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23 Alberto Candel and Lawrence Conlon, Foliations I, 2000

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http://dx.doi.org/10.1090/gsm/048

Introduction to the h-Principle

Y. Eliashberg

N. Mishachev

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 48

Introduction to the h-Principle

Editorial Board

Walter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 58Axx.

Abstract. The book is devoted to topological methods for solving differential equations andinequalities. Its content significantly overlaps with Gromov’s book “Partial differential relations”.However, the exposition is more elementary and intended for a broader mathematical audience,including graduate, and even advanced undergraduate students.

Library of Congress Cataloging-in-Publication Data

Eliashberg, Y., 1946–Introduction to the h-principle / Y. Eliashberg and N. Mishachev.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v 48)Includes bibliographical references.ISBN 0-8218-3227-1 (alk. paper)1. Geometry, Differential. 2. Differentiable manifolds. 3. Differential equations–Numerical

solutions. I. Mishachev, N (Nikolai M.) 1952– II. Title. III. Series.QA641.E62 2002516.3′6–dc21 2002019347

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10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02

To

Vladimir Igorevich Arnold

who introduced us to the world of singularities

and

Misha Gromov

who taught us how to get rid of them

Contents

Preface xv

Intrigue 1

Part 1. Holonomic Approximation

Chapter 1. Jets and Holonomy 7

§1.1. Maps and sections 7

§1.2. Coordinate definition of jets 7

§1.3. Invariant definition of jets 9

§1.4. The space X(1) 10

§1.5. Holonomic sections of the jet space X(r) 11

§1.6. Geometric representation of sections of X(r) 12

§1.7. Holonomic splitting 12

Chapter 2. Thom Transversality Theorem 15

§2.1. Generic properties and transversality 15

§2.2. Stratified sets and polyhedra 16

§2.3. Thom Transversality Theorem 17

Chapter 3. Holonomic Approximation 21

§3.1. Main theorem 21

§3.2. Holonomic approximation over a cube 23

§3.3. Fiberwise holonomic sections 24

§3.4. Inductive Lemma 25

ix

x Contents

§3.5. Proof of the Inductive Lemma 28

§3.6. Holonomic approximation over a cube 33

§3.7. Parametric case 34

Chapter 4. Applications 37

§4.1. Functions without critical points 37

§4.2. Smale’s sphere eversion 38

§4.3. Open manifolds 40

§4.4. Approximate integration of tangential homotopies 41

§4.5. Directed embeddings of open manifolds 44

§4.6. Directed embeddings of closed manifolds 45

§4.7. Approximation of differential forms by closed forms 47

Part 2. Differential Relations and Gromov’s h-Principle

Chapter 5. Differential Relations 53

§5.1. What is a differential relation? 53

§5.2. Open and closed differential relations 55

§5.3. Formal and genuine solutions of a differential relation 56

§5.4. Extension problem 56

§5.5. Approximate solutions to systems of differential equations 57

Chapter 6. Homotopy Principle 59

§6.1. Philosophy of the h-principle 59

§6.2. Different flavors of the h-principle 62

Chapter 7. Open Diff V -Invariant Differential Relations 65

§7.1. Diff V -invariant differential relations 65

§7.2. Local h-principle for open Diff V -invariant relations 66

Chapter 8. Applications to Closed Manifolds 69

§8.1. Microextension trick 69

§8.2. Smale-Hirsch h-principle 69

§8.3. Sections transversal to distribution 71

Part 3. The Homotopy Principle in Symplectic Geometry

Chapter 9. Symplectic and Contact Basics 75

§9.1. Linear symplectic and complex geometries 75

§9.2. Symplectic and complex manifolds 80

Contents xi

§9.3. Symplectic stability 85

§9.4. Contact manifolds 88

§9.5. Contact stability 94

§9.6. Lagrangian and Legendrian submanifolds 95

§9.7. Hamiltonian and contact vector fields 97

Chapter 10. Symplectic and Contact Structures on Open Manifolds 99

§10.1. Classification problem for symplectic and contact structures 99

§10.2. Symplectic structures on open manifolds 100

§10.3. Contact structures on open manifolds 102

§10.4. Two-forms of maximal rank on odd-dimensional manifolds 103

Chapter 11. Symplectic and Contact Structures on Closed Manifolds 105

§11.1. Symplectic structures on closed manifolds 105

§11.2. Contact structures on closed manifolds 107

Chapter 12. Embeddings into Symplectic and Contact Manifolds 111

§12.1. Isosymplectic embeddings 111

§12.2. Equidimensional isosymplectic immersions 118

§12.3. Isocontact embeddings 121

§12.4. Subcritical isotropic embeddings 128

