E . Since U is open, tliere is an open

l~all B with s E B c U. There is clearlv a neighborl~ood W of s witli tlie p~ 'o~er ty tllat foi- J . E W tlle point t j q is in B. ancl Iicnce iri U, Tllis means thar for. J. E W \ve have po . ) 2 t > p ( x ) - E. e:*

Even when p is discontinuous, it looks as if U should be difTeomorphic to R". Proving this turns out to be quite a feat, and we will be content with proving the following.

19. LEMAJA. If U is an open star-shaped set in Rn, then U k ( u ) =Z Hk(R") and U/ (U) U:(Rn) for al1 k.

PROOF. Tlie proof for U k is clear, since U is smoothly contractible to a point. \Ve also know t11at U," [U) -" R =Z H:[JRn). By Theorem 8-17, we just have to show that H ~ ( u ) = O for O 5 k < ir.

Let w be a closed k-form with compact support K c U. \Ve claim that there is a Cm function b : S"-' + JR such that < p and

K c V = {rx : x E Sn-' and O _< t < ~ I x ) ] .

This will prove the Len-irna, for then V is dfleoinorpl-iic to Rn, and consequentl!. w = dq \vlier.e q has compact support contained in V, and hence in U.

For each x E S"-', choose lx < p(x) sucl~ tliat al1 points in K of the form u x for O 5 u 5 p(x) actually llave u < 1,. Since K is closed and p is lower semi-

continuous: tliei-e is a neighborhood W, of x iii S"-' sucli that t, may also lie used as 1,. for al1 y E W. Let M~,,,.. . , W,, cover S"-', let d i , . . . ,#r be a pai.tition of unity subordiriate to tliis cover, and define

Any poini A. E S"-' is in a certain siibcollection of t l ~ e Wxi , say W,, , . . . , W,, for com~eilience. Tl-ien p/+r [x), . . . , pr [A-) are 0. Each r x l , . . . , fx, is < p(x). Siiice #] (A-) + . + $1 (x) = 1. it follows tl-iat b(x) < p [ x ) . Similarly, K C V. 43

\Ve can applv this last Lemma in the following way. Let M be a compact manifold: and clioose a Riemannian metric for M. According to Problem 9-32, every point has a neighborhood U which is geodesically convex; we can also choose U so tliat for any p E U tlie map exp, takes an open subset of Mp diffe~mor~hically onto U. k t {U', . . . , U,) be a finite cover by such open sets. If any V = Uil n. n Uil is non-empty, then V is clearly geodesically convex. If p E V, then expp establishes a diffeorn~r~hism of V with an open star-shaped set in Mp. 11 follows from k m m a 19 tliat V has the same HP and H: as Rn. In general, a manifold M will be called of finite type if there is a finite cover {UI,. . . , U,) such that each non-empty intersection has the same Hk and H/ as Rn; such a caver will be called nice.

It is fairly elear that if we consider N = {1,2,3,. . . ) as a subset of R2, then M = R2 - N is not of finite type. To prove this rigorously, \*.e fin< use thr Mayer-Vietoris sequence for W2 = M U V, wliere V is a disjoint union of balls around 1,2,3,. . . . We obtain

where M n 1' ]>as tlie same H' as a disjoint union of infinitely many copies of S'; tliis sliows that H 1 ( M ) is infinite dimensional (see Roblem 7 for more infoi.niation about tlle cohomology of M). O n the other hand:

20. PROMSITION. If M has fiiiite type, then HP [M) and H;[M) are finite dimensional for al1 k .

PROOF. BY iriduction on tlle number- of open sets i. in a ~iicc cover; It is clear for r = l . Suppose it is true for a certain r , and consider a nice cover. {U', . . . , U,, U] of M. T l ~ e n tlle tlleor-em is true for V = Ur U . - . U U, and

for U. It is also true for UnV, sii-ice thisl~as the nicecover {UnUi , . . . ,UnU;j . Now consider the Mayer-Vietoris sequence

The map o maps Hk(M) onto a finite dimensional vector space, and the kernel of o is also finite dimensional. So Hk(M) must be finite dimensional.

Tlie pioof for H: (M) is similar,

For ai-iy mariifold M we can define (see Problem 8-31) t l~e cup product map

w H ~ ( M ) x H/(M) -+ H ~ + I ( M )

by

We can also define

by tlie same formula, since w A q Iias compact support if q does. Now suppose that M" is connected a r~d oriented: witli orientation ,u. Tliere is tlien a uriiqur element of H:(M) repi-esented by any q E C , ( M ) witli

It is comler-iient to also use ,u to denote both tliis element of H:(M) and the isomorphism H:(M) + W which takes this element to 1 E R. Now every a E H" M) determines an element of the dual space H:-~ (M )* by

We denote this clement of H:-'(M)' by P! (o) , the "Poincaré duai" of o, so that we llave a map

One of tlle fundamental tlreorems of manifold tl~eory states that P ! is always an isomorpllisrn. We are al1 set up to prove this fact, but we shail restrict tlle tl~eoi-em to manifolds of finite type, in order not to plague ourselves with additional tecl~rlical details. As witl~ most big tl-ieorems of algebraic topology, tlle maiil part of the poof is called a Lemma, and the theorem itself is a simple corollan:

21. LEMMA. If M = U U V for open sets U and V and PD is an isomorpllism for al1 k on U, V: and U n V: then PD is also an isomorphism for al1 k on M.

PROOF. Let 1 = I r - k. Consider the following diagram, in wl~ich t l ~ e top row is the Mayer-Vietor-is sequence, and the bottom row is the dual of the Mayer- Vietoris sequence for compact supports.

By assuinption, al1 vertical maps, except possibly tlle middle one, are isomorpllisms. It is not hard to check (Problem 8) that every square in this diagram commutes up to sign, so tllat by changing some of tlle vertical isomor.pllisins to tlleir llegatives, we obtain a commutative diagiam. We now forget al1 abour our manifold and use a purely algebraic result .

"TH E Fl VE LEh/lMA". Consider tlle followil~g commutative diagram of vector spaces and linear- i-ilaps. Suppose that the rows are exact, and tllat #i, $2.

44, are isomorpl~isms. Then #3 is also ail isomorpliism.

PROOF. Supposc 4 3 ( ~ ) = O for some x E V3. Tllcn P3#3 (x) = O, so #4a3 (x) = O. Hence a3(x) = O, sirlce #4 is an isomorpl-iism. By exactness at V3: tllere i5

1. E V2 witl-i A- = a2(y). Thus O = # 3 ( ~ - ) = #3a2 (y) = P2#2 (y). Hencc #2()*) = r91 ( z ) for some 5 E Wl . Moreover, : = 41 (w) for some w E VI. Then

wllicll implies that y = al (w). Hence

So $3 is one-onc. Tllc ~i 'oof tl-iat #3 is onto is similar, and is left ro the reader. This proves thr,

original Lemma.

22. THEOREM (THE POINCARE DUALITY THEOREM). ~f M is a connected or-iented n-manifold of finite type, then the map

PD: H ~ ( M ) + H:-~(M)*

is an isomorpllism for ail k.

PROOE By induction on tl-ie number I. of open sets in a nice cover of M. The theorem is clearly true for r = 1. Suppose i t ic true for a certain r , and consider a nice cover {Ui,. . . , U,, U) of M. Let V = U1 U. U U,. The theorem is true for U, V, and for U n V (as in the proof of Proposition 19). By the Lemma, it is true for M. Tllis completes the induction step. 4%

23. COROLLARY. If M is a connected oriented n-manifold of finite type, tllen H k ( M ) and N:-'(M) have the same dimension.

PROOE Use tlle Theoren~ and fioposition 19, noting that V* is isomorphic to V if V is finite dimensional. 4%

Even though tlie Poincaré Duaiity Tlleorem holds for manifolds which are not of finite type, Coi-ollary 23 does not. In fact, Problem 7 sliows that N i (IR2 - N) and H,' (IR2 - N) have dúrerent (infinite) dimensions.

24. COROLLARY. If M is a compact corlr-iected orientable n-manifold, then H k (M) and H " - ~ ( M ) have the same dimension.

25. COROLLARY. If M is a compact or-ierltable odd-dimensional manifold, then x(M) = 0.

PROOF. In the expression for x (M) , tlle terms ( - 1 1 ~ dim H'(M) aiid

cancel in pairs. +:+

A more iinrolved use of Poirlcaré duality will aleritually allow us to say much more about tlle Euler characteristic of any compact connected oriented manifold M''. ]Ve begirl by consideiing a smootll k-dimensional orientable vector bundle 6 = x : E + M over M. Orientatjons p for M and v for 4 give an orientation p v for the (n + k)-mar~ifold E. since E is locally a product. If {Ui,. . . , U, 3 is a i~ice cover of M by geodesically convex sets so small that each bundle cIUi is ti-ivial: thei~ a slight modification of the proof for Lemma 19

sliows that {ñ-' ( U ] ) , . . . ,ñ l1 (U,)] is a nice cover of E , so E is a manifold of finite type. Notice also that for the maps

ñ o s = identity of M

s o ñ is smootllly homotopic to identity of E,

so ñ*: H ' ( M ) + H ' ( E ) is an isornorphism for al1 I . The hincaré dualih tl~eorem shows that there is a unique class U E H ~ ( E ) such that

This class U is called tlle Thom class of t. Our first goal will be to find a simpler property to characterize U.

Let Fp = ñ-' ( p ) be the fibre of 6 over any point p E M , and let jp : Fp + E be the inclusion map. Silice jp is proper, there is an element jp*U E H:(Fp). 0 1 1 the other hand, the orientation v for < determines an orientation vp for Fp: and hence an element vp E H: ( F ~ ) .

26. THEOREM. liet (M, p) be a compact connected oriented manifold, and .f = 7~ : E M an orienied k-plane bundle over M with orientation v. Theil the Thom class U is the unique elernent of H , ~ ( E ) with the property that for al1 p E M we have jpSU = vp. (This condition means that

wl~ere U is the class of the closed form o.)

PROOF. Pick some closed form o E C: ( E ) re~resenting U , and let r ) E C n ( M ) be a form representing F , so tllat q = 1. OUT definition of U states that

Let A c M be an open set which is dfleomorphic to IRn, so that A is smoothly conii-actible to any poini p E A . Also choose A so that there is an equivalence

This equivalente allows us to identify IT-' ( A ) with A x Fp. Undes this identification, the map jp : Fp + n-' ( A ) corresponds to the map e H ( p , e ) for e E Fp: which we will continue to denote by jp. We \vil1 also use nz : A x Fp + Fp to denote psojection on the second factor.

Let 1 1 1 1 be a norm on Fp. By choosing a smaller A if necessary, we can assume that tl~ere is some K r O such that: ui~der the identjfication of x - ' ( A ) with A x Fp, the support of wlrr-' ( A ) is coiitained in { ( q , e ) : q E A , lle 1 1 < K j .

, support w

Usir-ig the fact that A is smoothly contractible to p, it is easy to see that there is a smooth liomotopy U : ( A x Fp) x [O, 11 + A x Fp such that

we just pul1 tlie fibres aloiig the smootli liomoiopy which makes A contractible

to e . k r the H constructed in this way it follows t11ar

H ( e , r ) # support w if Ilell 2 K .

Consequentlx tlie form H*w on ( A x Fp) x [O, 11 has support contained in { [ q , e, r ) : lle 11 < K ) . A L. qlance at the definition of 1 (page 224) shows that the

form lH*w on A x Fp has support contained in {(q, e) : llell < K > . Theo- rem 7-14 sllows tllat

Now, on tlle one hand we have (Problem 8-1 7)

O n tlie otller llaild, we claim tllat tl-ie last integral in (3) is O. To prove tllis, i i

~.learly suffices to prove tllat tlle integral is O over A' x Fp for any closed ball A' c A . Since

x*p A cih = f d ( x * p A dh).

