INTRODUCTION TO THE H-PRINCIPLE
CONTENTS INTRIGUE PART 1 HOLONOMIC APPROXIMATION CHAPTER 1 JETS AND HOLONOMY 11 MAPS AND SECTIONS 12 COORDINATE DE?NITION OF JETS THE SPACE J R(RN, RQ) 13 INVARIANT DE?NITION OF JETS 14 THE SPACE X(1) 15 HOLONOMIC SECTIONS OF THE JET SPACE X(R) 16 GEOMETRIC REPRESENTATION OF SECTIONS V ?X(R) 17 HOLONOMIC SPLITTING CHAPTER 2 THOM TRANSVERSALITY THEOREM 21 GENERIC PROPERTIES AND TRANSVERSALITY 22 STRATI?ED SETS AND POLYHEDRA 23 THOM TRANSVERSALITY THEOREM CHAPTER 3 HOLONOMIC APPROXIMATION 31 MAIN THEOREM 32 HOLONOMIC APPROXIMATION OVER A CUBE 33 FIBERWISE HOLONOMIC SECTIONS 34 INDUCTIVE LEMMA CONTENTS 35 PROOF OF THE INDUCTIVE LEMMA 36 HOLONOMIC APPROXIMATION OVER A CUBE 37 PARAMETRIC CASE CHAPTER 4 APPLICATIONS 41 FUNCTIONS WITHOUT CRITICAL POINTS 42 SMALE’S SPHERE EVERSION 43 OPEN MANIFOLDS 44 APPROXIMATE INTEGRATION OF TANGENTIAL HOMOTOPIES 45 DIRECTED EMBEDDINGS OF OPEN MANIFOLDS 46 DIRECTED EMBEDDINGS OF CLOSED MANIFOLDS 47 APPROXIMATION OF DI?ERENTIAL FORMS BY CLOSED FORMS PART 2 DI?ERENTIAL RELATIONS AND GROMOV’S H-PRINCIPLE CHAPTER 5 DI?ERENTIAL RELATIONS 51 WHAT IS A DI?ERENTIAL RELATION? 52 OPEN AND CLOSED DI?ERENTIAL RELATIONS 53 FORMAL AND GENUINE SOLUTIONS OF A DI?ERENTIAL RELATION 54 EXTENSION PROBLEM 55 APPROXIMATE SOLUTIONS TO SYSTEMS OF DI?ERENTIAL EQUATIONS CHAPTER 6 HOMOTOPY PRINCIPLE 61 PHILOSOPHY OF THE H-PRINCIPLE 62 DI?ERENT ?AVORS OF THE H-PRINCIPLE CHAPTER 7 OPEN DI?V -INVARIANT DI?ERENTIAL RELATIONS 71 DI?V -INVARIANT DI?ERENTIAL RELATIONS 72 LOCAL H-PRINCIPLE FOR OPEN DI?V -INVARIANT RELATIONS CHAPTER 8 APPLICATIONS TO CLOSED MANIFOLDS 81 MICROEXTENSION TRICK 82 SMALE-HIRSCH H-PRINCIPLE 83 SECTIONS TRANSVERSAL TO DISTRIBUTION PART 3 HOMOTOPY PRINCIPLE IN SYMPLECTIC GEOMETRY CHAPTER 9 SYMPLECTIC AND CONTACT BASICS 91 LINEAR SYMPLECTIC AND COMPLEX GEOMETRIES 92 SYMPLECTIC AND COMPLEX MANIFOLDS CONTENTS 93 SYMPLECTIC STABILITY 94 CONTACT MANIFOLDS 95 CONTACT STABILITY 96 LAGRANGIAN AND LEGENDRIAN SUBMANIFOLDS 97 HAMILTONIAN AND CONTACT VECTOR ?ELDS CHAPTER 10 SYMPLECTIC AND CONTACT STRUCTURES ON OPEN MANIFOLDS 101 CLASSI?CATION PROBLEM FOR SYMPLECTIC AND CONTACT STRUCTURES 102 SYMPLECTIC STRUCTURES ON OPEN MANIFOLDS 103 CONTACT STRUCTURES ON OPEN MANIFOLDS 104 TWO-FORMS OF MAXIMAL RANK ON ODD-DIMENSIONAL MANIFOLDS CHAPTER 11 SYMPLECTIC AND CONTACT STRUCTURES ON CLOSED MANIFOLDS 111 SYMPLECTIC STRUCTURES ON CLOSED MANIFOLDS 112 CONTACT STRUCTURES