Symplectic and Contact Geometry - A Concise Introduction

Preface
Contents
1 Symplectic Linear Algebra
	1.1 Symplectic Forms
	1.2 Annihilator of a Linear Subspace
	1.3 Orthogonal Complement of a Linear Subspace
	1.4 Isotropic Subspaces
	1.5 Symplectic Basis
	1.6 The Lagrangian Grassmannian
	1.7 The Symplectic Linear Group and Its Lie Algebra
	1.8 Hermitian Forms
	1.9 Exterior Algebra
2 Symplectic Manifolds
	2.1 Definition of Symplectic Manifold
	2.2 Examples and Constructions of Symplectic Manifolds
		2.2.1 The Cotangent Bundle
		2.2.2 Reduced Symplectic Manifold
		2.2.3 Complex Projective Manifolds
	2.3 Almost Complex Structure
	2.4 Cohomology Classes
	2.5 Symplectomorphisms
	2.6 Lagrangian Submanifolds
	2.7 Geodesic Flow
	2.8 Local Theorems
	2.9 Symplectic Invariants
3 Hamiltonian Systems
	3.1 Invariant Form Under the Flow of a Vector Field
	3.2 Hamiltonian Vector Fields
	3.3 The Legendre Transform
	3.4 Classical Mechanics
	3.5 Poisson Brackets
	3.6 Darboux's Theorem
	3.7 Poisson Structures
	3.8 Hamiltonian Group Actions
	3.9 Symplectic Quotients
4 Contact Manifolds
	4.1 Definition of Contact Manifold
	4.2 Examples and Constructions
		4.2.1 Hypersurfaces in Symplectic Manifolds
		4.2.2 Contactization
	4.3 Contactomorphisms
	4.4 Complex Structure
	4.5 Darboux's Theorem
	4.6 Legendrian Submanifolds
	4.7 Geometric Optic
	4.8 Characteristic Foliation
	4.9 Legendrian Knots in Contact 3-Manifolds
	4.10 Contact Surgery
	4.11 Contact Invariants
	4.12 Legendrian Surgery Exact Sequences
References
Index