Introduction to the H-Principle

Introduction to the h-Principle

= )PMEWLFIVK 2 MMWLEGLIZ

GVEHYEXI SXYHMIW MR MEXLIQEXMGW :SPYQI 48

AQIVMGER MEXLIQEXMGEP SSGMIX] http://dx.doi.org/10.1090/gsm/048

Titles in This Series

48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, 2002 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 HershelM.FarkasandIrwinKra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 JamesF.DavisandPaulKirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and András I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

Introduction to the h-Principle

Y. Eliashberg N. Mishachev

Graduate Studies in Mathematics Volume 48

American Mathematical Society Providence, Rhode Island Editorial Board Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair)

2000 Mathematics Subject Classiﬁcation. Primary 58Axx.

Abstract. The book is devoted to topological methods for solving differential equations and inequalities. 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 2002 516.36–dc21 2002019347

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02940-6248 USA. Requests can also be made by e-mail to [email protected].

c 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 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

1 3

Part I 2 4

Part II 3

1756 8

Part III 9 6,7,8

10 11 12 14 16

4313 15

6 17 18

Part IV 21 19 20

Figure 0.1. The relations between chapters of the book Preface

A partial diﬀerential relation R is any condition imposed on the partial derivatives of an unknown function. A solution of R is any function which satisﬁes this relation.

The classical partial diﬀerential relations, mostly rooted in Physics, are usually described by (systems of) equations. Moreover, the corresponding systems of equations are mostly determined: the number of unknown functions is equal to the number of equations. Given appropriate boundary conditions, such a diﬀerential relation usually has a unique solution. In some cases this solution can be found using certain analytical methods (potential theory, Fourier method and so on).

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

xv xvi Preface

The notion of h-principle (under the name “w.h.e.-principle”) first appeared in [Gr71] and [GE71]. The term “h-principle” was introduced and pop- ularized by M. Gromov in his book [Gr86]. The h-principle for solutions of partial differential relations exposed the soft/hard (or flexible/rigid) di- chotomy for the problems formulated in terms of derivatives: a particular analytical problem is “soft” or “abides by the h-principle” if its solvability is determined by some underlying algebraic or geometric data. The softness phenomena was first discovered in the fifties by J. Nash [Na54] for isometric C1-immersions, and by S. Smale [Sm58, Sm59] for differential immersions. However, instances of soft problems appeared earlier (e.g. H. Whitney’s paper [Wh37]). In the sixties several new geometrically interesting examples of soft problems were discovered by M. Hirsch, V. Poénaru, 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], Gromov transformed Smale’s and Nash’s ideas into two powerful general methods for solving partial differential relations: continuous sheaves (or the covering homotopy) 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’s work to popularize” (see [Be00]). We have written our book in order to im- prove the situation. We consider here two geometrical methods: holonomic approximation, which is a version of the method of continuous sheaves, and convex 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 to look for hidden treasures there. On the other hand, the reader interested in applications will ﬁnd that with a few notable exceptions (e.g. Lohkamp’s theory [Lo95] of negative Ricci curvature and Donaldson’s theory [Do96] of approximately holomorphic sections) most instances of the h-principle which are known today can be treated by the methods considered in the present book.

The first three parts of the book are devoted to a quite general theorem about holonomic approximation of sections of jet-bundles and its applications. Given an arbitrary submanifold V0 ⊂ V of positive codimension, the Holonomic Approximation Theorem allows us to solve any open differential relations R near a slightly perturbed submanifold V0 = h(V )where h : V → V is a C0-small diffeomorphism. Gromov’s h-principle for open Diff V -invariant differential relations on open manifolds, his directed embedding theorem, as well as some other results in the spirit of the h-principle are 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 closed manifolds require an additional trick, called microextension. It was first used by M. Hirsch in [Hi59]. The holonomic approximation method also works 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. For convenience of the reader the basic notions of Symplectic Geometry are also reviewed 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. Spring in [Sp98]. In our exposition of convex integration we pursue a different goal. Rather than considering the sophisticated advanced version of convex integration presented in [Gr86], we explore only its simple version for first order differential relations, similar to the first exposition of the theory by Gromov in [Gr73]. Nevertheless, we prove here practically all the most interesting corollaries of the theory, including the Nash-Kuiper theorem on C1-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’s book [Ad93], Haefliger’s paper [Ha71], Poénaru’s paper [Po71] and, most recently, Geiges’ notes [Ge01]. Acknowledgements. The book was partially written while the second author visited the Department of Mathematics of Stanford University, and the first author visited the Mathematical Institute of Leiden University and the Institute for Advanced Study at Princeton. The authors thank the host institutions for their hospitality. While writing this book the authors were partially supported by the National Science Foundation. The first author also acknowledges the support of The Veblen Fund during his stay at the IAS. We are indebted to Ana Cannas da Silva, Hansjorg Geiges, Simon Gober- stein, Dusa McDuff and David Spring who read the preliminary version of this book and corrected numerous misprints and mistakes. We are very thankful to all the mathematicians who communicated to us their critical remarks and suggestions.

