The purpose of this post is to discuss a few basic facts about differentiable manifolds and state the Darboux theorem, which I will prove next time. (People who are looking for a more ambitious leap into symplectic geometry might want to try lewallen’s two posts over at Concrete Nonsense.)

A **symplectic manifold** is a smooth manifold equipped with a *closed* symplectic 2-form . In other words, is alternating and nondegenerate on each tangent space .

The basic example of a symplectic form is

on with coordinates .

This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle of any manifold into a symplectic manifold. First, we define a 1-form on . Let be the projection downwards. Given lying above , define

To make this clearer, here is an interpretation in local coordinates. Let be local coordinates for . Then coordinates for . Then

as is easily checked by working through the definitions. So we can define a **canonical 2-form** as ; this makes into a symplectic manifold. (more…)