A Brief Introduction To Symplectic And Contact Manifolds

by Augustin Banyaga, Djideme F Houenou

Overview

The book introduces the basic notions in Symplectic and Contact Geometry at the level of the second year graduate student. It also contains many exercises, some of which are solved only in the last chapter.We begin with the linear theory, then give the definition of symplectic manifolds and some basic examples, review advanced calculus, discuss Hamiltonian systems, tour rapidly group and the basics of contact geometry, and solve problems in chapter 8. The material just described can be used as a one semester course on Symplectic and Contact Geometry.The book contains also more advanced material, suitable to advanced graduate students and researchers.

Product Details

ISBN-13:	9789814696708
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/03/2016
Series:	Nankai Tracts In Mathematics , #15
Pages:	180
Product dimensions:	5.90(w) x 9.10(h) x 0.80(d)

Table of Contents

Introduction v

1 Symplectic vector spaces 1

1.1 Bilinear forms 1

1.2 Basis 1

1.3 Immediate consequence of Theorem 1.2 8

1.4 Another consequence of Theorem 1.2 8

1.5 Compatible complex structures 9

1.6 The symplectic group 11

2 Symplectic manifolds 13

2.1 Examples of symplectic manifolds 14

2.1.1 Euclidean spaces 14

2.1.2 Tori 14

2.1.3 Oriented surfaces 14

2.1.4 Product of symplectie manifolds 15

2.1.5 Cotangent bundles 15

2.2 The cohomology class of a symplectic form (see Appendix A) 16

2.3 Moser path method 17

2.3.1 Proof of Darboux-Weinstein Theorem 18

2.4 Symplectomorphisms 20

2.4.1 A general method for constructing symplectomorphisms 21

2.4.2 The Calabi homomorphism [Ban78], [Ban97] 25

2.5 Lagrangian submanifolds 27

2.5.1 Examples 27

2.6 Compatible almost complex structures 30

2.7 Almost Kaehler structures 31

3 Hamiltonian systems and Poisson algebra 33

3.1 Hamiltonian systems 33

3.2 A characterisation of symplectic diffeomorphisms 36

3.3 The Poisson bracket 37

3.4 Integrable Hamiltonian systems 41

3.5 Hamiltonian diffeomorphisms 43

3.6 Poisson manifolds 45

4 Group actions 47

4.1 Basic definitions 47

4.2 Examples 49

4.2.1 Examples of Lie group 49

4.2.2 Examples of group actions 49

4.3 Symplectic reduction 51

4.4 Convexity theorem 53

5 Contact manifolds 55

5.1 Examples 55

5.1.1 Basic examples 55

5.1.2 More examples 57

5.2 Relation, with, symplectic manifolds 59

5.2.1 Contactization of symplectic manifold 59

5.2.2 Symplectization SP of a contact manifold (P, α) 60

5.2.3 Hypersurfaces of contact type in a symplectic manifold 60

5.3 The Reeb field of a contact form 61

5.3.1 Contact dynamics 63

5.3.2 The Weinstein's conjecture 63

5.3.3 Regular contact flows 64

5.4 Contact structures 68

5.5 Two basic theorems 70

5.6 Contactomorphisms 73

5.6.1 Applications 76

5.6.2 Some properties of the group of contactomorphisms 77

5.7 Contact metric structures 78

6 Solutions of selected exercises 82

7 Epilogue: The C⁰-symplectic and contact topology 92

7.1 The Hofer norm [Hof90] 93

7.2 Contact rigidity 99

A Review of calculus on manifolds 102

A.1 Differential forms and de Rham cohomology 102

A.2 Hodge-de Rham decomposition theorem [War71] 105

B Complete integrability in contact geometry 107

B.1 Introduction 107

B.2 Complete integrability in symplectic geometry 112

B.2.1 The classical Arnold-Liouville theorem [1], [23] 112

B.2.2 A unified theory including both global and singular properties 113

B.3 Contact angle-action coordinates 114

B.3.1 Contact geometry preliminaries [7], [8], [23] 114

B.3.2 The regular Case 117

B.3.3 The singular case 123

B.4 The manifold of invariant Tori 133

B.4.1 Admissible change of action coordinates 133

B.4.2 Legendre lattices 134

B.4.3 The Chern class of the singular fibration π: P → W 138

B.4.4 The classification theorem 143

B.5 Global Tⁿ⁺¹ actions 148

B.5.1 The convexity and realization theorems 148

B.5.2 Miscellaneous and applications to K-contact structures 153

Bibliography 159

Index 165