Bibliography

[Ad93] M. Adachi, Embeddings and Immersions, American Mathematical Society, Providence, RI, 1993. [Ar65] V. Arnold, Sur une propriété topologique des applications globalement canon- iques de la mécanique classique,C.R.Acad.Sci.Paris,261(1965), 3719–3722. [Ar78] V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978. [Ar86] V. Arnold, First steps of symplectic topology, Russian Math. Surveys, 41(1986) no. 6(251), 1–21. [AG90] V. Arnold and A. Givental, Symplectic Geometry, Dynamical Systems IV, Springer-Verlag, 1990. [Ba78] A. Banyaga, Sur la structure du groupe des diffeomorphismes qui préservent une forme symplectique,Comment.Math.Helv.,53(1978), 174–227. [Be83] D. Bennequin, Entrelacements et équations de Pfaff,Astérisque, 107– 108(1983), 87-161. [Be00] M. Berger, Encounter with a geometer, Notices of Amer. Math. Soc., 47(2000), 183–194. [Bo01] V. Borrelli, On totally real isotopy classes, preprint, 2001. [BH93] R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions,Invent. Math.,114(1993), 435–461. [Ca70] E. Calabi, On the group of automorphisms of a symplectic manifold,Problem in analysis, 1–26, Princeton University Press, 1970. [CdS01] Ana Cannas da Silva, Lectures on Symplectic Geometry, Lect. Notes in Math., 1764(2001), Springer-Verlag. [Da97] M. Datta, Homotopy classification of strict contact immersions, Ann. Global Anal. Geom., 15(1997), 211–219. [DA00] G. D’Ambra, An application of the h-principle to C1-isometric immersions in contact manifolds, Comm. Anal. Geom., 8(2000), 347–373. [DL01] G. D’Ambra, A. Loi, A symplectic version of the Nash C1-isometric embedding theorem, preprint 2001, to appear in Diff. Geom. and Appl.

199 200 Bibliography

[Do96] S.K. Donaldson, Symplectic submanifolds and almost complex geometry,J.Diff. Geom., 44(1996), 666–705. [Du84] T. Duchamp, The classification of Legendre immersions, preprint, 1984. [Ee66] J. Eells, A setting for global analysis, Bull. Amer. Math. Soc., 72(1966), 751– 807. [El72] Y. Eliashberg, Surgery of singularities of smooth maps, Izv. Akad. Nauk SSSR Ser. Mat., 36(1972), 1321–1347. [El85] Y. Eliashberg, Complexification of contact structures on 3-dimensional manifolds, Uspekhi Mat. Nauk, 40(1985) no. 6(246), 161–162. [El89] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math., 98(1989), 623–637. [El91] Y. Eliashberg, Filling by holomorphic discs and its applications, London Math. Soc. Lecture Note Ser., 151(1991), 45–67. [EGH00] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. 2000, Special Volume, Part II, 560–673. [EM01] Y. Eliashberg and N. Mishachev, Holonomic approximation and Gromov’s h- principle, L’Enseignement Math., 47(2001). [Et99] J. Etnyre, Tight contact structures on lens spaces, Commun. Contemp. Math., 2(2000), 559–577. [Fe69] S. Feit, k-mersions of manifolds,ActaMath.,122(1969), 173–195. 2

[Fd87] T. Fiedler, Totally real embeddings of the torus in C , Ann. Glob. Anal. Geom., 5(1987), 117–121. [Fi67] A.F. Filippov, Classical solutions of differential equations with multi-valued righ-hand side, SIAM J. Control, 6(1967), pp. 609–621. [Ge01] H. Geiges, h-principle and flexibility in geometry, preprint 2001. [GT99] H. Geiges and C.B. Thomas, Contact structures on 7-manifolds,CRMProc. Lecture Notes, 15(1998), 53–67. [Gi97] V. Ginzburg, A smooth counterexample to the Hamiltonian Seifert conjecture in R6, Internat. Math. Res. Notices, 13(1997), 641–650. [Gi99] E. Giroux, Une infinité de structures de contact tendues sur une infinitéde variétés, Invent. Math. 135(1999), 789–802. [Gi01] E. Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76(2001), 218–262. [GM87] M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, 1988. [Gr59] A. Gray, Some global properties of contact structures, Ann. of Math.(2) 69(1959), 421-450. [Gr69] M. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 33(1969), 707–734. [Gr71] M. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, ICM 1970, Nice, vol. 2, 221–225. [Gr73] M. Gromov, Convex integration of partial differential relations. I,Izv.Akad. Nauk SSSR Ser. Mat., 37(1973), 329–343. [Gr85] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307–347. [Gr86] M. Gromov, Partial Differential Relations, Springer-Verlag, 1986. Bibliography 201

