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

 $x=(x_{1},\ldots,x_{2n}):U\to\mathbb{R}^{2n},$

such that

 $\omega=\sum_{i=1}^{n}dx_{i}\wedge dx_{n+i}.$

These are called canonical or Darboux coordinates. On $U$, $\omega$ is the pullback by $X$ of the standard symplectic form on $\mathbb{R}^{2n}$, 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) DarbouxsTheoremsymplecticGeometry 2013-03-22 13:15:31 2013-03-22 13:15:31 bwebste (988) bwebste (988) 6 bwebste (988) Theorem msc 53D05 Darboux coordinates