S where n*p A A has

(5) J x*p ~ d h = d ( x * p ~ h ) compactsupporton A' x Fp A' x Fp A' x Fp by (2)

= f a ' p n h bv Stokes' Tl-ieorem a A ' x Fl,

= o.

becaiisc tlle form x*p A h is cleai-ly O on a A' x Fp (sincc a A' is (n - 1)-dimeiisional).

Conlbii~ing (3), (41, (5) we see tllai

This shows that JF jpio is independent of p, for p E A. Using connectedness. P

it is easy to see that it is inde~endent of p for al1 p E M, so we will denote i t simply by JF j'o. Thus

Con-iparing with equation (l) , and utilizing partitions of un ig we conclude tl-iat

wliich proves t l~e first part of tlie theorem. Now suppose we llave another class U' E H,k (E). Since

it follows tliat U' = cU for some c E R. Consequently,

Hence U' l~a s the sarne propcrty as U only if c = 1. 4 3

The Tl-iom class U of $ = n: E + M can now be used to determine an element of Hk (M). Let S : M + E be any section; tliere always is one (namel): i11e O-sectior-i) and ar-iv two are clearly smoothly liomotopic. We define t l ~ e Euler class ~ ( $ 1 E H k ( ~ ) of 6 h\.

R'otice that if 6 has a non-zero section S: M + E, and o E C;(E) rep- r-esents U, then a suitable multiple c . s of S takes M to the complement of support o. Hence, in tllis case

T l ~ e termiiiology "Euler class" is connected witli the special case of the bundle TM, wliose sections are, of course, vector fields ori M. If X is a vector field on M wl-iich has aii isolaced O at some point p (tl~at is, X(p) = 0, but X(q) # O for q # p iri a rieighborl-iood of p), tl~en, quite independently of our- previous considei.atioi~s, \ve can define an "ii~dex" of X at p. Consider first a vectoi-

field X on an open set U c W n with an isolared zero at O E U. We can define a function fx: U - {O) + S"-' by fx(p) = X(p)/lX(p)l. If i: S"-' + U is i(p) = EP, mapping S"-' into U, then the map fx o i : S"-' + S"-' has a certain degree; it is independent of E, for small E , since the maps i ~ , i2 : S"-' + U corresponding to E] and ~2 will be smoothly homotopic. This degree is called the index of X at O.

index O index O index 1 index I

index - 1 index 2 index -2

iildex 1 in R" index (-1)" in Rn

Now consider a diffeom~r~hism 8 : U + V c Wn with h(0) = O. Recall thar h,X is the vector field on V with

Clearly O is also an isolated zero of h , X

27. LEMMA. If h : U + V c R" is a diffeomorpllism with hIO) = 0, and X has ai-i isolated O at O, tllen tlle ii-idex of Ir,X at O equals the index of X at O.

PROOF. Suppose first that h is orientation preserving. Define

U : IR" x [0,1]+ Rn

This is a smooth homotopy; to prove that it is smooth at O we use Lemma 3-2 (compare Problem 3-32). Each map U, = A- t+ H I x , t ) is clearly a dfleomorphism, O < r < 1 . Note that Ui E SO(n), since h is orientation preserving. There is also a smooth homotopy {Ut], 1 _< I < 2 with each Ht E SO(n) and Hz = identity, since SO(n) is connected. So (see koblem 8-25), tlle map h is smoothly llomotopic to the identity, via maps wliich are dfieomorphisms. This sliows that fh,x is smoothly homotopic to fx on a sufficiently small region of IRn - { O ] . Hence the degree of _fh,x o i is the same as the degree of fx o i.

To deal witll non-orientation preseniing h, it obviously suffices to check the theorem for h(x ) = ( x l , . . . , xn-l, -xn). In this case

which shows that degree h , ~ O i = degree fx o i. 4 3

As a consequence of Lemma 27, we can now define tlie iiidex of a vector field on a manifold. If X is a vector field on a manifold M , with an isolated zero at p E M , we clloose a coordinate system ( x , U ) with x ( p ) = 0 , and define the index of X at p to be the index of x,X at O.

28. THEOREM. Let M be a compact connected manifold with an orientation F , whicll is, by definition, also an orientation for the taiigent bundle 6 = IT: TM + M . Let X : M + T M be a vector field with only a finite iiumber of zeros, and let a be the sum of the indices of X at these zeros. Then

PROOF. k t pl , . . . , p, be t he zeros of X. Clloose disjoint coordiiiate systems ( U ] , x l ) , . . . , (Ur, xr ) with xi(pi) = 0, and let

If w E CF(E) is a closed form representiilg the Thom class U of $, then we are trying to prove that

f

\Ve can clearly suppose tliat X(q) # Support w for q # Ui Bi. So

tlius it suffices to prove that

(*) X *(u) = index of X at pi.

It will be conueilient to drop the subscript i from now on. We can assume tliat T M is trivial over 3: so that n f l (B) can be identified

with B x M,. Let j p aiid n2 llave tlie same meaning as in the proof of Tlie- orem 26. Also clioose a iiorm 11 11 on M p . We can assume tllat uiider tlie identification of n-' (3) ~ i t h B x Mp, the su1ipo1-1 of wln-' (3) is contained iii {(y, u) : y E A , llvll 5 11. Recall fiom tlle proof of Theorem 26 tliat

Since we can assume tliat X(q) support h for y E 8 3 , we llave

O n tlle maiiifold Mp we llave

p aii [ I T - 1 )-form oii Mp jp* w = dp (witll non-compact support).

If D c Mp is tlic unii disc (witli respect to tlle iiorni 11 1 1 ) and S"-' deiiotes 8D c M,, tliei-r

- by Tlieorem 26, and the fact - 1. tllat support jP*w C D.

Now, for q E B - {p}, we can define

and f : aB + T M is smoothly homotopic to X : al3 + T M . So

(31 S, x*x2*( jp*u) = J, x*x2* d p

X*x2*p by Stokes' Theorem

From tlle defiilition of tlle iildex of a vector field, together with equation (2), it follows tllat

L B ( x 2 0 Z * ) p = index of X at p.

Equations (1), (31, (4) togetlier imply (*). 43

29. COROLLARJ7. If X aiid Y are two vector fields witli only finitely many zems on a compact orieiitable maiiifold? tllen the sum of the indices of X equals tlie sum oi' tlie indices of Y.

At tlie monicilt, we do not even kilow that there is a vector field on M with finitely nianv zeros, iloi- do we know what this constant sum of the indices is (althougli our terniiilology certainly suggests a good guess). To resolve these questions, we consider once again a triangulation of M . MJe can then find a vector field X witll just oile zero in each k-simplex of the triangulation. We begin by drawing tlie integral curves of X along the 1 -simplexes, with a zero at cacli O-simplex and at oiie point in eacll 1-siinplex. \)Ve then extend this picture

to ir~clude tlie integral cunres of X on the 2-simplexes, producing a zero at oní

point in cach of tliem. We tlien continue similarly until tlie n-simplexes are filled.

30. THEOREM (POINCARÉ-HOPQ. Tlie sum of tlie indices of t his vectoi. field (and Iience of aiiy vector field) on M is tlie Euler characteristic X( M) . Tlius, for $ = n : TM + M we have x($) = x(M) p.

PROOF. At each O-simplex of tlie triangulation, the vector field looks like

with index 1. Now coiisider tlle \lector field in a iieigliborhood of tlie place where it is zero

on a 1-simplex. The vector field looks like a vector field on W" = W1 x IR"-' wliicli points djrecOy iiiwairls on W1 x {O) and directly outwards on {O) x W"-' .

t (b) j? = 3

For 11 = 2, the index is clearly - l . To compute the index in general, we note that fx takes the "north pole" N = (0, . . . ,O, 1) to itself and no other point qoes to N. By Theorem 8-12 we just have to compute signN fx. NOW at N we L

can pick projection on Wn-' x (O) as the coordinate system. Along the inversc image of the x'-axis the vector field looks exacily like figure (a) above, where w already know the degree is - 1, so fx. takes the subspace of S"-'N consisting of tangent vectors to this cuive into the same subspace, in an orientation reversing way. Along the inverse image of the x2-, . . . , xn-'-axes the vector field looks likc

so fx. iakes the correspoiiding subpaces of S n - I N into themselves in an orientation preservjng way Thus signN fx = -1: which is tlierefore the index of the vector field.

In general, aear a ze1.o within a k-simplex, X looks like a vector field on W" = IRk x Wn-hrliicli points directly inlvards on JRk x (0) and directly outwaids on (O) x IRn-'. The same argument shows that the index is ( - 1 )k.

Consequently, the sum of the indices is

a0 - al + a2 - . = X ( M). 43

\Ve end tliis chapter with one more obsenlation, which we will need in the last chapter of Volume V ! Let 6 = n : E + M be a smooth oriented k-plane bundle over a compact conliected oriented 12-manifold M, and let ( , ) be a Riemannian metric for 6 . Then we can form tlie "associated disc bundle" and "associated sphere bundle"

D = (e : (e,e) 5 1 )

S = ( e : (e,e) = 1 ) .

It is easy to see tl-iat D is a compact oriented (iz + k)-manifold, wjth aD = S; mor.eover, tlie D constructed for any other Rieniannian metric is diffeorn~r~liic to this one. We Iet no ; S + M be 17 1s.

31. THEOREM. A class a E H k ( ~ ) satisfies no*(a) = O if and only if a is a multiple of ~ ( 6 ) .

PROOF. Consider the following picture. The top row is the exact sequence

H k ( M )

for ( D , S ) @ven by Theorem 13. The map S : M +- D - S is the O-section: while i: M + D is tlie same O-section. Note that everything commutes.

no* = i* o (n 1 D)* since no = ( n I D ) o i;

S* = S* o C) since extending a form to D does not affect its value on S ( M ) ,

and that S* o (n lD)* = identity of H k ( M ) ,

since [n 1 D ) o i is smootlily homotopic to the identity Now lei a E H ~ ( M ) satisfy no*(a) = O. Then i*(n 1 D)*a = O, so (nID) 'a E

iinage L.. Since D - S is diffeomorpl~ic to E, aiid every element of H: ( D - S ) is a niultiple of the TIiom class U of t, we conclude that

Tlie proof of the coiiverst: is similar.

Iixirn:rio?i i?r lhr. Ren hn of . A Lyth?uic C 7Új)ologl.

PROBLEMS

1. Find Hk(S' x . x S') by induction on the number rr offactors. [Answei.: dim H' = (;).]

2. (a) Use the Mayer-Vieioris sequence to determine Hk ( M - { p ) ) in terms of H k ( ~ ) , for a connected manifold M. (b) If M and N are two connected 11-manifolds, let M # N be obtained by joining M and N as shown below. Find the c o h o m o l o ~ of M # N in terms of that of M and lb'.

(c) find x for ihe rt-holed torus. [Answer: 2 - 21n.J 3. (a) Find ~ ~ ( A 4 o b i u s strip). (b) Find Hk (lP2). (c) Iind Hk (P"). (Use Problem 1-15(b); it is necessary to consider wliether a neighborliood of BW-' iii B" is orientable 01. not.) [Answer: dim Hk(P") = 1 if k even and 5 11: = O othenuise.] (d) Rnd Hk ( ~ l e i n bot de). (e) Fiiid tlie co l ion io lo~ of M # (Mobius strip) and M # (Kleiri bottle) if M is the n-l-ioled torus.

4. (a) Tlie fipr+e I~eIow is a triangulation of a rectangle. If we perform the indicated ident&catioris of edges we do ?iol obtain a triangulation of the torus. Why not?