[GE71] M. Gromov and Y. Eliashberg, Removal of singularities of smooth maps,Izv. Akad. Nauk SSSR Ser. Mat., 35(1971), 600–627. [GE73] M. Gromov and Y. Eliashberg, Construction of a smooth map with prescribed- Jacobian, Funk. Anal. and Applications, 5:2(1973), 27–32 [Ha71] A. Haefliger, Lectures on the theorem of Gromov, Lecture Notes in Math., 209(1971), 128–141. [Hi59] M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc., 93(1959), 242– 276. [HZ94] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, 1994. [Ho00] Ko Honda, On the classification of tight contact structures. I, Geom. Topol., 4(2000), 309–368. [Ku55] N. Kuiper, On C1-isometric imbeddings, Indag. Math., 17(1955), 545–556. [Le76] J. Lees, On the classification of Lagrange immersion, Duke Math. J. 43(1976), 217–224 [Lo95] J. Lohkamp, Curvature h-principles, Ann. of Math., 142 (1995), 457–498. [Lu77] R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27(1977) no. 3, 1–15. [Ma71] J. Martinet, Formes de contact sur les variétés de dimension 3, Lecture Notes in Math., 209(1971), 142–163. [MD87a] D. McDuff, Application of convex integration to symplectic and contact geometry Ann. Inst. Fourier, (Grenoble) 37(1987), no 1, 107–133. [MD87b] D. McDuff, Examples of symplectic structures, Invent. Math., 89(1987), 13–36. [MS98] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Clarendon Press. Oxford, 1998. [MT99] C. McMullen and C.H. Taubes, 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations, Math. Res. Lett., 6 (1999), 681– 696. [Na54] J. Nash, C1-isometric imbeddings, Ann. of Math.(2), 60(1954), 383–396. [NN57] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math.(2), 65(1957), 391–404. [Pa66] R. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5(1966), 1–16. [Ph67] A. Phillips, Submersions of open manifolds, Topology 6(1967), 171–206. [Po66] V. Poénaru, On regular homotopy in codimension 1, Ann. of Math.(2), 83(1966), 257–265. [Po71] V. Poénaru, Homotopy theory and differentiable singularities, Lecture Notes in Math., 197(1971), 106–132. [Pt88] L. Polterovich, New invariants of embedded totally-real tori and one problem of Hamiltonian mechanics, in the book “Methods of qualitative theory and the Theory of bifurcations” (in Russian), Gorki, 1988, 84–90. [RS97] C. Rourke and B. Sanderson, The compression theorem – I,II,Geom.Topol., 5(2001), 399–429, 431–440. [RS00] C. Rourke and B. Sanderson, Directed embeddings: a short proof of Gromov’s theorem, preprint, 2000. 202 Bibliography

[Ru94] Y. Ruan, Symplectic topology on algebraic 3-folds, Differential. Geom., 39(1994), 215–227. [Sa42] A. Sard, The measure of the critical values of differentiable maps, Bull. Ann. Math. Soc. 48(1942), 8883–8890. [Si64] C. L. Siegel, Symplectic Geometry, Academic Press, 1964 [Sm58] S. Smale, A classification of immersions of the two-sphere, Trans. Am. Math. Soc. 90(1958), 281-290. [Sm59] S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math.(2) 69(1959), 327–344. [Sp98] D. Spring, Convex Integration Theory, Birkhauser, 1998. [Sp00] D. Spring, Directed embeddings and the simplification of singularities,preprint, 2000. [Ta94] C. H. Taubes, The Seiberg-Witten invariants and symplectic forms,Math.Res. Lett., 1 (1994), 809–822. [Th76] W.P. Thurston, Some simple examples of symplectic manifolds,Proc.Amer. Math. Soc., 55(1976), 467–468. [Us99] I. Ustilovsky, Infinitely many contact structures on S4m+1, Internat. Math. Res. Notices, 14(1999), 781–791. [Wh37] H. Whitney, On regular closed curves in the plane, Compositio Math., 4(1937), 276–284. Index

