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.
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
See also
A version for supergeometry is due to
A version for prequantized symplectic manifolds (prequantum bundles) is discussed in
A version for shifted symplectic structures (BV-brackets) on stacks is discussed in
Christopher Brav, Vittoria Bussi, Dominic Joyce, A ‘Darboux theorem’ for derived schemes with shifted symplectic structure (arXiv:1305.6302)
Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, Dominic Joyce, A ‘Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications (arXiv:1312.0090)
Last revised on November 17, 2017 at 13:15:27. See the history of this page for a list of all contributions to it.