Chapter 3 Symplectic geometry

Abstract

This chapter discusses symplectic geometry—the geometry of symplectic manifolds . From a language for classical mechanics in the XVIII century, symplectic geometry has matured since the 1960s to a rich and central branch of differential geometry and topology. The chapter describes symplectic manifolds, Lagrangian submanifolds, complex structures or almost complex structures abound in symplectic geometry, symplectic geography, Hamiltonian geometry, and symplectic reduction. Symplectic manifolds are manifolds equipped with symplectic forms. A symplectic form is a closed nondegenerate 2-form. Basic properties, major classical examples, equivalence notions, local normal forms of symplectic manifolds and symplectic submanifolds are discussed in the chapter.