Symplectic Vector Spaces

Abstract

This chapter is a brief introduction to symplectic manifolds. We will start this chapter by defining a symplectic vector space (Sect. 1.1). After briefly reviewing the notion of an almost complex structure on a vector space, we will see how the compatibility condition between the symplectic form and an almost complex structure gives rise to an inner product. In Sect. 1.3, we will discuss the definition of symplectic manifolds, describe some of their basic properties and will finally see some examples in Sect. 1.4. Section 1.2 contains a review of results from differential topology which are essential material for what follows.