# nLab Darboux's theorem

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

Darboux’s theorem states that a smooth manifold $X$ equipped with a differential 1-form $\theta$ which is sufficiently non-degenerate admits local coordinate charts $\phi_i \colon\mathbb{R}^{2n} \to X$ on which $\theta$ takes the canonical form $\phi_i^\ast \theta = \sum_{k} x^{2k} \mathbf{d} x^{2k+1}$.

In particular

• for $(X, \omega)$ a symplectic manifold there are local charts in which the symplectic form $\omega$ takes the form $\phi_i^\ast \omega = \sum_{k} \mathbf{d} x^{2k} \wedge \mathbf{d} x^{2k+1}$;

• similarly, for contact manifolds there are adapated local charts.

(e.g. Arnold 78, p. 362)

For the case of symplectic manifolds the Darboux theorem may also be read as saying that a G-structure for $G = Sp(2n)$ the symplectic group (hence an almost symplectic structure) is an integrable G-structure already when it is first-order integrable, i.e. torsion-free, i.e. symplectic. See at integrability of G-structures – Examples – Symplectic structure.

## References

Lecture notes include

• Andreas Čap, section 1.8 of Differential Geometry 2, 2011/2012 (pdf)

• Federica Pasquotto, Linear $G$-structures by example (pdf)

Textbook accounts include