Symplectic Manifolds

Abstract

We say that a smooth manifold M is a symplectic manifold if it is given together with a 2-form ω ∈ Ω ²(M) which is

$$\displaystyle{\begin{array}{rrl} (i)& \ \mathrm{Closed}:&d\omega = 0,\\ (\mathit{ii } )&\ \mathrm{Non\ degenerate:}&\langle u \wedge v,\omega \rangle = 0\ \ \forall v\ \Rightarrow \ u = 0, \\ \end{array} }$$

we denote with $\langle v_{1} \wedge \ldots \wedge v_{k},f\rangle$ the evaluation of a k-form f on k vectors v ₁, …, v _k .

The name complex group formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word complex in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective symplectic.

Hermann Weyl, The classical Groups