Brief Introduction To Symplectic And Contact Manifolds, A

Augustin Banyaga, Djideme F Houenou

• Publisher: WSPC
• Year: 2016
• eBook ISBN: 9789814696722
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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 g

Chapter 1

Symplectic vector spaces

1.1Bilinear forms

A bilinear form on a (real) vector space V is a map b : V × V

R which is linear in each variable, i.e:

and analogously in v.

A bilinear form is symmetric if b(u, v) = b(v, u) and antisymmetric if

A bilinear form b : V × V

R determines a linear map

: V

V* (where V* is the dual of V, i.e. the space of linear maps V

R), by

The rank of b is the dimension of the image of

. The bilinear form b is said to be non-degenerate if

is an isomorphism.

Definition 1.1

A symplectic form on a vector space V is a non-degenerate antisymmetric bilinear form ω : V × V

R.

The couple (V, ω) of a vector space and a symplectic form on V is called a symplectic vector space.

1.2Basis

If we fix a basis E = (e₁, · · · , e_2n) of the vector space V, then any bilinear form b : V × V

R can be represented by a matrix M_b = (α_ij) where α_ij = b(e_i, e_j).

If b is symmetric, then M_b is a symmetric matrix, i.e ^tM_b = M_b (where ^tM_b = (α_ji) stands for the transpose of M_b).

If b is antisymmetric, then M_b is a skew-symmetric matrix, i.e ^tM_b = −M_b or α_ji = −α_ij, ∀ i ≠ j.

The bilinear form b is non-degenerate if and only if M_b is an invertible matrix, i.e. the determinant of M_b is different of zero (det M_b ≠ 0).

Hence the matrix M_ω of a symplectic form satisfies:

1.^tM_ω = −M_ω

2.det M_ω ≠ 0.

Proposition 1...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Copyright
• Introduction
• Contents
• 1 Symplectic vector spaces
• 2 Symplectic manifolds
• 3 Hamiltonian systems and Poisson algebra
• 4 Group actions
• 5 Contact manifolds
• 6 Solutions of selected exercises
• 7 Epilogue: The C0-symplectic and contact topology
• A Review of calculus on manifolds
• B Complete integrability in contact geometry
• Bibliography
• Index