(b) Tlie figure below does give a triangulation of the torus when sides are identified. Find ao, u ' , u2 for this triangulation; compare with Theorem 5 and Problem l .

5. (a) For any trianplation of a compact 2-manifold M, show that

(b) Sliow that for triangulations of s2 and tlie torus T~ = S' x S' we have

Find iriangrilatioiis for wlijch iliese incq uaiities are al1 equalities.

6. (a) Tiiid H! (S" x R"') 11y iiiductioii oii n, using tlie Mayer-Vietoris sequenci for compact supports. (h) Use tlie exact sequeiice of ille pair [S" x Rm, { p ) x Rm) to compute the same vector spaces. (c) Compute H ~ S " x snl-' ), using Tlieorem 13.

7. (a) The vector space ~ ' ( 3 - N) may be described as tlie set of ail sc- queiices of real numbers. Using the exact sequeiice of the pair ( R 2 , N ) , show tliat H!(IR2 - N ) may l ~ e corisidered as the set of al1 real sequeiices ( a , ) such tliat a, = O for al1 but fiiiitely many 1 1 .

(h) Describe tlie map PD: H1(IR2 - N ) + H,'(IR2 - N)* in terms of these desci-iptions of H' (LR2 - N ) and H: ( IR2 -N ) . aiid show thai it is an isomoi-phism. (c) Clearly H:(LR2 - N ) has a countable basis. Show that H1(LR2 - N) does iiot. Hiiil: If vi = { u i j ) E H 1 ( R 2 - N ) , choose (b1 ,b2) E R2 linearly iiidependent of (a11 ,a12) ; tllen choose (b3 , b4, b s ) E R3 linearly independent of boih ( a l 3 , a l4 , a l S ) and ( í123, nz4, nz5); etc.

8. Show that the squares in the diagram in tlie proof of Lemma 21 commute, except for the squaw

wliich commutes up to tlie sign ( - I ) ~ . (It will be iiecessary to recal] howvarious maps are defined, which is a good exercise; the only slightly difficult maps are tlie ones involved in the above diagram.)

9. (a) Let M = Mi U M2 U M3 U. - be a disjoint union of oriented n-manifolds. Show that H! ( M ) H: ( Mi) , this "direct sum" consisting of al1 sequences

( a l , a2 , a3 , . . . ) with aj E H:(Mi) and al1 but finitely rnany ai = O E H$ ( M i ) . (b) Sliow that H k ( M ) % ni H k ( M ~ ) : this "direct product" consisting of nll sequences ( a l , a2, a3, . . . ) with ai E H k ( M i ) . (c) Show that if tlie Poincaré duality theorem holds for each Mi, then it holds for M. (d) The figure below shows a decomposition of a triangulated 2-manifold into tliree open sets U*, U1, and U2. Use an analogous decomposition in n dimensions to prove that Poincarii duality holds for any triangulated manifold.

U. is union of shaded (-1) U2 is union of shaded .::.ii;i::' i\ ,,;,

10. Let 6 = rr : E + M and 6' = rr' : E' + M be oriented k-plane bundles. over a compact orieiited manifold M , and (f, f ) a bundle map from t' to t \vliicli is an isomorphism on each fibre.

(a) If U E H: ( E ) and U' E H: ( E') are the Thom classes, then f * ( ~ ) = U'. (b) . [* ( x ( c ) ) = ~ ( 6 ' ) . (Using tlie iiotation ofProbIem 3-23, we Iiave f* Ix IE) ) = xrf*rt>>.) 11. (a) Let 6 = T : E + M be an oriented k-plar-ie bundle over an oiiented rnalijfold M , witli Tliom class U. Using Poincaré duality, prove tlie Tliom Isomorpliism Theorem: Tlie map H i ( E ) + H:+~ ( E ) @ven by a t+ a v U is an jsomoi-phism for a11 i . (11) Since we can also coiisider U as being in H k ( E ) , we can form U v U E

H : ~ ( E ) . Using anticominutativity of A, sliow that tliis is O for k odd. Concludr tliat U represents O E H ~ E ) ) , so that ~ ( t ) = 0. It follows, in particular, tliai ~ ( t ) = O wheii 6 = rr : TM + M foi- M of odd dimension, providing anothel proof that x ( M ) = O in tliis case.

12. IS a \rector field X has aii isolated singularity at p E Mn, show that thc iiidex of - X at p js (-1)" times tlie index of X at p. This provides ariothel- proof that x ( M ) = O for odd n.

13. (a) Let pr, . . . , pr E M. Usir-ig Problem 8-26, sl-iow that there is a subsei D c M diffeomor.pliic to the closed ball, sucli tliat al1 p; E interior D. (h) If M is compact, tlien tliere is a vector field X oli M with only one siiiyu- laritx (c) is a fact that a Cm niap j. : S"-' + Sn-' of degi-ee O is sinootlily Iio- niotopic to a coiistant map. Usii-ig this, show tliat if x ( M ) = O, theii tliere is a lio~rliere O vector field on M . (d) lf M is connected ai-id not compact, then there is a nowl-iere O vector field ciii M. (Begiri wjth a tria1i~gu1atioi-i to obtaiii a vector field with a discrete set oj zeros. join tliese by a ray goirig to infirlity, enclose tliis ray in a cone, and pus11 ever-ything off to irifiriit\:j

(e) If M is a coririected riianjfold-witli-boiriidary. with aM # 0, tlien tliere is a rioivher-e zero vector field on M .

14. This Problem proves de Rliam's Theorem. Basic knowledge of singular col~omoIogy is rcquired. l e wiil denote tlie group of singular k-chains of X by Sk(X). For a manifold M, we let S r ( M ) denote the Cm singular k-chains, aiid let i : S r ( M ) + S k ( M ) be the inclusion. 11 is not hard to show that theiac is a cliair-i map r : Sk(M) + Sr( M ) so that r o i = identity of Sr( M), wliile i o r is chain homotopic to the identity of Sk( M ) [basically, r is approximatioi~ by a Cm cbain]. Tliis mearis that we obtain the correct singular cohomology of M if we consider the complex H o m ( S r ( M), R).

(a) If w is a closed k-fo~-m on M, let Hl(w) E Hom(Sr [M) , W) be

Show tliat Rli is a cllain iiiap from ( c ~ ( M ) ) to ( H o m ( S ~ ( M ) , R ) ) . (Hitii: Stokes' Tlieol-em.) It follows t hat t here is an iriduced map Rii from the de Rliam cohomology of M to tlie singular cohomology of M. (b) Sliow tliat Rii is aii isomorphism on a smootlily coritractible manifold (Lem- mas 17, 18, and 19 will not be necessary for this.) (c) Imitate the proof of Theorem 21, using the Mayer-Vietoris sequence for singular. col-iomology, to show t hat if Rii is an isoniorphism for U, V, and U n V: tlieti it is an isomorphism for U U V. (d) Coiiclude that Rii is aii isomorpliism if M is of finite type. (Using the method of Problem 9: it follo\vs that Rii is an isomorpl-iism for any trianLgulated manifold.) (e) Check that tlie cup product defined using A corresponds to tlie cup product defined in singular cohomology.

APPENDTX A

CHAPTER 1

Following the sugpestions in tl-iis chapcer: we wiII no-\v define a manifold to b i a topological space M such that

(1) M is Hausdofi

(2) For eacli x E M there is a neighborliood U of x and an integer n > O such that U is homeomorphic to IR".

Condition (1) is necessary, for there is eveii a l-dimensional "manifold" which is i-iot Hausdorff. l t consists of R U (*) where * # ~ 4 t h tlie following topo lo^-\.: A set U is open if and only if

(1) U n R is opeii:

(2) If * E U. thcn (U n R) U (O: is a neighborliood of O (in R).

Thus tlie neigliborlioods of * look just like neigl~borhoods of O. Tliis space mav also be obtained by idei~tif~ring al1 points except O in one copy of R with thr correspondirig point in anotlier copv of R. Alt l-iougli non-H ausdofl manifold? are important in certain cases, we ~ri l l not consider tllem.

We llave just seen that the Hausdoflproperty is not a "local property", but local compactiiess is, so every maiiifold is locally compact. hgoreover, a Haus- dorfflocally compact space is regular. so every manifold is regular. (By the wa)i this argunient does i~o t wor-k fol. "infiiiite dimensional" manifolds, which are lo- c a l ] ~ like Banacli spaces; these iieed iiot be regular even if they are Hausdofl.) On tlie otlier liaild, there are manifolds whicli are not iiormal (Problem 6). E\:- ery mariifold is also clearly locally coniiected, so eveq component is open, and ihiis a mariifold iiself. Before exhibitjiis noii-rnetrizable manifolds, we first note tl-iat alnlost al1 "nice" propertie.; of a manifold are equivalerit.

THEOREhl. The following pr-operties are equivalei-it for any manifold M:

(a) Each cornponent of M is a-compact.

(b) Eacli coinpoiieilt of M is secoild coui-itable (lias a countablc base for t he t o p o l o ~ j .

(c) M is metrizable.

(d) M is paracompact.

(In particiilai: a compact mariifold is metriza ble.)

FIRST PROOF. (ai a (b) foIlo\m immediately fi-om tlie simple propositioil tliai a o-compact IocaIlv second countable space is second countable.

(11) a (e) follo-\vs fi-om tlie Urysohn metrization tlieorem. (cj 4 (d) because any metric space is paiaacoinpact (Kelley. Ckieraf 7Üpofog.i..

pg. 160). TIie second pr-oof does iiot rely on this dificult theorem. (dj + (a) is a consequeiice of the follouliii?,

LEMMA. A coiiiiecied. locally coinpact, ~~ai.;icoiiipact space is o-compaci.

Ao@ T1iei.e is a locally fiiiite covei' of tlie space by open sets with coinpact closure. If U. is oiie ofthese. then U. can interseci oiily a finite numl~ei- L'i, . . . , L',,, - of ilie otbers. Siiiiilai.ly Üo U Üi U - - U U,I, iiiiersects only Un,+i. . . . , U,,: and so on. The uiiioii

is clcarl!; opeil. It is also closed, foi- if A- is i i l ilir closure! then x níiisi be i i i

the closure of a finiie iiilion of these Ui. becausc s has a neighboi.liood \vliich iiitei.sects oiily fiilitel!~ niany. Tlius x is iii the uilioii.

Siilce the spacc js coililecied, ii equals this couiitable unioii oi' coiilpact sets. Tliis proIres tlie Leninia aild the Tlieorcm.

SECO.ArR l'R001': (a) d (b) 3 (c) aiid (d) d (a) as befoi-e. (c) a (a) is Theorem 1-2. (a) d (d). Le1 M = C1 U C7 U. a a . \v l~c i .~ each Ci is conlpact. Clcai-lv C1 ha'

an opeii iieiglil~orliood Ui \r?iili compaci closure. Tlieii Ül U C2 lias a11 ol~eil iiriyliborhood fi wjtll ('OiiilIact closui*e. Coniiniiiiig iii thjs way, we obtaiii open seis Ui witli Ui roiiqlaci aiid G. c Ui+i. whosr uiiioii coniaiiis al1 Ci. aiid lieiicr is M. It ir; easv to sliow li-onl thjs tliat M is pai-acoml~act. 43

It iuriis out thai t l i ~ i - ~ aiLr eveii 1 -mailifolds wliich a1.e noi paracoinpact. Thr- coilst ruction of ihrse exain~~lcis I-equires t11e ordiilal iiumbers, whicli are bricfl~. cxplaiiied herc.. (Oi.diiial iiiiiii11ei.s will iioi 11c needed for a 2-diineiisioiial example io come latei:

ORDINAL NUMBERS

Recall tliat a n orderine < on a set A is a i~latioil sucli tliai

(1) a < b and h < í- implies a < c for al1 a , b, r, E A (transitivit?.~

(2) For al1 a , b E A : one aild onlv one of the follo1~41ig holds:

(i) a = h (ii) a < (trichotom?~).