Chapter 13. Microflexibility and Holonomic R-Approximation 129

§13.1. Local integrability 129

§13.2. Homotopy extension property for formal solutions 131

§13.3. Microflexibility 131

§13.4. Theorem on holonomic R-approximation 133

§13.5. Local h-principle for microflexible Diff V -invariant relations 133

Chapter 14. First Applications of Microflexibility 135

§14.1. Subcritical isotropic immersions 135

§14.2. Maps transversal to a contact structure 136

Chapter 15. Microflexible A-Invariant Differential Relations 139

§15.1. A-invariant differential relations 139

§15.2. Local h-principle for microflexible A-invariant relations 140

Chapter 16. Further Applications to Symplectic Geometry 143

§16.1. Legendrian and isocontact immersions 143

§16.2. Generalized isocontact immersions 144

xii Contents

§16.3. Lagrangian immersions 146

§16.4. Isosymplectic immersions 147

§16.5. Generalized isosymplectic immersions 149

Part 4. Convex Integration

Chapter 17. One-Dimensional Convex Integration 153

§17.1. Example 153

§17.2. Convex hulls and ampleness 154

§17.3. Main lemma 155

§17.4. Proof of the main lemma 156

§17.5. Parametric version of the main lemma 161

§17.6. Proof of the parametric version of the main lemma 162

Chapter 18. Homotopy Principle for Ample Differential Relations 167

§18.1. Ampleness in coordinate directions 167

§18.2. Iterated convex integration 168

§18.3. Principal subspaces and ample differential relations in X(1) 170

§18.4. Convex integration of ample differential relations 171

Chapter 19. Directed Immersions and Embeddings 173

§19.1. Criterion of ampleness for directed immersions 173

§19.2. Directed immersions into almost symplectic manifolds 174

§19.3. Directed immersions into almost complex manifolds 175

§19.4. Directed embeddings 176

Chapter 20. First Order Linear Differential Operators 179

§20.1. Formal inverse of a linear differential operator 179

§20.2. Homotopy principle for D-sections 180

§20.3. Non-vanishing D-sections 181

§20.4. Systems of linearly independent D-sections 182

§20.5. Two-forms of maximal rank on odd-dimensional manifolds 184

§20.6. One-forms of maximal rank on even-dimensional manifolds 186

Chapter 21. Nash-Kuiper Theorem 189

§21.1. Isometric immersions and short immersions 189

§21.2. Nash-Kuiper theorem 190

§21.3. Decomposition of a metric into a sum of primitive metrics 191

§21.4. Approximation Theorem 191

Contents xiii

§21.5. One-dimensional Approximation Theorem 193

§21.6. Adding a primitive metric 194

§21.7. End of the proof of the approximation theorem 196

§21.8. Proof of the Nash-Kuiper theorem 196

Bibliography 199

Index 203

xiv Contents

4 3

11 12 14 16

13 15

17 18

19 2021

6

Part I

Part II

Part III

Part IV

1

2

3

4

3

1 7 85 6

6,7,89

10

Figure 0.1. The relations between chapters of the book

Preface

A partial differential relation R is any condition imposed on the partialderivatives of an unknown function. A solution of R is any function whichsatisfies this relation.

The classical partial differential relations, mostly rooted in Physics, are usu-ally described by (systems of) equations. Moreover, the corresponding sys-tems of equations are mostly determined: the number of unknown functionsis equal to the number of equations. Given appropriate boundary condi-tions, such a differential relation usually has a unique solution. In somecases this solution can be found using certain analytical methods (potentialtheory, Fourier method and so on).

In differential geometry and topology one often deals with systems of partialdifferential equations, as well as partial differential inequalities, which haveinfinitely many solutions whatever boundary conditions are imposed. More-over, sometimes solutions of these differential relations are C0-dense in thecorresponding space of functions or mappings. The systems of differentialequations in question are usually (but not necessarily) underdetermined. Wediscuss in this book homotopical methods for solving this kind of differen-tial relations. Any differential relation has an underlying algebraic relationwhich one gets by substituting derivatives by new independent variables. Asolution of the corresponding algebraic relation is called a formal solution ofthe original differential relation R. Its existence is a necessary condition forthe solvability of R, and it is a natural starting point for exploring R. Thenone can try to deform the formal solution into a genuine solution. We saythat the h-principle holds for a differential relation R if any formal solutionof R can be deformed into a genuine solution.

