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Tables of Contents for Introduction to the H-Principle
Chapter/Section Title
Page #
Page Count
Preface
xv
Intrigue
1
6
Part 1. Holonomic Approximation
Jets and Holonomy
7
8
Maps and sections
7
1
Coordinate definition of jets
7
2
Invariant definition of jets
9
1
The space X(1)
10
1
Holonomic sections of the jet space X(r)
11
1
Geometric representation of sections of X(r)
12
1
Holonomic splitting
12
3
Thom Transversality Theorem
15
6
Generic properties and transversality
15
1
Stratified sets and polyhedra
16
1
Thom Transversality Theorem
17
4
Holonomic Approximation
21
16
Main theorem
21
2
Holonomic approximation over a cube
23
1
Fiberwise holonomic sections
24
1
Inductive Lemma
25
3
Proof of the Inductive Lemma
28
5
Holonomic approximation over a cube
33
1
Parametric case
34
3
Applications
37
16
Functions without critical points
37
1
Smale's sphere eversion
38
2
Open manifolds
40
1
Approximate integration of tangential homotopies
41
3
Directed embeddings of open manifolds
44
1
Directed embeddings of closed manifolds
45
2
Approximation of differential forms by closed forms
47
6
Part 2. Differential Relations and Gromov's h-Principle
Differential Relations
53
6
What is a differential relation?
53
2
Open and closed differential relations
55
1
Formal and genuine solutions of a differential relation
56
1
Extension problem
56
1
Approximate solutions to systems of differential equations
57
2
Homotopy Principle
59
6
Philosophy of the h-principle
59
3
Different flavors of the h-principle
62
3
Open Diff V-Invariant Differential Relations
65
4
Diff V-invariant differential relations
65
1
Local h-principle for open Diff V-invariant relations
66
3
Applications to Closed Manifolds
69
6
Microextension trick
69
1
Smale-Hirsch h-principle
69
2
Sections transversal to distribution
71
4
Part 3. The Homotopy Principle in Symplectic Geometry
Symplectic and Contact Basics
75
24
Linear symplectic and complex geometries
75
5
Symplectic and complex manifolds
80
5
Symplectic stability
85
3
Contact manifolds
88
6
Contact stability
94
1
Lagrangian and Legendrian submanifolds
95
2
Hamiltonian and contact vector fields
97
2
Symplectic and Contact Structures on Open Manifolds
99
6
Classification problem for symplectic and contact structures
99
1
Symplectic structures on open manifolds
100
2
Contact structures on open manifolds
102
1
Two-forms of maximal rank on odd-dimensional manifolds
103
2
Symplectic and Contact Structures on Closed Manifolds
105
6
Symplectic structures on closed manifolds
105
2
Contact structures on closed manifolds
107
4
Embeddings into Symplectic and Contact Manifolds
111
18
Isosymplectic embeddings
111
7
Equidimensional isosymplectic immersions
118
3
Isocontact embeddings
121
7
Subcritical isotropic embeddings
128
1
Microflexibility and Holonomic R-Approximation
129
6
Local integrability
129
2
Homotopy extension property for formal solutions
131
1
Microflexibility
131
2
Theorem on holonomic R-approximation
133
1
Local h-principle for microflexible Diff V-invariant relations
133
2
First Applications of Microflexibility
135
4
Subcritical isotropic immersions
135
1
Maps transversal to a contact structure
136
3
Microflexible U-Invariant Differential Relations
139
4
U-invariant differential relations
139
1
Local h-principle for microflexible U-invariant relations
140
3
Further Applications to Symplectic Geometry
143
10
Legendrian and isocontact immersions
143
1
Generalized isocontact immersions
144
2
Lagrangian immersions
146
1
Isosymplectic immersions
147
2
Generalized isosymplectic immersions
149
4
Part 4. Convex Integration
One-Dimensional Convex Integration
153
14
Example
153
1
Convex hulls and ampleness
154
1
Main lemma
155
1
Proof of the main lemma
156
5
Parametric version of the main lemma
161
1
Proof of the parametric version of the main lemma
162
5
Homotopy Principle for Ample Differential Relations
167
6
Ampleness in coordinate directions
167
1
Iterated convex integration
168
2
Principal subspaces and ample differential relations in X(1)
170
1
Convex integration of ample differential relations
171
2
Directed Immersions and Embeddings
173
6
Criterion of ampleness for directed immersions
173
1
Directed immersions into almost symplectic manifolds
174
1
Directed immersions into almost complex manifolds
175
1
Directed embeddings
176
3
First Order Linear Differential Operators
179
10
Formal inverse of a linear differential operator
179
1
Homotopy principle for D-sections
180
1
Non-vanishing D-sections
181
1
Systems of linearly independent D-sections
182
2
Two-forms of maximal rank on odd-dimensional manifolds
184
2
One-forms of maximal rank on even-dimensional manifolds
186
3
Nash-Kuiper Theorem
189
10
Isometric immersions and short immersions
189
1
Nash-Kuiper theorem
190
1
Decomposition of a metric into a sum of primitive metrics
191
1
Approximation Theorem
191
2
One-dimensional Approximation Theorem
193
1
Adding a primitive metric
194
2
End of the proof of the approximation theorem
196
1
Proof of the Nash-Kuiper theorem
196
3
Bibliography
199
4
Index
203