(iii) b < a (also written cr > h q

An ordered set is iust a pair ( A : i ) wliere < is an orclering; on A . 1-1410 oi-dered sets ( A , <) aiid ( 3 , + ) are order isomorphic i f there is a one-one onto hiictioii , f ' : A + 3 sucli that a < b implies / ( a ) < .f(b): t11e map $ itself is called al1 order isomorphism. aiid , f - 5 s easih: seeii to be an oi.dei- isomorphism also.

An orderine < on A is a well-ordering if every non-eimpty subset 3 c A has a,prs/ clement. tl~at is. ail element b such tliat b 5 b' for al1 b' E B, Somc well-orda-ed sets are illustrated below: in tliis scl~eme we do 1101 list anv of the <

rrlations whicl-i are co-iiseqiieiices of the olies already listed.

0

(01

O < I ( A = ( 0 , l ) )

O < ] < ? ( A = (0, 1,3)!

O i 1 < 3 < . 7 etc.

0 < 1 < 3 < . 7 < . . .

O < l < 3 < . - . < w (w is some set # O , ] , 2,3.. . . ,

(W + 1 is? fol. the pi-esei~i. O < l < 3 < 4 . . < w < w + I just a set distjnct from

tliose already meiitioii ed 1

An! subset of a IY~II-ordered set is. of course, also a well-ordered set wjtli tl-il same 01-derjiig. 111 particulai: a subset B of a 1ve11-oi.dered set A is called aii (initial) s e p e n t i f b E B and a < h imply a E B. It is easv to see that if B is a

serinent of A , then eitlier 3 = A 01. else there is some a E A such thai

B = (a' E A : a' < a)

in fact. a is tl-ie first element of A - B. Notice tliat each set on our list is ci

s c ~ ~ i ~ e i ~ t of the succeeding oiies. It is not liard to s ic tl-iat 1-10 two sets oii our lis1 are 01-der isomorpliic. For example.

O < l < . . - < o and O < l < . - . < o < # + ]

al-c 1101 ordei- isomoi-pliic because the second Iias botl-i a last and a ~iext to lasi elei-ileiit. while tlie firsi does 1101. But thei-e is a mucli inore geiieral proliositioii whic11 wjll settle al1 cascs ai oi-~cc:

1. PROPOSITION. If B # A As a seLment of A . theil B is iiot ordei isomoi- plijc to A . 11-1 fact, tlii oiily oi-der isomoi.]ilijsni kom B '10 a ~egirleni of A is tlii jdeiititv.

I'ROOF. If J ' : B + 3' c A is ai-i oi.dei. jsornoi.pliism aiid 3' is a seLmei~i of A . tlien fol. tlie fii-si rlcn-iei-it b of B (aiid Iieiice of A ) we cleai-Iy I ~ U S I lia\:c. J ( h ) = b. Tlien J'(b1) n~ust be b'. where b' is tlie second element. And so oii. i \ re~i foi- the " w ' ~ " eleiiieiit (tlie fii-SI oiie after tlie Iirst, second, tliird, etc.)! TIir .\va!- 1 % ~ pi-OIT t liis iigoi.ou.sly is alnaziiigly simlile: 11- J'Ib) # b for some b E B. just coiisider tlie first eleinent of ( b E B : f (b ) # b ) : aii ouiriglit contradjctioii aplicai+s almost immediatel\r. Q

Pi.oliogtioii 1 lias a com~~a~i io i i . wliicli makes tlie study of well-ordei-ed set? simplv delightful.

2. PROPOSITION. If ( A , <) aiid (3, <) are ~rell-01-dered sets, theli oiie il; oider- iso~norphic to a segment of the other:

PROOF. W7e match tlic hrst elemeiit of A m i t h tl-ie first of 3, the second witli tlie secoiid, . . . , the "oth" aitli tlie "oth", eic., uiiiil we ].un out of one sri. To do iliis rigorously, coi-isjder order isomorphjsms fi-om segments of A oiito sepnieiiis o f B. Ir is easy to S ~ O M ~ tliat any two sucli oi-del. isomor]iliisins ag.rcs o11 tlic sniallei. of tlieii. t\vo domains (just co~isider- the smallest element \vliei.r

tliev don't). So al1 sucli ordei. isomorpliisms caii be yut together to give anotliei: ~vhicli is clearlv the largest of all. If it is defined oil al1 of A we are done. l f ir js iiot: tlien its raiige must be al1 of B (or we could easjlv extend it) and iue ai-c still do11 e. *:*

Suppose wc defiiie a relation < bebveen iuell-oi-dei-ed sets by stipulating tliar ( A . <) < (B,<) wlieii ( A . <) is oi.der isoinorphic to a proper segment oí (B. <). Ti-aiisitivihr of < is ob\lious, and Propositions 1 and 2 show tliat we al- inost liave tricliotom. "Niiiost", because the coiiditioii "(A, <) == (B, <)" musr be i4eplaced by "(A, <) order isomorphic to (B. <)". TO obviate this djficult?. we need oiilv nrork witli 01-dei- isomorplijsm classes of nrell-ordered sets, instead of witli tlie weII-oi.dei.ed sets theniselves. These ol-del- isomorphism classes arc. called ordinal numbers. They are beautiful:*

3. PROPOSITION. is a i~rell-orderiiig of tlie ordinal numbers.

I'ROOF. Giireii a non-empty set A of ordiilal ntimbers, let ( A , <) be a well- oi-dered set repi-esciiting one of its elcmerits a . To pi.oducc a smallest elemenr of A \ve can obviouslv ipiiore elemeilts > cx. E\~erv element < cx is repi.esented lxj aii 01-dei-ed set wliicli is order i.soinorp1iic io soine proper segmeii t of A : eacli of tliese is tlie sepileiit coi-isisting of eleiileiits of A less tlrat some u E A . Coi-isider tlre least of tliese a's. It detei-mines a segmeiit which represents some f i E ,A. Tliis #l is tlie snldlest elemeilt of A, 4 4

Kotice tliat if a is aii oi-dinal xiumber, i-epreseiited by a well-ordered set ( A . <), tlieri tlie \vell-01-dered set of al1 oi-diiials f i < cx lias a particularly simple i.epi.esentation: it is ordei- isomorphic to the set { A . <)! Rouglily speaking: AII oi.dinal irumhi- is oidei. jsoxiioi-pliic to tlie set of al1 oi-diilals less tlian it.

If a is an ordinal iiumbei; we will deriote by cx + 1 tlie smallest ordii~al after cx (if cx is iaepresented by dre well-01-dered set (A, <). tlien cx + 1 is represented I n ; a \krell-ordered set witli just one moi-e eleniait. larger. thail al1 member.~ of A) . Notice tliat some ordjnals are not of tlie forin a + 1 for any a ; these are called limit ordinals, wl-iile those of tlie form cx + 1 are cailed successor

*OilIi. oiie feature niars the beauty of tlie ordina1 iiuiiilriers as preseiited here. Eacl-i oiliiiá~ nurnber js a horribly larg set; it would he mur11 iiicei. to rhoose one specific well- oi.dei-ecl ser fmn-i eacli oi.dei- jsoinorphisrn rlas, aild drfine these specjfic sets to he tI3c 01-cliilal iiumhcrs. Tliei-e is a panic.ularJy eIegaiit way to do t h i ~ due to Lron Nairnaiiii, ~ b h i c h can be fouiid in tlie Appei~dix to Kelley, C;Encral IÓpolog~.

ordinals. \Ve \vil1 also denote some ordinals by the svmbols appearing before: 0, 1,2.3 ...., w,w + 1 ,... , etc.

Oui- list of well-ordered sets only begins to suggest the complexity which well- ordered sets caii acliieve. \dTjtli a little tliouglit, one caii see how the syrnbolc w w . . . . would appear (symbols hke w 3 + w2 . 3 + w 4 + 6 would be used soinewliere between w3 aild w4): after al1 tliese oiie ~ lou ld need

and after al1 tliese the synbol pops up. After

one comes to

and this is oi~ly the begiiiiiiiic! Al/ tlic wcll-oi.dcied sets iiieiit ioiied so fai- are rori~iinb/r. Thei-e are indeed aii

~1io1.1110~~ nuinl~er of couii talde well-ordered sets:

4. PROI'OSI TlOn'. Let R be tlir collectioii of al1 countable ordiiials (ordinalc 1.v-esrilted bv a countable wcll-oi-del-ed set). Theii R js iii~countablc.

I-'ROOE BV P~.opositioii 3. ( R. <) is a well-ordei-ed set. l f jt were countable. ji ~vould i-epresent a countal~lc 01-djlial cx E R . By tlie remark after Pi-oposition 3. thjs would nieaii iliat 52 is ol.clri. isoinorpliic to tlie collcctioii of orclinals < a . i.c.. to a prolxr scgmeiit of jtsclS. conti*adicting* Proposition l . e:+

\Vr hm;c tlius e~tablished tlie existente of an ulicouiitable ordinal. Oui. spt-- ciíic rxainplc. i.cpl.csellted l ~ y R , js clcai-lg tiic fii'st ulirouiitable oi.dii~al; aiiy mrmbei- of R is countable, and consequentl\~ has onlv countablv mariy p1.r- clec.essors. (lt is liopdess to ti.? to "reacli" R bv co iitiiiuin~ the listing of well- ordei-ed sets begiln abmle. Soi. oiie ~ ~ o u l d liave to go u~icountably far, aild cil- roLiiitrl- S C ~ S with an uiicouiitaI>le ~iuinber of degrees oi- complexity. A leap oí fai tli ir; required.)

Altl~oii~li tli(% r-oiiiit;iMc oi.dnials exliibjt ulícountabl\: manv degrees of coiil- l~lcxit,; tliey are eacli simple j i i one wa.1

* 13y drlrtii~q tlle words coiiiiial~lr+ aild iinroui~taI~le ili this ~-ii.ool'one obiaiiis thc "Burali- Fo i-t i l'iii.~iclos": thr se i Ord ol' al1 oi-cli iial nurnhers js well-oi-dei-ed, so it represeil ts ñn oi-dii-ial a E M. aiicl hencr is oi-clvi isoinorpl-iir to ail iiiitjal sCgiiirilt of (M. ]coi- i-esoliii ioii of iliis pai-ados, ser Kclley's Appei~clís.

5. PROPOSITION. If a E R is a limit ordinal, tlien there is a sequence #?t < #?2 < P3 < - . . < a, such that every #? < a satisfies #? < #?, Sor some n (we sal. that {#?,) is "cofinal" in a).

PROOF. Sirice a is countable: all its members cari be listed (in not-necessarilv iiicreasing order) y,, y2, y ? : . . . . Let #?t = yt and let #?,+i be the first y in the list wliicli comes after #?,. +

6. COROLLARY. If a E R. then cx is represeii ted by some well-ordered subset of R . However, iio subset of R is order isomorpliic to R .

I'ROOF. Suppose the1.e were orie, and heiice a smallest, cx E S2 not repi-esentcd bv some subset of R. l t carinot happen that a = #? + 1, Sor then #? would bc 1-epi-esented by a subset of R t thus also by a subset of (-ca,O) and a could b\: 1-epi-esented by a subset of R. So by Proposition 5, there is a sequence jl1 < #?* < P3 < - ' - < a cofinal in a . Tlien Di is represented by a suhset oi (-m. i). and we can easilv arrailqe that the subset representing Di is a seL~ei- i i of tlie subset represeritiiig #?, fol. i < j . Tlie union of al1 these sets would then represent a , a contradiction.

