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

Format:

Book

Author/Creator:

Eliashberg, Y., 1946-

Contributor:

Mishachev, N. (Nikolai M.), 1952-

Series:

Graduate studies in mathematics 1065-7339 ; v. 48.

Graduate studies in mathematics, 1065-7339 ; v. 48

Language:

English

Subjects (All):

Geometry, Differential.

Differentiable manifolds.

Differential equations--Numerical solutions.

Differential equations.

Physical Description:

xvii, 206 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

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[superscript (1)] 10

1.5. Holonomic sections of the jet space X[superscript (r)] 11

1.6. Geometric representation of sections of X[superscript (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

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

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 U-Invariant Differential Relations 139

15.1. U-invariant differential relations 139

15.2. Local h-principle for microflexible U-invariant relations 140

Chapter 16. Further Applications to Symplectic Geometry 143

16.1. Legendrian and isocontact immersions 143

16.2. Generalized isocontact immersions 144

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

17.2. 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[superscript (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

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.

Notes:

Includes bibliographical references (pages 199-202) and index.

ISBN:

0821832271

OCLC:

49312496