xv

xvi Preface

The notion of h-principle (under the name “w.h.e.-principle”) first appearedin [Gr71] and [GE71]. The term “h-principle” was introduced and pop-ularized by M. Gromov in his book [Gr86]. The h-principle for solutionsof partial differential relations exposed the soft/hard (or flexible/rigid) di-chotomy for the problems formulated in terms of derivatives: a particularanalytical problem is “soft” or “abides by the h-principle” if its solvabilityis determined by some underlying algebraic or geometric data. The softnessphenomena was first discovered in the fifties by J. Nash [Na54] for isometricC1-immersions, and by S. Smale [Sm58, Sm59] for differential immersions.However, instances of soft problems appeared earlier (e.g. H. Whitney’s pa-per [Wh37]). In the sixties several new geometrically interesting examplesof soft problems were discovered by M. Hirsch, V. Poenaru, A. Phillips, S.Feit and other authors (see [Hi59], [Po66], [Ph67], [Fe69]). In his disser-tation [Gr69], in the paper [Gr73] and later in his book [Gr86], Gromovtransformed Smale’s and Nash’s ideas into two powerful general methodsfor solving partial differential relations: continuous sheaves (or the coveringhomotopy) method and the convex integration method. The third method,called removal of singularities, was first introduced and explored in [GE71].

There is an opinion that “the h-principle is the hardest part of Gromov’swork to popularize” (see [Be00]). We have written our book in order to im-prove the situation. We consider here two geometrical methods: holonomicapproximation, which is a version of the method of continuous sheaves, andconvex integration. We do not pretend to cover here the content of Gro-mov’s book [Gr86], but rather want to prepare and motivate the reader tolook for hidden treasures there. On the other hand, the reader interestedin applications will find that with a few notable exceptions (e.g. Lohkamp’stheory [Lo95] of negative Ricci curvature and Donaldson’s theory [Do96]of approximately holomorphic sections) most instances of the h-principlewhich are known today can be treated by the methods considered in thepresent book.

The first three parts of the book are devoted to a quite general theoremabout holonomic approximation of sections of jet-bundles and its applica-tions. Given an arbitrary submanifold V0 ⊂ V of positive codimension, theHolonomic Approximation Theorem allows us to solve any open differen-

tial relations R near a slightly perturbed submanifold V0 = h(V ) whereh : V → V is a C0-small diffeomorphism. Gromov’s h-principle for openDiff V -invariant differential relations on open manifolds, his directed embed-ding theorem, as well as some other results in the spirit of the h-principleare immediate corollaries of the Holonomic Approximation Theorem.

Preface xvii

The method for proving the h-principle based on the Holonomic Approx-imation Theorem works well for open manifolds. Applications to closedmanifolds require an additional trick, called microextension. It was firstused by M. Hirsch in [Hi59]. The holonomic approximation method alsoworks well for differential relations which are not open, but microflexible.The most interesting applications of this type come from Symplectic Geom-etry. These applications are discussed in the third part of the book. Forconvenience of the reader the basic notions of Symplectic Geometry are alsoreviewed in that part of the book.

The fourth part of the book is devoted to convex integration theory. Gro-mov’s convex integration theory was treated in great detail by D. Springin [Sp98]. In our exposition of convex integration we pursue a differentgoal. Rather than considering the sophisticated advanced version of convexintegration presented in [Gr86], we explore only its simple version for firstorder differential relations, similar to the first exposition of the theory byGromov in [Gr73]. Nevertheless, we prove here practically all the mostinteresting corollaries of the theory, including the Nash-Kuiper theorem onC1-isometric embeddings.

Let us list here some available books and survey papers about the h-principle.Besides Gromov’s book [Gr86], these are: Spring’s book [Sp98], Adachi’sbook [Ad93], Haefliger’s paper [Ha71], Poenaru’s paper [Po71] and, mostrecently, Geiges’ notes [Ge01].

Acknowledgements. The book was partially written while the secondauthor visited the Department of Mathematics of Stanford University, andthe first author visited the Mathematical Institute of Leiden University andthe Institute for Advanced Study at Princeton. The authors thank the hostinstitutions for their hospitality. While writing this book the authors werepartially supported by the National Science Foundation. The first authoralso acknowledges the support of The Veblen Fund during his stay at theIAS.

We are indebted to Ana Cannas da Silva, Hansjorg Geiges, Simon Gober-stein, Dusa McDuff and David Spring who read the preliminary version ofthis book and corrected numerous misprints and mistakes. We are verythankful to all the mathematicians who communicated to us their criticalremarks and suggestions.