If a subset of R wei-e order isomorpliic to R ? tliei-i there would be uncouiltablv niariv disjoint inteivals in R, iianieiy tliose betlveeii tlie poiiits r-epresentin~ a aiid a + 1 Sor al1 a E R. Thjs is impossible. O

The first example ol' a noil-metrizable mar-iifold is defined in terms of S2. Consider R x [O, l ) ? witli tlie order < defined as follows:

b , ~ ) < ( # ? J ) ifcx < #? or ii'cx = #? ai ids < t .

'Tliis can be pictured as follows:

Tlie set R x [O, 1 ) \vitli tlre order topology (a suM3ase coilsists of sets of tlie forin (.Y : < xo) aiid {S ; .v > -yO:) is called tlie closed longray (witl-i "origin" (O, 0)). aiid L+ = R x [O, 1) - ((0, O)) is t l~e (open) long ray. Tlie disjoint union of two copies of the closed long rav with tlieii- origins identified is tlie long line L. 7-0 distiiiguish L+ and L. the iiames "half-loiig liric" aiid "long h e ' ' may also bc irsed. Tlie Coro11ai-v to Pi-oposition 5 ii-ilplies easil!. tliat the long ray and tlir lorig liiie are 1-din~ensional rnatiifolds; aside fi-om tlie line and the circle, t1iei.c are iio otlier coniiected t -manifolds.

Quite a femr new Zmaiiifolds can now be constructed:

L+ x S' (hall-long cvlinderj, L x S' (long c~linder).

L+ x W (half-Ion9 strip,. L x Jli. (loilg strip),

L x L (big ~lanc;. L x L+ (big hall-plane),

L+ x L+ (big quadrani).

ldeiitifying al1 points ( (0 , O ) , 8) iii the product of ihe closed long ray and S' pi.oduces another 2-manifold. w11jch might be called the "big disc".

Tliere is aiiotliei- way of producing a non-metrizable 2-manifold which d o e ~ not use R at ell. \Ve begin 14th tlle open upper hall-plane R: = ( ( x , y ) E IE2 : J . > 0 ) and another copy W 2 x ( O } of the plane; we will denote thjs set by W i . aiid denote the poini (.Y, }.,O) by (A-, Define a map / o : (w: )+ + IR: b!.

Coiisider the disjojiii unjoii of IE: and IE;, wiih p E (IR:)+ and h ( p ) E IP;; ideiitified. Tliis is a Hausdorfl' manifold; the following djagram sliows two open seis liomeomorphjc to W 2 . The manifold itself js in fact, homeomorphir

io IR2: wc could Iiave thrown away R: to begin \vi111 siiice it is idcnijfied ly a lion~íomorphism with ( W i ) + .

Bui roiisidrr iiow, for eacli a E W , another copy of W 2 , say W 2 x { a } , whicli we \vil1 denoie by W z . Deliiie f.: (P:)+ + W: b!.

In tlie disjoint union of IR: and o!! IR:, a E W mre wish to identify each p E (IR:)+ with f,(p) E IR:. \+e may dispense irrith IR: complerely, and in the djsjoini uiiion of al1 IR: ideniify each ( x , g ) , and ( ~ ' , p ' ) ~ for which y = 11' > O and x!.+a - x ' ~ ~ + b . The equivalence classes, of course, are a space homeomorphir io IR:, so we will consider IR: a subset of tlie resulting space. This space is still a Hausdorff manifold, but it cannot be: second countable, for it has aii uncouri table discrete subset, riamely tl-ie set ((0, O),:. This manifold, the Prüfer manifoId, and related manifolds? have some ver' strange properties, developed in tl-ie problems.

PROBLEMS

1, (a) A well-ordered set cannot contain a deci-easing infinite sequence A-t > X2 > Xg > . . . . (b) If Irre denote (cx + 1 ) + 1 by cx + 2: (cx + 3) + 1 by cx + 3, etc., theii any cx iquals /3 + I I for a unique limit ordinal /3 axid iiiteger ,I >_ O. (Tl-ius one can define even and odd 01-diiials.)

2, Let c be a "clioice functioil", i.e., c ( A ) is defined for each set A # 0, and c ( A ) E A for al1 A . Giveii a set X, a well-oi-dei-iiig < on a subset Y of X \vil1 be called "distjnguisl-ied" if for al1 y E Y,

(a) Shomr that of aiiv two distinguished well-orderings, one is an extension of tl-ie othei: (b) Shomr that there is a ~vell-ordering on X. (Zorn's Lemma may be deduced from t his fact fairly easil!r.l (c) Given two sets, sliow tliat one of them is equivalent to (can be put in one-one coi-1-espoiidence with) a subset of the other. (d) SIiow that on any iiifinite set tha-e is a well-ordering whicl-i represents a limit ordinal. (e) From (d), aiid Problem 1, show that jf X aiid Y are disjoint equivaleni infinite sets, then X U Y is equivaleni to 1:

3. (a) L+ and L are iiot metrizable. .

(b) II' xl 5 x 2 5 .y3 5 . ' . is a sequelice ii-i L+. tlien (x,) converges to some poiii t. Cor~sequently, aiiy sequence I-ias a coiivergeri t su bsequence (but L + is i ~ o t compact!).

(c) If {A-,] and {}r,j are sequences in L+ with x, 5 y, 5 x,+~ for al1 n: tlien hoth sequences converge to the same point.

(d) L+ (and also L) are normal. (Use (c)). (e) hilore ~enerally, anv order topology is normal (completely dflerent proofj. ( f ) If f : L + + R is contii~uous, and 7' > S, then one of the sets f-' ( [ -m, S ] )

and . f -' (Ir, m) ) is countablt. (g) If .f : L+ + R is contjnuous, then f is eventually constant.

4. (a) L' is iiot contractible. N&: Given W : L+x [O, 11 + L+ with W ( x , O ) =

x for a11 x ! show that for ever-y I we have { H ( ~ , I ) ) = L+. (b) ni ( L + ) = nl ( L ) = O. Siniilarly for L+ x R, L x R, L x L, L x L+, L+ x L' .

(c) q ( L + X S I ) = n1(L x Si) = Z.

5. (a) L+ and L are not Aomt~omorphic. Nini: Imitating ProbIem 1-19, defini "paracompact ends".

(b) L + x and L x R are iiot homeomorphic; L' x S' and L x S' are iioi h omeomorphic. (c) Of the 2-manifolds constructed from L+ or L with ni = O and one paracompact end, oi11y L+ x R Iias the homotopy type of L+. (d) Tlie ~ione-Cech compactHcations of L x L , L+ x L, L+ x L+, and thca l i g disc arc al1 distjiict. (Usji-ig Rnblem 3(g), one can explicitly cor-istruct thesc ~ i o n e - ~ e c h ~om~actificationr.

6. (a) S l~ow tl~at the Prüier manifold P js Hausdorff: (b) P does not llave a countable dense subset.

(c ) Let U l ~ e aii open set in whjch js ihe uiiion of "wedges" ceiitered ai (cr.0) foi e w q r ir~.ational u . Show tliat U iiicludes a whole rectangle of the forni

( u . b ) x ( O . & ) . N~T~I: Let A , = {u : the wedge centered at a has width >_ 1 / 7 7 ) .

Silice R = $ U U, A,. son-ie A , is not nowhere denst.

(d) Let C1,C2 c P be

Ci - ( ( 0 , O ) , : a irrationai]

C2 = ( ( O , O ) , : a rational) .

Show tliat Cl and C2 are closed, but that tliey are not contained in disjoint open seis. (e) Define W : P x [0,1]+ P b y

Sliour that H is well-defined and that H ( p , 1 ) E U ( (0 , O),) for al1 p E P. Conclude that P is contractible. (f) P - ((x, y ) , : y O ) is a manifold-with-bouildary P', whose boundary is a disjoint union of uncountably many copies of R. (g) The disjoint uiiion of t w ~ copies of P', uitli con-esponding points on thc boundary identified, is a manifold which is not metrizable, but which has a countable dense subset. Its fundamental group is uncountable.

7. It is known that every second countable coiitractible 2-manifold is or IR2. Hence ihe i-esuli of constiucting the Prüfer manifold using only copies W: for i-ational u must be hoineomorphic to P2. Desciibe a homeomorphism of thiz manifold onto P2.

8. Let M be a connected Hausdofl manifold which is not a point.

(a) If A C M has cardinality c (the cardinalitv of IR)' then the closure 2 lis‘ cardinaiity c. (b) If C c M is closed and has cardinality then C has an open neighborliood witli cardinality r. (e) Let p E M. There is a function f : R + (set of subsets oí" M ) such that -f ( a ) has cai-diiialits c foi. al1 rr E R , and suc11 thar

f ( 0 ) = ( p l

/ ( a ) is an open iieighborhood of tlie closure of f ( B ) .

(Consider functions defined oii iiii tia1 segnients of R wit1.i tllese same properties, and apply Zorn's Lemma. Alteriiatively, oile caii I-equire J ' ( c x ) to be the result of applying tlie elloice fuiiction to tlie set of al1 open iieigl~borhoods of the closurc

of UBo f ( B ) ~ i t h cardinality C. Then there is a unique f with the required properties. This is an example of defining a function by "transfinite induction".) (d) A function f : Q + (set of subsets of [O, 11) with the properties of the function in part (c) is eventuaIIy constant. (e) M has cardinality r, (Given p' c M : consider an arc from p to p'.)

9. (a) A coniiected 1-maiiifold whose topology is the order topology for some order, is Iiomeomorphic to either the real line, the Iong line, or the half-long Iine. (h) Every 1-manifold M contains a maximal open submanifold N whose topology is the order topology for some ordei: (c) If M is connected and N # M, then M is homeomorphic to S'.

CHAPTER 2

The long ray L+ can be given a Cm structure, and even a CW structure. To see this we need the result of Problem 9-24-anv Cm [or CW] structurc on a maiiifold M homeomorphic to R is d8eomor;hic to R with the usual structure. Thjs implies that it is also dSeomorphic to (O, l ) , and consequently that tlie structui-e on M can be extended if M is a proper subset of L+. An easy applicatioii of Zorn's Lemma then shows that Cm and CW structures exist on L+.

1 do not know whether al1 Cm structures on L+ are diff'eomorphic. It is klio~.n tliat tliei-e ale uncountably many inequivalent CW structures on L+, If p E L+. and L+p deiiotes al1 points 5 p , then L+ - L+p is clearly homeomorpliic to L+. If O is a CW structure for L+, then it yields a CW structure foi L+ - L+p' and Iience for L+. These are al1 distinct, in other words, there is 110 CW map

f : L + - L ~ + L + - L + , Y f'

~ 4 t h a CW inverse. In fact, wre must Iiave f (q) > q, and then jt is easy to ser

that we must also have J'(f ( Y ) ) 3 f (Y), f (f (f ( Y ) ) ) 3 f (f (Y)) , etc. The iiicreasing sequence y, J'(y ), f (j '(q )), . . . has a limit point xo E L+ - L+p. and .f[xo) = xo. Now j' cannot be the identity on al1 points 3 xo (for then it would be tlie ideiitity werywliere, siiice it is CW). So foi. some ql 3 xo we have j '(qi ) # ql; we can assume J'(ql) > ql, since we can consider f -' in tlie coiltrary case. Reasoni~ig as befoi-e, we obtain xl 3 x.0 with f (x l ) = xl. Colitinujng iil tliis wax we obtain xo < xl < x2 < . - with f (x,) = x,. This sequence has a limit jn L+ - L+p: but this implies that .f[x) = x for al1 x, a contradictioii.

