Table Of Content

Introduction to the h-Principle Y. Eliashberg N. Mishachev iii To Vladimir Igorevich Arnold who introduced us to the world of singularities and Misha Gromov who taught us how to get rid of them iv Contents Preface XI Intrigue 1 Part 1. Holonomic Approximation Chapter 1. Jets and Holonomy 7 1.1. Maps and sections. 7 § 1.2. Coordinate definition of jets. The space Jr(Rn,Rq) 8 § 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 V X(r) 11 § → 1.7. Holonomic splitting 12 § Chapter 2. Thom Transversality Theorem 13 2.1. Generic Properties and Transversality 13 § 2.2. Stratified sets and polyhedra 14 § 2.3. Thom Transversality Theorem 15 § Chapter 3. Holonomic Approximation 19 3.1. Main theorem 19 § 3.2. Holonomic approximation over a cube 21 § 3.3. Fiberwise holonomic sections 22 § 3.4. Inductive Lemma 22 § V VI Contents 3.5. Proof of the Inductive Lemma 26 § 3.6. Holonomic approximation over a cube 31 § 3.7. Parametric case. 32 § Chapter 4. Applications 33 4.1. Functions without critical points 33 § 4.2. Smale’s sphere eversion 34 § 4.3. Open manifolds 35 § 4.4. Approximate integration of tangential homotopies 37 § 4.5. Directed embeddings of open manifolds 39 § 4.6. Directed embeddings of closed manifolds 41 § 4.7. Approximation of differential forms by closed forms 42 § Part 2. Differential Relations and Gromov’s h-Principle Chapter 5. Differential Relations 47 5.1. What is a differential relation? 47 § 5.2. Open and closed differential relations 49 § 5.3. Formal and genuine solutions of a differential relation 50 § 5.4. Extension problem 50 § 5.5. Approximate solutions to systems of differential equations 51 § Chapter 6. Homotopy Principle 53 6.1. Philosophy of the h-principle 53 § 6.2. Different flavors of the h-principle 56 § Chapter 7. Open DiffV-Invariant Differential Relations 59 7.1. DiffV-invariant differential relations 59 § 7.2. Local h-principle for open DiffV-invariant relations 60 § Chapter 8. Applications to Closed Manifolds 63 8.1. Microextension trick 63 § 8.2. Smale-Hirsch h-principle 63 § 8.3. Sections transversal to distribution 65 § Part 3. Homotopy Principle in Symplectic Geometry Chapter 9. Symplectic and Contact Basics 69 9.1. Linear symplectic and complex geometries 69 § 9.2. Symplectic and complex manifolds 74 § Contents VII 9.3. Symplectic stability 79 § 9.4. Contact manifolds 82 § 9.5. Contact stability 87 § 9.6. Lagrangian and Legendrian submanifolds 89 § 9.7. Hamiltonian and contact vector fields 91 § Chapter 10. Symplectic and Contact Structures on Open Manifolds 93 10.1. Classification problem for symplectic and contact structures 93 § 10.2. Symplectic structures on open manifolds 94 § 10.3. Contact structures on open manifolds 96 § 10.4. Two-forms of maximal rank on odd-dimensional manifolds 97 § Chapter 11. Symplectic and Contact Structures on Closed Manifolds 99 11.1. Symplectic structures on closed manifolds 99 § 11.2. Contact structures on closed manifolds 101 § Chapter 12. Embeddings into Symplectic and Contact Manifolds 105 12.1. Isosymplectic embeddings 105 § 12.2. Equidimensional isosymplectic immersions 112 § 12.3. Isocontact embeddings 115 § 12.4. Subcritical isotropic embeddings 122 § Chapter 13. Microflexibility and Holonomic -approximation 123 R 13.1. Local integrability 123 § 13.2. Homotopy extension property for formal solutions 125 § 13.3. Microflexibility 125 § 13.4. Theorem on holonomic -approximation 127 § R 13.5. Local h-principle for microflexible DiffV-invariant relations 128 § Chapter 14. First Applications of Microflexibility 129 14.1. Subcritical isotropic immersions 129 § 14.2. Maps transversal to a contact structure 130 § Chapter 15. Microflexible A-invariant Differential Relations 133 15.1. A-invariant differential relations 133 § 15.2. Local h-principle for microflexible A-invariant relations 134 § Chapter 16. Further Applications to Symplectic Geometry 137 16.1. Legendrian and isocontact immersions 137 § 16.2. Generalized isocontact immersions 138 § VIII Contents 16.3. Lagrangian immersions 140 § 16.4. Isosymplectic immersions 141 § 16.5. Generalized isosymplectic immersions 143 § Part 4. Convex Integration Chapter 17. One-Dimensional Convex Integration 147 17.1. Example 147 § 17.2. Convex hulls and ampleness 148 § 17.3. Main lemma 149 § 17.4. Proof of the main lemma 150 § 17.5. Parametric version of the main lemma 155 § 17.6. Proof of the parametric version of the main lemma 155 § Chapter 18. Homotopy Principle for Ample Differential Relations 159 18.1. Ampleness in coordinate directions 159 § 18.2. Iterated convex integration 160 § 18.3. Principal subspaces and ample differential relations in X(1) 162 § 18.4. Convex integration of ample differential relations 163 § Chapter 19. Directed Immersions and Embeddings 165 19.1. Criterion of ampleness for directed immersions 165 § 19.2. Directed immersions into almost symplectic manifolds 166 § 19.3. Directed immersions into almost complex manifolds 167 § 19.4. Directed embeddings 168 § Chapter 20. First Order Linear Differential Operators 171 20.1. Formal inverse of a linear differential operator 171 § 20.2. Homotopy principle for -sections 172 § D 20.3. Non-vanishing -sections 173 § D 20.4. Systems of linearly independent -sections 174 § D 20.5. Two-forms of maximal rank on odd-dimensional manifolds 177 § 20.6. One-forms of maximal rank on even-dimensional manifolds 178 § Chapter 21. Nash-Kuiper Theorem 181 21.1. Isometric immersions and short immersions 181 § 21.2. Nash-Kuiper theorem 182 § 21.3. Decomposition of a metric into a sum of primitive metrics 183 § 21.4. Approximation theorem 183 § Contents IX 21.5. One-dimensional approximation theorem 185 § 21.6. Adding a primitive metric 186 § 21.7. End of the proof of the approximation theorem 188 § 21.8. Proof of the Nash-Kuiper theorem 188 § Bibliography 191 Index 195 X Contents 1 3 2 4 Part I Part II 3 1 5 6 7 8 Part III 9 6,7,8 10 11 12 14 16 4 3 13 15 6 17 18 21 19 20 Part IV Figure 0.1. The relations between chapters of the book