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Index

Jr(V,W ), 10

Jr(Rn , Rq ), 8

Jrf , 8

P i(z), 167

Pt,y , 154

S⊥ω , 76

r(eg, g), 191

CS(ξ), 89

CS(ξ+), 89

CloaR, 100

ExaR, 102

GrnW , 41

H(L), 79

H(X), 80

HolX(r), 11

J , 99

J (L), 78

J (X), 80

LX , 85

ΛpV , 47

Ω(t, y), 154

OpA, 9

RεLag, 175

Rεcoisot, 175

Rεcomp, 176

Rεisot, 175

RA, 173

RLag, 130, 174

RLeg, 93

Rclo, 55

Rcoisot, 174

Rcomp, 176

Rcont, 93

Rcoreal, 176

Rhol, 132

Rimm−trans, 137

Rimm, 54

Risocont, 93

Risosymp, 130

Risot, 174

Riso, 55

Rreal, 176

Rsub−isotr, 130

Rsub, 54

Rsymp, 174

Rtang, 136

Rtrans, 136

Rk−mers, 168

S(L), 76S(X), 80

S+cont, 100

Scont, 100

Snon−deg, 103

Ssymp, 99

SecX(r), 11

X(r), 9

S+cont, 100

Sanon−deg, 103

Sasymp, 99

Scont, 100

Ssymp, 99

bsF , 11

GF , Gdf , 41

θ-pair, 131

dg(f, ef), 192

k-mersion, 68

pr, 9

pr0, 9

prs, 10

pJ , 79

203

204 Index

pS , 79Conny Ω, 154ConvR, 155ConvF R , 155

Almost complex structure, 81integrable, 81

Almost symplectic structure, 81integrable, 81

Ampleness criterion, 173

Balanced path, 159

Canonical symplectic structure (form)

on R2n , 82on a cotangent bundle, 83

Capacious Lie subgroup, 139Characteristic foliation, 82

Compatible complex and symplectic struc-tures, 79

Complexmanifold, 81, 82

structure, 77subspace, 78vector space, 77

Contact

cutting-off, 98distribution, 88form, 88Hamiltonian, 98

manifold, 88monomorphism, 93structure, 4, 88

cooriented, 89

overtwisted, 108vector field, 98

Contact structuresformally homotopic, 108

homotopic, 108isotopic, 108

Contactization, 90Contactomorphisms, 89

Convex integration, xviiterated, 168one-dimensional, 155

parametric, 161

Coordinate principal subspace, 167, 170CR-structure, 82

Darboux contact form, 88

Darboux’ chart, 82Diffeotopy

δ-small, 22Differential condition, 53

Differential inclusion, 154Differential relation, xv, 53

Diff V -invariant, 66

A-invariant, 139k-flexible, 132k-microflexible, 131, 132ample, 154, 171

ample in the coordinate directions, 167closed, 55determined, xv, 55fibered, 64

fiberwise path-connected, 155locally integrable, 129, 130microflexible, 132open, 55overdetermined, 55

underdetermined, xv, 55Distribution, 47

Embeddingε-Lagrangian, 3co-real, 178

contact, 121directed, 176isocontact, 121isosymplectic, 111

Lagrangian, 3real, 3, 178symplectic, 111

Engel structure, 4Epimorphism, 54

ExactLagrangian immersion, 96Lagrangian submanifold, 96symplectic manifold, 146

Family of sections, 7

continuous, 7smooth, 7

Fiber bundle, 7Fibered map, 64

Fibration, 7natural, 65trivial, 7

Flower, 157abstract, 156

fibered, 162Formal inverse, 180Formal primitive, 48

Hamiltonian function, 97time dependent, 97

Hamiltonian isotopy, 97Hermitian structure, 79, 81

integrable, 81Holonomic R-Approximation Theorem, 133Holonomic approximation, 21

Holonomic Approximation Theorem, 22Homotopy

holonomic, 11

Index 205

regular, 1, 38tangential, 41

Homotopy principle (h-principle), xv, 2, 60C0-dense, 64(multi) parametric, 62

fibered, 64for

D-sections, 180C1-isometric immersions, 190ample differential relations, 171ample differential relations over a cube,

170

contact structures on open manifolds,102

directed embeddings, 176divergence free vector fields, 182immersions transversal to contact struc-

ture, 137immersions transversal to distribution,

71

isocontact embeddings, 121isocontact immersions, 143isosymplectic embeddings, 112isosymplectic immersions, 148Lagrangian immersions, 146Legendrian immersions, 144linearly-independent D-sections, 182

maps transversal to contact structure,136

maximally non-degenerate two-forms onodd-dimensional manifolds, 103, 185

microflexible Diff V -invariant relations,133

microflexible A-invariant relations, 140non-integrable hyperplane distributions

on even-dimensional manifolds, 138,

186non-vanishing D-sections, 181open Diff V -invariant relations, 66real and co-real embeddings, 178real and co-real immersions, 176sections transversal to distribution, 71

subcritical isotropic embeddings, 128symplectic forms on open manifold, 101systems of divergence free vector fields,