ON CLOSED MANIFOLDS CHAPTER 12 EMBEDDINGS INTO SYMPLECTIC AND CONTACT MANIFOLDS 121 ISOSYMPLECTIC EMBEDDINGS 122 EQUIDIMENSIONAL ISOSYMPLECTIC IMMERSIONS 123 ISOCONTACT EMBEDDINGS 124 SUBCRITICAL ISOTROPIC EMBEDDINGS CHAPTER 13 MICRO?EXIBILITY AND HOLONOMIC R-APPROXIMATION 131 LOCAL INTEGRABILITY 132 HOMOTOPY EXTENSION PROPERTY FOR FORMAL SOLUTIONS 133 MICRO?EXIBILITY 134 THEOREM ON HOLONOMIC R-APPROXIMATION 135 LOCAL H-PRINCIPLE FOR MICRO?EXIBLE DI?V -INVARIANT RELATIONS CHAPTER 14 FIRST APPLICATIONS OF MICRO?EXIBILITY 141 SUBCRITICAL ISOTROPIC IMMERSIONS 142 MAPS TRANSVERSAL TO A CONTACT STRUCTURE CHAPTER 15 MICRO?EXIBLE A-INVARIANT DI?ERENTIAL RELATIONS 151 A-INVARIANT DI?ERENTIAL RELATIONS 152 LOCAL H-PRINCIPLE FOR MICRO?EXIBLE A-INVARIANT RELATIONS CHAPTER 16 FURTHER APPLICATIONS TO SYMPLECTIC GEOMETRY 161 LEGENDRIAN AND ISOCONTACT IMMERSIONS 162 GENERALIZED ISOCONTACT IMMERSIONS CONTENTS 163 LAGRANGIAN IMMERSIONS 164 ISOSYMPLECTIC IMMERSIONS 165 GENERALIZED ISOSYMPLECTIC IMMERSIONS PART 4 CONVEX INTEGRATION CHAPTER 17 ONE-DIMENSIONAL CONVEX INTEGRATION 171 EXAMPLE 172 CONVEX HULLS AND AMPLENESS 173 MAIN LEMMA 174 PROOF OF THE MAIN LEMMA 175 PARAMETRIC VERSION OF THE MAIN LEMMA 176 PROOF OF THE PARAMETRIC VERSION OF THE MAIN LEMMA CHAPTER 18 HOMOTOPY PRINCIPLE FOR AMPLE DI?ERENTIAL RELATIONS 181 AMPLENESS IN COORDINATE DIRECTIONS 182 ITERATED CONVEX INTEGRATION 183 PRINCIPAL SUBSPACES AND AMPLE DI?ERENTIAL RELATIONS IN X (1) 184 CONVEX INTEGRATION OF AMPLE DI?ERENTIAL RELATIONS CHAPTER 19 DIRECTED IMMERSIONS AND EMBEDDINGS 191 CRITERION OF AMPLENESS FOR DIRECTED IMMERSIONS 192 DIRECTED IMMERSIONS INTO ALMOST SYMPLECTIC MANIFOLDS 193 DIRECTED IMMERSIONS INTO ALMOST COMPLEX MANIFOLDS 194 DIRECTED EMBEDDINGS CHAPTER 20 FIRST ORDER LINEAR DI?ERENTIAL OPERATORS 201 FORMAL INVERSE OF A LINEAR DI?ERENTIAL OPERATOR 202 HOMOTOPY PRINCIPLE FOR D-SECTIONS 203 NON-VANISHING D-SECTIONS 204 SYSTEMS OF LINEARLY INDEPENDENT D-SECTIONS 205 TWO-FORMS OF MAXIMAL RANK ON ODD-DIMENSIONAL MANIFOLDS 206 ONE-FORMS OF MAXIMAL RANK ON EVEN-DIMENSIONAL MANIFOLDS CHAPTER 21 NASH-KUIPER THEOREM 211 ISOMETRIC IMMERSIONS AND SHORT IMMERSIONS 212 NASH-KUIPER THEOREM 213 DECOMPOSITION OF A METRIC INTO A SUM OF PRIMITIVE METRICS 214 APPROXIMATION THEOREM CONTENTS 215 ONE-DIMENSIONAL APPROXIMATION THEOREM 216 ADDING A PRIMITIVE METRIC 217 END OF THE PROOF OF THE APPROXIMATION THEOREM 218 PROOF OF THE NASH-KUIPER THEOREM