A CW sti.uctui-e exists oii tlie Prüfer manifold; this follows immediately from tlir fact iliat ilie iiiaps /.. used for ideniifying poiiiis in various (R:)+ wiili points in R;, are al1 CW. 1 do not know whether eveqr ?-manifold has a Cm stiuctui-c.

Using the CW structui.e on L+, we can get a CW structure on L+ x L+. How- ever, tlie ine thod used foi- obtainiiig a CW structui-e on L+ will not yield a comphx ariaiiyiic ,rirwciu?r oii L+ x L+; tlie prohlem is that a complex analytic structure on R2 niav bc con for nially eq uivalent to the disc, aiid lieiice extendable, bu t it

may also be equivalent to the complex plane, and not extendable. In fact, i i

is a classical t11eo1-em of Rado that every Riemannian surface (2-manifold mith a complex analvtic structure) is second countable. On the other hand, a modification of the 'Priifer manifold yields a non-me~izable manifold of complex dimension 2. Refei-ences to these matters are to be found i i ~

CaIabi and Rosenlicht, Comklex Anabf ic Manfo ldr widiour Coutifable Base, Proc. Arner. h4ath. Soc. 4 (19531, pp. 335-340.

H. Kneser. A~iaJírfiscJie S f ruck fu~ und Ab<aJilbark&f, Ann. Acad. Sic. Fennicae Se- ries A: 1 25 1i5 (19581, pp. 1-8.

10. Prove thai íor y > p there is no non-constant C W map /: L+ - L.+, + L+ - L+ 4 -

1 1. Let (Y, p) be a n~etric spacc a11d let .f : X + Y be a continuous locall!. one-one map. where X is Hausdorff: connected, Iocdly coni~ected, and locallv compac t.

(a) EWI-y two pojilts A-,]? E X are coi~tained in a compact connected C c X. (h) Let d(a, J.) be the greatest lower bouild of the diameters of f (C ) (in the p-metric) Sor a11 compact connected C containing A- and J.. Show that d is a n ~ e t ~ i r on X which @ves the same topology for X.

12. Of tbe 1-al-ious manifolds mnltioned in the previous section, try to determine w11ich can be immersed in which.

CHAPTER 6

Problem A-G(g) describes a non-paracompact 2-manifold in which two open half-planes are a dense set. \Ve will now describe a 3-dimensional version with a twist.

Let A = {(x, y, z ) E W 3 : j~ # 0): and for each a E W let W: be a copy of W3. points in P: being deiloted by (x, y, r),. In the disjoint union of A and al1 P:. a E R we identify

(x, y, E), for y > O with (a + yx, y, r + a)

(A-, y, L), for y < O with (a + yx, y, z - a)

TIie equivalente classes form a 3-dimensional Hausdoflmanifold M. On this manifold tbere is an obvious function "í", and the sets r = constant form a foliation of M by a 2-dimensional manifold N. The remarkable fact about t h i ~ 2-dimei~sional manifold N is that it is connected. For: tlie set of points (x, y, c) E

A with y > O is ideniified ~ i t h tlie set of points (x, y. r - a), E W: with y > 0. Now the folium contairling {(A-, y, c - a),) contains the points (x, y, c - a), with y < 0. and these are identified with the set of points (x,y,c - 20) E A with y < O. Since we can choose a = c/2, we see t l~at al1 leaves of the foliatioi~ are the same as the leaf coiltaiiling ((x, y, O) : y < 0) c A .

This example is due io M. Kneser, Baipiel ail~i diinensio~zserllohenden anal)iíic- clieii A lrbildutig ptvlsclieíi ülrerabpalilbal-en A4a1i?li&alt$hztm. Archiv. Math. 1 1 (1 960); pp. 288-281.

C H A P T E R S 7, 9, 10

l. 1% llave seen t l~at any pai-acompact Cm manifold has a Riemannian metric. The converse also holds, sjnce a Riemannian metric determines an ordinan- metric.

2. Problem A-1 l jmplies that a manifold N immersed jn a paracompact manifold M js paracompact, but a much easjer poof is now available: Let ( , ) bc a Riemannian metrjc on M; if f: N + M is an immersion, then N has the Riemannian metric f * ( , ).

\+e can now dispense witll the argument in the proof of Theorem 6-6 whicli was used to show that each folium of a distributjon on a metrizable manifold j >

also metrizable, for the folium is a submanifold, and hence paracompact.

3. Since there is no Riemannian metric on a non-paracompact manifold M. the tangent bundle TM cannot be trjvial. Thus the tangent bundle of the long line is i ~ o t trivial, ilor is the tangent bundle of the Prüfer manifold, even thougli the Prüfer manifold is contractible. (On the other hand, a basic result about bundles says that a bundle ovei- a paracompact contractjble space is trivial. Compare pg. V. 2 7 2 .)

4. The tangent bundle of tlre long line L is clearly orientable, so there canno, be a iioivhere zero 1-form w o11 LI for w and the orientation would dr- termille a i~owhere zero vector field, contradicting tlle fact that the tangeni bundle is not trivial. Thus, Theorem 7-9 fails for L. Notice also that if M is non-pai-acompact, then TM is definitely not equivalent to T*M, since an eqiiii~aIei~ce ivould determine a Riemanilian metric. So there are at least two ii~e~uiiraleil t non-trivial bundles over M.

5. Altliougli the results in the Addendum to Chapter 9 can be extended to closed. not iiecessarily compact. submanifolds, they cannot be extended to non- pai-acompact manifolds, as can be seen by consjdering the O-dimensional submanifold ( ( O . O ) a j of the Prüfer manifold.

6. A Lje group is automatjcally paracompact, sii~ce its tangent bundle is trivial. Adore gei~eralh; a locally compact connected topolocgical group is a-compaci (Problein 10-4 ;.

- 1 . 11 is not clear that a non-paracompact mailifold cannot have an ji~deíii~jich met ic (a non-degenerate iniler product o11 each tangent space). Tl-iis will bc proved in Volume 11 (Chapter 8, Addendum 1).

PROBLEM

13. 1s there a nowhere zero 2-form on the various non-paracompact 2-mailifolds which have been described?

CHAPTER 1

CHAPTER 2

A' C ' C0 C" C"

D i f (0) GL(n, R)

R(n) (Rn, U i

R)

CHAPTER 4

?I d.v ' End(V) ,f * c Hom(V, W) T*M

CHAPTER 3 T @ S 116

CHAPTER 9

cosh

cosh -' d ( p . q ) ds d t/ Euc( V ) Eucít)

exp

f*( , ) (giJ ) (U, 13 Lt

2 68 J

285 sin11 29 1 tanh

264, 291 M/L

2 64 Xl ( v ) 264 5 t 263 6 1 ~ ) 248, 285 S I 285 ¿ -1 260 r 246

r!. 1.1

I . ) 2 39 r 3 )<

r 3 ),-

243, 245, 246, 248. I ) I

r - 257, 259, 288 ( ) *

CHAPTER 10

1, 40 1 i - 0 374 L u 379 J ( G ) 376 a l 3 ) 388 oín) 376 P 41 1 P-1 41 1 8 , 374 SO(n) - 373 A' C

376 A' 3'79 @* 380 Y' 395

A v) 403, 410 o (natural g-valuecl

I -form, 403 1 - 1 376 111 A A1 40.';

CHAPTER 11

Ahelian Lie alpehra, 376, 382, 395 Adams, J. E, 100 Adjoint T* of a linear transformatioi~

T , 103 Ado, J. D., 380 Alexander's Horned Sphere, 55 Algehra. Fundamental Theorem of.

285, 293 Algehraic inequalities, principie of

irrelevante of, 233 Alternatin~

covariant tensor field, 207 multilinear function, 201

Alternation, 202 Analytic mai-ijfold, 34 Annihilator, 228 Annulus, F: Antipodal

map, 27l: point, 11

Arclength, 31 2 fuiiction, 313, 332

Arcwise coi~nected, 20 suhgroup of a Lie group, 409

Area, generalized, 246 Associated

disc huiidle, 451 sphere bui-idle, 45 1

Atlas, 28 maximal, 24

Auslai-ider, L., 106

Rariach space. 145 Rase spacc, 7 1 Basi!

dual, 1 O7 for Mp*: 208 for R~ ( p ) , 208

Beloiip to a distributioii, 191 Besic.ovitc11, A. S., 174 Rie

djsc, 460 half-l~lai~e. 466 plane, 460

quadrant, 466 Bi-invarian t metric, 40 1 Boundary, 19, 248, 252 Bouiided manifold, 1 Boy's Surface, 60 Bracket, 154

in gr(n,R), 3TE; in o(n, R), 359

Bundle cotancentr 1 O? dual, 108 fihre, 309 ii-iduced, 101 map, 73 n-plane, 71 i-iormal, 344 of contra\nariant teasors, 120 of covariant tensors, 1 1'7 - - tangerit, / j

trivial, 72, 210 \lector, 71

Burali-Forti Paradox, 464

Calahi, E., 472 Calculus of variations, 31 6 Cartan, Elie, 39, 348, 361) Cartan's Lemma, 230 Cauc hy-Riemani-i equations, 200 Cayley iiumhers, 100 Chain, 248, 285 Chain Rule, 35, 38 Change, infiiiitely small, 1 11 Chart, 28 Choice, 283 Choice fui-iction, 46; Circle, 6 Closed

form, 21 8, 253 geodesic, 367 half-space, 19 long ray, 465 manifold, 19 suhgroup of a Lie group, 391 su hmanifold, 4ci

Cl osed (conli~rued, up to first order: 16f;

Cofinal, 465 Col1omology, 41 9

de Rliam, 263 group of M tttith real coefficients.

263 of a complex, 421

Coinmutative diagrain, 65, 420 Comrnutati\7e h e al_q-bi.a. 371: Complete, geodesically. 341 Complex, 421

ai-ialytic structure, 4 7 1 i~uml)ers of norm 1 . 37 3

Conjugate, 358 Constants of structure, 391) Coi~tinuous homomorplijsm, 387 Coi-itractihle, 220, 235. 236 Coi-itraction, 121, 139. 227

h m m a , 139 Contravariani

fuiictor, 130 tensor field, 120 lector field, 1 13

Conves ~eodesjcally, 363 polyhedroi-i, 429

Coordiaate lii-ies, 159 Coordinate systeim, 28. 158 Coordinates, 28 Cotangent bundle. 1 09 Co\yai.iai-ii

functor, 130 icnsor field, 1 1; vector field, 1 13

C;O\~PI locallv finite, .5O point-finite, 60 rcfinement of, S(1

Ciamer's Rule, 372 Critica1 poini, 40

j11 the calculus of \.ariatioi-is, 3120 Critica1 value, 40 Crass section, 227 Cross-cap, 14 Ci-oss-produc t, 299 Cul~e, sin-gular, 24ú

Cup product, 299, 439 Curl, 238 Cginder, 8 Cr manifold, 34 C0 manifold, 34 C"

distrihution, 179 form, 207 fuilction, 32 manifold, 29 manifold-with-houndary, 32 Riemaiinian nietric, 308 structure on TM, 82

Cm-related, 28

Darbou': integrahle, 283 integral, 283

DarBoux's Theorem, 284 Dehaucti of iildices, 39, 123 Decomposahle, 228 Definitioi-i, invariant, 2 t 4 Delormatjon retractioii, 259 Degenerate, 286 Degree, 275

mod 2, 295 Dei~sin-

evei; sealar, 133, 209 odd scalar, 133, 259 relative scalai-, 23 1 scalar, 133

Derj\xtjoi~, 39, 78 of a riiig, 83

Derived set, 25 Descartes-Euler Tlieoi-em, 429 Detei-miilant, 232 Diffkomorphic, 30 Diileoinoi-pliism, 30

one-parameter .y-oup of, 148 Djflkrentiable, 27, 28, 31, 39

at a point, 3 1 mai-ijfold, 29 structure, 30

oil the loi-ig line, 47 1 011 P", 33

Dflerent iahle (cm/inirru' (structure conlinrc~u',

on IR", 21) on S", 30

Dflerential, 2 10 equation, 136, 164

deperidjng on parameters, 169 linear, 165

forms, 201 of a functjon, 109

Djmensioil, 4 Direct sum, 421 Disc bundle, associated. 45 1 Djscrjminant, 233 Djsjoint union, 4, 20 Djstrihution, 179, 181

ideal of, 2 15 on torus, 180

Divergente, 238 Theorem, 352

Domajn, 3 Du Bois Reyniond's Lemma, 355 Dual

hasis, 10; space, 105 vector bundle, 108

Ejnsteiii summatjoi~ coi~\lciitjon, 39 Elemeiits of norm 1 , 308 Elliptjcal non-Eucljdeai~ geometry, 367 Emhedding, 49 Ei-id, 23

paracompact, 466 Ei~domorplijsm, 12 1 Eiiergy, 324 Enwlope, 358 Equa tioiis depeiidjiip o i ~ paranle ter>.

