Darboux’s theorem (symplectic geometry)

If $(M,\omega )$ is a $2n$-dimensional symplectic manifold, and $m\in M$, then there exists aneighborhood $U$ 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 theoremimplies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry,where there is curvature.