Darboux thoerem

Importance

Darboux’s theorem shows the stark contrast between a symplectic manifold and a Riemannian manifold. A symplectic manifold has no interesting local invariants up to symplectomorphism (the only one is dimension).

Said another way, every symplectic manifold is locally symplectomorphic to .

Statement

Let be a -dimensional symplectic manifold and let . There is a coordinate chart centered at such that on

A chart as above is called a Darboux chart.

Proof