Darboux’s theorem (symplectic geometry)

If $(M,\omega )$ is a $2n$-dimensional symplectic manifold, and $m\in M$, then there exists a neighborhood $U$ of $m$ with a coordinate chart

$$x=(x_1,\mathrm{\dots },x_{2n}):U\to {ℝ}^{2n},$$

such that

$$\omega =\sum _{i=1}^{n}d{x}_i\wedge d{x}_{n+i}.$$

These are called canonical or Darboux coordinates. On $U$, $\omega$ is the pullback by $X$ of the standard symplectic form on ${ℝ}^{2n}$, so $x$ is a symplectomorphism. Darboux’s theorem implies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry, where there is curvature.

Title	Darboux’s theorem (symplectic geometry)
Canonical name	DarbouxsTheoremsymplecticGeometry
Date of creation	2013-03-22 13:15:31
Last modified on	2013-03-22 13:15:31
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Theorem
Classification	msc 53D05
Synonym	Darboux coordinates