169 Equatjons of structure. 404 Equi\laleiice (01' vector bundles), 72

weak, 96 Eu clj deai~

inetric, 305, 315 motjoii. 374 11-space, 1

Euler, 429 characterjstjc, 428 class, 445

Euler's Equatioii, 320 Eveii

ordinal, 467 relative scalar, 23 1 relative tensor, 134, 231 scalar density, 133, 209

Exac i form, 21-8 sequence, 419, 422

of a pajr, 433 of vector bundles, 1 03

Exponential map, 334, 383 Exponential of matrices, 384 Extension, 432 Extrema], 320

Fajth, leap of, 464 Fihre, 64, 68, 5 1 Fjnite

characteristic, 205 qTe, 438

First elemeiit, 461 Fjrst varjatioii, 3 19, 32; Fjve Lemma, 440 Fjxed point, 139 Foliatioi~, 194 Foljum, 194 Force field, 240 Form, 207

differei-it ial, 20 1 left invariant, 374 righ t ii-ivarjaiit, 400

.f-related, 190 Frol~ei~ius Intep-ahility Theorem, 192,

215 Fuhiiii's tl~eorem, 254 Fuiictor, 130 Functorites, 89 Fui~damei-ital Theorem of Algebra.

285, 293 Fui~damental Tlieorem of Calculus,

254

Gauss's Lemma, 33, General linear group, 61, 372 Gei-ieraljzed area, 246 Geodesic, 333

closed, 36; reversiiig map, 401

Geodesjcallv complete, 34 1 ~eodesjcallk coi-i\lex. 363 Geodesy, 333 Germs of k-forms, 432 Glol~al theorv of integral mai-iifoldi

194 Gradient , 23; Grarn-Schinidt ort11oiioi.iiialjzatioi-i

process, 304 G I P ~ . 8-1 Gioup

Lie. 371 n-iatrix, 373 op~osite, 407 or~thogonal, 372 topolo~ical, 3 7 1

Guillemiri, M!, 106

Hal-in-Rai~acl-i theorem, 14 5 Hair-, 69 H al f-1oi-i~

cvliiider. 466 liile, 46.5 strip, 466

Half-space, 19 Handle, H Hardy, G. H., 179 Has oi-ie eiid, 23 Heiiilcin, Robert A., 84 HausdodT, 459 Ho~no~eiieou S, 7 Homomoipl-iism

conti~iuous, 387 of Lje al~ebras, 380

Homotopic, 104, 277 Hon~otoly, 104, 277 Hopf, H., 342, 450 Hopf-Rii~ow-de Rliam Tl~eor.em, 342

Hyperholir cosine, 356 sine, 356 tai-igent, 356

Ideal of a Lie algehra, 410 Jdentificatioi~, 10 Jmbeddinc: 49

topological, 14 Immersed submai-iifold, 47 Jmrnersjoii, 46

iopolo@cal, 14, 46 Inipljcit fuiictjoi~ theorem, 60 Jndefiniie metric, 350 Jndependerlt ii-ifii-iitesimals, 3 14 Iiidex of jni-ier ]>roduct, 349 Ii-idex of \rector field

on a manifold, 447 on Rn, 44 G

Ii-idices dehauch of, 39, 123 raisii-ig and lowering, 35 1

Ii-iduced l~uiidle, 101 orie~ltation, 260

Jr-ieqiiali t ies, principle of irrele~rance oI algebraic? 235

Jnertia, Syl~lester's Law of, 349 Jnfinite volunie, 312 Ji-ifiniiely small cl-iarige, 1 1 1 Ji-ifinitely small displacemenis, 3 14 Ji-ifinjtesimal generator, 148 Jnfinitesimals, iriclependent, 314 Ji-iitial conditions, 136

of integral clirve, 136 Jnjtjal segmcnt, 462 Jnner product, 227, 301

preserviiig, 304, 372 usual, 301

lnside, 21 Jntegahility conditioiis, 189 Jntegrahle disti.il>utioi.i, 192 Jiiiegable functioii

Darhoux, 283 Riemann, 283

In tepral c u m , 136 Darhoux, 283 Ijne, 239, 243 manifold, 179, 181

maximal, 194 of a djfferentjal equatioi-i, 136 Riemann, 283 surface, 245

Intepration, 136, 226, 339 Tnvariance of Domain, ? Invai-iant, 128, 232

definjijon, 2 1 4 Trrelevance of algebraic inequaljtiea,

prii-iciple of, 23 3 Isometry, 340 Ison-iorphic Lie groups. locally, 382 lsomorphjcm, natural. 10P lsotopjc: 294

Jacobi jdei-itity, 155, 3TG for tlie hi-ac.kei in ai-iyl i'ji~g, 37 8

'jacobjan mairjx, 40 Jordan C u m Tl-ieoren~, 21, 435

Kelley, J., 460, 463, 464 Kji~k, 366 Klej11 bottle, 18, 435 Kneser, H., 472 Kneser, M., 473

Lanc, S., 145 Laplace's expai-ision, 2311 Laplacian, 58 1,aw of 1 nertia, Sylvesier's, 34 9 Leaf. 194 Leap of faith, 464 ~ e f ; jiwai-jaii~

form, 394 n form, 400

vector field, 374 Left trar~slation, 374 Lei-igh, 243, 305, 312

of a curve, 59 Lie algebra: 376

ahelian, 376, 382, 395 coinmutative, 376 homomorphism of, 380 ideal of, 410 opposi te, 40 T

Lie derivaiive, 150 Lie group, 37 1

arrwjse connected suhgmup of. 409 closed si~bgroup of, 391 local, 415 ilormal subgi-oup of, 410 topologically isom orp1.i jc, 388

Lje suhproup, 373 Lie's fui~damei~tal theorem

firsi, 414 second, 415 third. 41 6

Limh ordj~~al , 463 set. 60

Lii-ie integral, 239, 243 Linear differential equations, 165

svstems of, I T 1 ~ji lkar trai-isformatjoi~

adjojilt of. 103 contractjoi~ of, 121 positive defiiiite, 104 positiw seinj-defii-iite, 104

Linkji-iF i~umber, 296 Ljpschitz condition, 138 Littlewood, J. E., 159 Lives at points, 1 19 Lobacl.ie\~sliia~-i non-Euclideaii geome-

tr)i; 368 Local

ffow, 144 L e g'oup, 41 5 one-parameter group of local dfleo-

morphism, 148 spani-ied locally, 17 9 u-jvjaljty, 7 f

Local ilieory of inte~ral manifolds, 190

Locall! compact, 20 coiii-iected, 20 finite cover, 50 isomorphic Lje groups, 382 Lipschitz. 1 39 oilc-oi1e, 13 pathwise connecied, 20

Long cyfinder, 466 ljne, 465 ray, 462

closed. 46.7 opeil, 465

Lo\ver sum, 283

MacKenzie. R. E., 1 O(; Magic, 2 14 Mailjfold, 1. 459

aiia1ytic. 3-1 atlas foi-. 28 l~oui~dary of, 19 l>oui~ded. 19 closed. l!i C', 34 CO, 34 CW, 21' rlifl'erciitial)le, 3!i dimei~sjoii of. 4 imheddji~g in IRN. 52 ji~tegi.aI. 159, 18 1

niasjmal, 19.3 noii-nictrizalJe: 465, 4 GTi oi-jciitat ion of. Ht, smooil~. 29

Manjfold-witli-l~ounda~r. 19 C", 32

Map I~etwreii coniplexes. 43 1 1)uiidlc. 73 rai~k of. 4 ( i

Massev. 1V.S.. 3 hlaii-jx groiips. 37: h4asii11:il integral mai~ifold, 194

Mayer-Vietoris Sequence, 424 for compact su pports, 43 1

Measure zero, 40: 41 hksh, 239 Meiric

hi-invariant, 401 Euclidean, 305, 315 indefiníte, 350 Riemannian, 308, 31 1

usual, 3 12 spaces, disjoii-it union of, 4, 20

Milnor, J. M!, 43 h4od 2 degree, 29.5 Mohius sirjp, 10

~eiieralized, 1 Ofl h4ulii-index, 208 M uItiIjnear fui-ic~ioi~. 1 15 Munkres, J. R., 34, 106

11-dimetisi oiial, 4 11-forms, left iiivariai-it, 400 n-holed torus, 9 11-maiiifoId, 4 11-plane buiidIe, 3 1 n-sphere: 7 17-torus. 7 Natural Q-valued 1-forni, 403 Natural isomorpl~isin, 108 Neighhorhood, tul~ular, 345 Newmail, M . H. A.. 3 Nice cmrer, 438 NOII-houi~ded, 1') Noii-deperierate, 30 1 NOII-Euclidean geoineti.?.

elljptical, 36; LoI)achevskiai~. 368

Noi-i-meirizable inanifold, 465, 466 No~~-orje~~iablc-

hui~dle, 86 manifold, 8h

Norrn, 303 pi-ese~jilg, 304 , 372

Normal bui~dle, 344

Normal (cmlinzud} space, 459 s u b p u p of a Lie gmup, 410 outward unit, 351

Nowl~ere zero section, 209

Odd ordinal, 465 relative tensor, 134, 288 scalar density, 133, 259

One-dimensional distribution, 179 One-dimensional sphere: 6 01ie-parameter p u p

of diffeomorphisms, 148 of local diffeomorphisms, local. 148

One-parameter subgmup, 384 O pe i-i

loiig ray, 465 map, 60 submanifold, 2

Opposiie gmup, 407 Lie algebra, 407

Order isomorphic, 461 isomorphism, 461 topology, 465

Ordered set, 461 Ordering, 460 Ordinal iiumhers, 463 Orientablr

I)uiidIe, 86 manifold. 86

0iieritatioi-i of a bundle, 85 of a manifold, 86 of a vector space, 84 presenring, 84, 85, 88, 105, 248 reversiiig, 84, 88, 248

Orthogonal gmup, 61, 352 Orthoiiormal, 304, 348 0i.tl-ioiiormalizatioii pmcess, Giaiii-

Sclimidt, 304 Os~ood's Theorem, 284

Outside, 21 Outward pointing, 260 Outward unit iiormal, 351

Palais, R. S., 100, 225 Paracompact, 210, 459

end, 468 Parameter curves, special, 165 Parameterized by arclength, 3 13 Partid derivatives, 35 Partition, 239, 245

of ui-ijty, 52 Pathwise coni~ected, 20 Piece~ljse smootli, 3 12 Pig, yellow, 434 Poincaré, H., 450 Poincaré dual, 439 Poincaré Duality Theorem, 441 Poincaré-Hopf Theorem, 450 Poincaré Lemma, 225 Poiricaré upper half-plane, 367 Poini

inward, 98 ourward, 98, 260

Point-derivation, 39 Poiiikfiniie cover, 60 Polar coordinates, 36

iiitegration in, 266 Polarization, 304 Pollack, A., 106 Positive definite, 104, 301 Positive element of norm 1, 308 Positive senii-definite, 104 Pmduct

of vector huiidles, 102 tensor, 1 16

Pmjection, 5, 30, 32 Pmjectivr

plaiie, 11, 435 space, 19, 88

PI-oper map, 60, 275 Prü fer nianifold, 46 7 Pseudometric. 95

Quaternions, 100 of norm 1, 373

Radial function, 435 Rado, T., 473 Rai-ik

of a b rm, 229 of a map, 40, 98

Rectifiahle, 59 Refinemeni of a cwer, 50 Regular

point, 40 space, 459 value, 40

Related vector fields. 190 Rclative

scalar, 134, 23 1 tensor, 134, 231, 288

Reparameterizatiori, 244, 248 Retraction, 264

deformation, 259 Revolutioii, surface of, 8, 321 de Rham, G., 343 de Rliam cohom01o~gy vector spaces.

