3 - Symplectic Geometry

Summary

This chapter is a crash course on symplectic geometry. We first introduce symplectic manifolds and their Lagrangian submanifolds and give some examples. Then we discuss the Moser method [Mos-65], and use it to prove results about the existence of local normal forms and of standard tubular neighbourhoods. This is of course just the beginning of the vast area of symplectic geometry. A more extensive account is contained, for example, in the book of McDuff-Salamon [McS-95]; see especially Chapter 3 of that book. Other useful references for this material include the writings of Weinstein [Wei-71, Wei-77].