184systems of exact forms, 184

local, 63one-parametric, 60

relative, 63Smale-Hirsch, 69

Immersion, 1, 38, 54A-directed, 44ε-Lagrangian, 175ε-coisotropic, 175ε-isotropic, 175

co-real, 84coisotropic, 84complex, 84, 176

contact, 93isocomplex, 84isocontact, 93, 144isometric, 1, 189

isosymplectic, 84, 149isotropic, 84, 93Lagrangian, 84Legendrian, 93

real, 84, 176subcritical, 93symplectic, 84

Immersion relation, 54Isotopy

Hamiltonian, 97Legendrian, 97

Kahler manifold, 81Kahler metric, 81

Liouville structure, 96

Manifoldalmost complex, 81almost Kahler, 81

almost symplectic, 81complex, 81, 82contact, 88

Hermitian, 81Kahler, 81open, 40symplectic, 81, 82

Mapfibered, 64free, 4short, 189

strictly short, 189transversal to a distribution, 71transversal to a stratified set, 17

transversal to a submanifold, 15Microextension trick, 69Monomorphism, 41

contact, 93

isocontact, 93, 145isosymplectic, 112, 149symplectic, 112

Nash-Kuiper theorem, 190Nijenhuis tensor, 81

Operator

formally invertible, 180pure differential, 180

Petals, 156

206 Index

Polyhedra, 16Positivity condition, 107Primitive quadratic form, 191Primitive semi-Riemannian metric, 191

Principal direction, 170Principal subspace, 170Product of paths

uniform, 158weighted, 158

Projectivization, 92

r-jet, 8, 9r-jet extension, 8, 10Reeb foliation, 90Reeb vector field, 90Removal of singularities, xvi

Riemannian Cr-manifolds, 189Riemannian Cr-metric, 189

Section, 7Σ-non-singular, 55fiberwise holonomic, 24holonomic, 11

transversal to a distribution, 71Semi-Riemannian metric, 189Set

m-complete, 45ample, 154

stratified, 16Short formal solution, 155Short path, 153Singularity, 55

thin, 171

thin in the coordinate directions, 168Smale’s sphere eversion, 39Solution, 56, 57

r-extended, 56formal, xv, 56, 57

genuine, 56Space

of r-jets, 8of complex structures, 78of symplectic structures, 76

Stability theorems, 86, 95Standard contact structure on R2n−1 , 88Stem, 156Stratification, 16Submanifold

(s, p), 81almost complex, 81almost symplectic, 81co-isotropic, 81co-real, 81

complex, 82isotropic, 81Lagrangian, 81Legendrian, 90

subcritical, 90symplectic, 82totally real, 81

Submersion, 54Submersion relation, 54Subspace

(s, p), 77co-real, 78coisotropic, 77isotropic, 77Lagrangian, 77symplectic, 77totally real, 78

Symplecticbasis, 76cutting-off, 98form, 75manifold, 81, 82

ortogonal complement, 76structure, 75subspace, 77twisting, 115vector field, 97vector space, 75

Symplectic formsformally homotopic, 105homotopic, 105isotopic, 105

Symplectization, 92Symplectomorphism, 82

linear, 76

Thom Transversality Theorem, 17Transversal subbundles, 47

Vector bundlecomplex, 80Hermitian, 80symplectic, 80

Vector fieldcontact, 98Hamiltonian, 97Liouville, 92Reeb, 90symplectic, 97

www.ams.orgAMS on the WebGSM/48

In differential geometry and topology one often deals with systems of partial differential e uations as well as partial differential ine ualities that ha e in nitely many solutions what-e er oundary conditions are imposed It was disco ered in the fties that the sol a ility of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corre-sponding differential relation satis es the h-principle. Two famous examples of the h-principle, the Nash-Kuiper C1-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the h-principle.

The authors cover two main methods for proving the h-principle: holonomic approxima-tion and convex integration. The reader will nd that, with a few notable exceptions, most instances of the h-principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry.

Gromov’s famous book “Partial Differential Relations”, which is devoted to the same subject, is an encyclopedia of the h-principle, written for experts, while the present book is the rst broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differ-ential equations and inequalities. Geometers, topologists and analysts will also nd much value in this very readable exposition of an important and remarkable topic.