r J (V,W), 10 Rimm−trans, 137

r n q R J ( R , ), 8 Rimm,54 r R Jf ,8 isocont,93 P i(z), 167 Risosymp, 130 R Pt,y, 154 isot, 174 ⊥ R S ω ,76 iso,55 R , 176 r(g,e g), 191 real R CS(ξ), 89 sub−isotr, 130 R CS(ξ+), 89 sub,54 R CloaR, 100 symp, 174 Exa R, 102 Rtang, 136 R GrnW ,41 trans, 136 R H(L), 79 k−mers, 168 H(X), 80 S(L), 76 Hol X(r),11 S(X), 80 S+ J ,99 cont, 100 J (L), 78 Scont, 100 J (X), 80 Snon−deg, 103 S LX ,85 symp,99 ΛpV ,47 Sec X(r),11 Ω(t, y), 154 X(r),9 + Op A,9 Scont, 100 ε a R S , 103 Lag, 175 non−deg ε a R S ,99 coisot, 175 symp ε R S comp, 176 cont, 100 ε R S ,99 isot, 175 symp RA, 173 bs F ,11 RLag, 130, 174 GF ,Gdf ,41 RLeg,93 θ-pair, 131

Rclo,55 dg(f,fe), 192 Rcoisot, 174 k-mersion, 68 r Rcomp, 176 p ,9 R r cont,93 p0,9 r Rcoreal, 176 ps,10 Rhol, 132 pJ ,79

203 204 Index

pS ,79 A-invariant, 139 Conny Ω, 154 k-flexible, 132 Conv R, 155 k-microflexible, 131, 132 ConvF R , 155 ample, 154, 171 ample in the coordinate directions, 167 Almost complex structure, 81 closed, 55 integrable, 81 determined, xv, 55 Almost symplectic structure, 81 fibered, 64 integrable, 81 fiberwise path-connected, 155 Ampleness criterion, 173 locally integrable, 129, 130 microflexible, 132 Balanced path, 159 open, 55 overdetermined, 55 Canonical symplectic structure (form) underdetermined, xv, 55 2n on R ,82 Distribution, 47 on a cotangent bundle, 83 Capacious Lie subgroup, 139 Embedding Characteristic foliation, 82 ε-Lagrangian, 3 Compatible complex and symplectic struc- co-real, 178 tures, 79 contact, 121 Complex directed, 176 manifold, 81, 82 isocontact, 121 structure, 77 isosymplectic, 111 subspace, 78 Lagrangian, 3 vector space, 77 real, 3, 178 Contact symplectic, 111 cutting-off, 98 Engel structure, 4 distribution, 88 Epimorphism, 54 form, 88 Exact Hamiltonian, 98 Lagrangian immersion, 96 manifold, 88 Lagrangian submanifold, 96 monomorphism, 93 symplectic manifold, 146 structure, 4, 88 cooriented, 89 Family of sections, 7 overtwisted, 108 continuous, 7 vector field, 98 smooth, 7 Contact structures Fiber bundle, 7 formally homotopic, 108 Fibered map, 64 homotopic, 108 Fibration, 7 isotopic, 108 natural, 65 Contactization, 90 trivial, 7 Contactomorphisms, 89 Flower, 157 Convex integration, xvi abstract, 156 iterated, 168 fibered, 162 one-dimensional, 155 Formal inverse, 180 parametric, 161 Formal primitive, 48 Coordinate principal subspace, 167, 170 CR-structure, 82 Hamiltonian function, 97 time dependent, 97 Darboux contact form, 88 Hamiltonian isotopy, 97 Darboux’ chart, 82 Hermitian structure, 79, 81 Diffeotopy integrable, 81 δ-small, 22 Holonomic R-Approximation Theorem, 133 Differential condition, 53 Holonomic approximation, 21 Differential inclusion, 154 Holonomic Approximation Theorem, 22 Differential relation, xv, 53 Homotopy Diff V -invariant, 66 holonomic, 11 Index 205