263 with compact supports, 268

de Rham's Theorem, 263, 455 Ricmaiii~

integrahle, 283 integral, 283 sum, 283

Riemaiinian meiric, 308, 31 1 iisual, 31 2

Righ t iiivariant n-fom, 400 Rigl~t translatioii, 3 74 Riiioy M!, 343 Roman surface, 17. 26 Rosciiliclit, M., 472

Sard's Theorem, 42, 294 Scalar, relative, 134, 231 Scalar density, 133 Schwarz, H., 354 Schwarz inequaljty, 303, 362 Second countahle, 459 Section of a vector bundle, 73

zem, 96 Segment, initial, 462 Se1 f-adjoint linear trarisformation, 104 Semi-definite, positive, 1 04 Separate points and closed sets, 95 Sequence

exact, 419, 422 of vector bundles, 103

Mayer-Vietoris, 424 fbr compact supports , 4.31

of a pair, 433 SIirjnking Lemma, 5 1 Shrinking Lemma, 60 Shuffle perrnutatioii, 227 Simplex

of a triaiigulation, 427 singular, 285

Simply-connectcd, 287 Lie p u p , 382

Siiigular- cuhe, 246 simplex, 285

Skew-svmmetric, 201, 378 Slice, 194 Slice maps, 54 Smooth, 28

homotopy, 277 maiiifbld, 29 piecewise, 3 12

Smoothly contractihle, 220 llomotopic, 277 isotopic, 294

Solid angIe, 290 Space filling curve, 58 Spar-ined locally, 179 Special liiiear gmup, G l Spccial ortliogoiial gmu p? 6'1 Sphere, 7

Rotatioii ~ r o u p . 62 Si11iei.e bundle, associated, 45 1

Standard n-simplex, 426 singular cuhe, 246

Star-shaped, 22 1 Steiner's surface, 17, 26 Steriiherg, S., 42, 106 Siokes' Theorem, 253, 261, 285, 352 ~ i o i i e - ~ e c h mmpactification, 468 Srructure constants, 396 Suhalgehra of a Lie algebra, 37 cl Suhhundle, 198 Suhcover, 50 Suhgmup

Lie, 373 or-ie-parameter, 384

Sul~manifold, 49 cm, 49 closed, 4:) immersed. 4 7 open, 2

Su ccessor ordir-ial, 464 Sum of vector hundles, M'hitney, 101 Support, 33, 147 Surface, 5

area, 354 integral, 239 of revolution, 8, 321

Sylvester's Law of Inertia, 349 Svmmetric hiliriear form, 301 Spstem of linear differei-itial equatioiis.

171 o-cornpact, 4, 459

Tangelit bundle, 57 Tangerir space of IR", 64 Tai-igent vectoi-

ir-iward pointing, 98 of a manifold, 76 of IW" 64 ou tward poii-itirig, 98, 260 to a curve, 63, 66

contravariaii t, 120 covariant, 1 1 3 even relative, 134, 231 odd relative, 134, 288

Tensor field classical definition of, 123 contravariani, 120 covariant, 1 13 mixed, 121, 122

Tensor produ ct: 1 1 6 Thom class, 442 Thom Isomorphism Tlieorem, 456 Topological

group, 371 imhedding, 14 immersion, 14, 46

Topologically isomorphic Lie groups, 388

Torus, 5, 8 wholed, 7, 9

Total space, 7 1 Totally disconiiected, 25 Transitivity, 4 60 Translation

left, 374 right, 374

Triangle ii-iequ ality, 303 Triangulation, 426

simplex of, 427 Trichotomy, 461 Trivial vector bundle, 72 Tubular neighborhood, 345 Two-holed torus, 8

Vick,j. W., 3

Wedge pmduct, 203 Whitney, H., 106 \Yhitney sum, 101

T hese books wei-e typeset using Doi~ald E. Knutli's typesetting system, together ~ 4 t l - i Berthold H orn's DV3PSONE PostScript driver. The figures

were produced with Adobe Illustrator. and new or modified fonts were created using Fontographei.

The text font is 1 I point h/lonotype Baskenrille-though the em-dash has beei-i modified-together with its italic. The elegant swashes of the italicy and f cause problems in words like iopology and apolog31, so a special gy ligature was added; special and gj ligatures w r e also required.

Altliough a Baskenrille bold face is ui-iliistorical, bold ty-pe was useful in special circumstances-niainly for indicating defined terms. The bold face supplied by Monotype: even the "semi-bold", is obtrusi\?ely extended, so a non-extended version was created.

Tl-ie somewliat bold appearaiice of cl-iaptei. l-ieadings, in 1 6 point ty-pe, results from the linear scaling, as well as the fact that t l~e upper case Baskerville letters are of somewliat Iieavier weight tlian tl-ie lowei- case. On the other hand, the tal1 initiai letters beginning eacl-i cl-iapter were designed specially, since simple scaling would llave made them u npleasantly heavy

A tl-iicker set of iiumerals was constructed for use witl-i tlie upper case lettering in chapter headings and statements of theorems, and special parentheses and other punctuation symbols were also required. Numerous otlier modifications of this sort, including additional kerns and alterations of set widths, were made for V ~ ~ O U S pu rposes.

The niathematics fonts are a variation of the Mutl?7Ime fonts: now based on the h4onotype Times New Roman famdy, togetliei. with the Monorype Times N R Seven and Times Small Text famdies-presumably these tliree families are based on the original designs for Times New Roman, whicl-i was created in tlii-ee essential sizes: 9, 7 and 51 point.

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The covers, paiiited by the author iil his spare moments, are loosely based on Samuel Taylor Coleridge's poem nte Rime ofilre Attcimt A4an%e;l.

CORRECTIONS FOR VOLUME 1

pg. 3, line 3-: change di(x, y ) < 1 to di (x, i 1. pg. 14: relabel the lower left part of the central figure as

pg. 19: replace the next-to-last paragraph with the following: .

The set of points in a manifold-with-boundary that do not have a neighborhood homeomorphic to R" (but only one homeomorphic to W") is caiied the boundary of M and is denoted by aM. Equivaiently, x E a M if and only if there is a neighborhood V of x and a homeomorphism t$ : V + Wn such that t$(x) = O. If M is actuaily a manifold, then a M = 0, and a M itself is aiways a manifold (without boundarjr.

pg. 22: Replace the top lefi figure with

pg. 43: Replace the last line and displayed equation with the foliowing:

Since rank f = k in a neighborhood of p, the lower rectangle in the ma&

pg. 60, Problem 30: Change part (f) and add part (8):

(f) If M is a connected manifold, there is a proper map f : M -, R; the function f can be made Cm if M iS a CM manifold. (g) The same is true if M has at most countably many components.

pg. 61, Problem 32: For clarity, restate part (c) as foiiows:

(c) This is faise if f : MI + R is replaced with f : Mi -, N for a disconnected manifold N.

pg. 70: Replace the last two lines of page 70 and the first two lines of page 7 1 with the following:

theorem of topology). If there were a way to map T(M,i), fibre by fibre, homeomorphicaiiy onto M x IR2, then each vp would correspond to (p , v(p)) for some v(p) E R2, and we could continuously pick w(p) E R2, corresponding to a dashed vector, by using the criterion that w(p) should make a positive angle with v(p).

pg. 78: the third display should read:

pg. 103, Problem 29(d). Add the hypothesis that M is orientable.

pg. 11 7: After the next to last display, ~ ( X I , . . . , Xk ) ( p ) = A (p) (XI (p), . . . , Xk (p)), add:

If A is Cm, then Ais Cm, in the sense that ~ ( x I , . . . , Xk) is a Cm function for al1 Cm vector fields Xi , . . . , Xk.

pg. 118: Add the following to tlie statement of the theorem: If A is Cm, then A is also.

pg. 119: Add the foliowing at the end of the proof:

Smoothness of A foiiows from the fact that the function Ail...ik is A(a/axrl,. . . , a/axik).

pg. 131, Problem 9: Let F be a covariant functor from V, . . . . pg. 133. Though there is considerable variation in terminology, what are here calIed "odd scalar densities" should probably simply be called "scalar densities"; what are c d e d "even scaiar densities" might best be c a k d "signed scalar dmsities".

In part (c) of Problem 10, we should be considenng the h of part (a), not the h of part (b)! Thus conclude that the bundle of signed scalar densities (mt the scalar densities) is not triviai if M is not orientable.

pg, 134. Extending the changed terminology from pg. 133, we should probably speak of the bundle of "signed tensor densities of type (f) and weight w" (though sometimes the term relative tensor is used instead, restricting densities to hose of weight l), when the transformation mle involves (de{ A)W, omitting the modifier "signed" when it involves [ det Alw.

pg. 143. The hypothesis of Theorem 3 should be changed so that it reads:

Let x E U and let a l , a2 be w o maps on some open i n t e d I such that al(¡), (Y*(¡) C U,

( Y ) = f ((Y (t)) i = 1,2

and ai(to) = a2(t0) for some t~ E 1.

And the first sentence of the proof shouid be deleted.

pg. 177. Problem 17, part (d) should begin:

(d) Let f : M N, and suppose that f., = 0. For X,, Y, E M, and . . . . pg. 198. In Problem 5, we must also assume that each Ai @ Aj is integrable.

pg. 226. In the comutative diagram, the lower right entry should be "1-forms on N".

pg. 233. The reference "pg. V375" refers to pg. 375 ofVolume V. pg. 237. In Problem 26, replace par& (b) and (c) with:

(b) Determine the ih component of vl x . x u,-1 in terms of the (n - 1) x (n - 1) submatrices of the matrix

In particular, for lR3, show that U X UI = (u2w3 - u3w2, u3w1 - u1w3, u1w2 - u2w1).

pg. 292. In Problem 20, the condition Ui n U, # 0 should be Ui n Ui+l # 0. pg. 408. Problem 16 (b) should read: "For any Lie group G, show that ... ". pp. 408-410. For consistency with standard usage, Aut should be replaced with Aut , and then replace End with End. In part (g) of Problem 19, add the hypothesis that H is a connected Lie subgroup.

pg. 41 1. The display in Problem 2 1, part (c) shouM read:

(-I)~"[w A [q A A]] + ( - ~ ) ~ ' [ q A [A A&]] + (-1)lm[h A [o A ql] = 0.

Spivak: A comprehensive Introduction to Differential Geometry Vol 1

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Transcript of Spivak: A comprehensive Introduction to Differential Geometry Vol 1