regular, 1, 38 co-real, 84 tangential, 41 coisotropic, 84 Homotopy principle (h-principle), xv, 2, 60 complex, 84, 176 C0-dense, 64 contact, 93 (multi) parametric, 62 isocomplex, 84 fibered, 64 isocontact, 93, 144 for isometric, 1, 189 D-sections, 180 isosymplectic, 84, 149 C1-isometric immersions, 190 isotropic, 84, 93 ample differential relations, 171 Lagrangian, 84 ample differential relations over a cube, Legendrian, 93 170 real, 84, 176 contact structures on open manifolds, subcritical, 93 102 symplectic, 84 directed embeddings, 176 Immersion relation, 54 divergence free vector fields, 182 Isotopy immersions transversal to contact struc- Hamiltonian, 97 ture, 137 Legendrian, 97 immersions transversal to distribution, 71 Kähler manifold, 81 isocontact embeddings, 121 Kähler metric, 81 isocontact immersions, 143 isosymplectic embeddings, 112 Liouville structure, 96 isosymplectic immersions, 148 Lagrangian immersions, 146 Manifold Legendrian immersions, 144 almost complex, 81 linearly-independent D-sections, 182 almost Kähler, 81 maps transversal to contact structure, almost symplectic, 81 136 complex, 81, 82 maximally non-degenerate two-forms on contact, 88 odd-dimensional manifolds, 103, 185 Hermitian, 81 microflexible Diff V -invariant relations, Kähler, 81 133 open, 40

microflexible A-invariant relations, 140 symplectic, 81, 82 non-integrable hyperplane distributions Map on even-dimensional manifolds, 138, fibered, 64 186 free, 4 non-vanishing D-sections, 181 short, 189 open Diff V -invariant relations, 66 strictly short, 189 real and co-real embeddings, 178 transversal to a distribution, 71 real and co-real immersions, 176 transversal to a stratified set, 17 sections transversal to distribution, 71 transversal to a submanifold, 15 subcritical isotropic embeddings, 128 Microextension trick, 69 symplectic forms on open manifold, 101 Monomorphism, 41 systems of divergence free vector fields, contact, 93 184 isocontact, 93, 145 systems of exact forms, 184 isosymplectic, 112, 149 local, 63 symplectic, 112 one-parametric, 60 relative, 63 Nash-Kuiper theorem, 190 Smale-Hirsch, 69 Nijenhuis tensor, 81

Immersion, 1, 38, 54 Operator A-directed, 44 formally invertible, 180 ε-Lagrangian, 175 pure diﬀerential, 180 ε-coisotropic, 175 ε-isotropic, 175 Petals, 156 206 Index

Polyhedra, 16 subcritical, 90 Positivity condition, 107 symplectic, 82 Primitive quadratic form, 191 totally real, 81 Primitive semi-Riemannian metric, 191 Submersion, 54 Principal direction, 170 Submersion relation, 54 Principal subspace, 170 Subspace Product of paths (s, p), 77 uniform, 158 co-real, 78 weighted, 158 coisotropic, 77 Projectivization, 92 isotropic, 77 Lagrangian, 77 r-jet, 8, 9 symplectic, 77 r-jetextension,8,10 totally real, 78 Reeb foliation, 90 Symplectic Reeb vector field, 90 basis, 76 Removal of singularities, xvi cutting-off, 98 Riemannian Cr-manifolds, 189 form, 75 Riemannian Cr-metric, 189 manifold, 81, 82 ortogonal complement, 76 Section, 7 structure, 75 Σ-non-singular, 55 subspace, 77 fiberwise holonomic, 24 twisting, 115 holonomic, 11 vector field, 97 transversal to a distribution, 71 vector space, 75 Semi-Riemannian metric, 189 Symplectic forms Set formally homotopic, 105 m-complete, 45 homotopic, 105 ample, 154 isotopic, 105 stratified, 16 Symplectization, 92 Short formal solution, 155 Symplectomorphism, 82 Short path, 153 linear, 76 Singularity, 55 thin, 171 Thom Transversality Theorem, 17 thin in the coordinate directions, 168 Transversal subbundles, 47 Smale’s sphere eversion, 39 Solution, 56, 57 Vector bundle r-extended, 56 complex, 80 formal, xv, 56, 57 Hermitian, 80 genuine, 56 symplectic, 80 Space Vector field of r-jets, 8 contact, 98 of complex structures, 78 Hamiltonian, 97 of symplectic structures, 76 Liouville, 92 Stability theorems, 86, 95 Reeb, 90 2n−1

Standard contact structure on R ,88 symplectic, 97 Stem, 156 Stratiﬁcation, 16 Submanifold (s, p), 81 almost complex, 81 almost symplectic, 81 co-isotropic, 81 co-real, 81 complex, 82 isotropic, 81 Lagrangian, 81 Legendrian, 90 In differential geometry and topology one often deals with systems of partial differential eUuations as well as partial differential ineUualities that haZe in½nitely many solutions what- eZer Foundary conditions are imposed It was discoZered in the ½fties that the solZaFility 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 corresponding differential relation satis½es 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 approximation and convex integration. The reader will ½nd 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 ½rst 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 differential equations and inequalities. Geometers, topologists and analysts will also ½nd much value in this very readable exposition of an important and remarkable topic.

GSM/48 AMS on the Web www.ams.org