Editing Darboux's theorem (section)

== Darboux's theorem for symplectic manifolds ==
Suppose that <math>\omega </math> is a [[Symplectic form|symplectic 2-form]] on an <math>n=2m </math>-dimensional manifold ''<math>M </math>''. In a neighborhood of each point ''<math>p </math>'' of ''<math>M </math>'', by the [[Poincaré lemma]], there is a 1-form <math>\theta </math> with <math>\mathrm{d} \theta = \omega</math>. Moreover, <math>\theta </math> satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a [[coordinate chart]] ''<math>U </math>'' near ''<math>p </math>'' in which<math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_m\,\mathrm{d}y_m. </math>

Taking an [[exterior derivative]] now shows
: <math display="block"> \omega = \mathrm{d} \theta = \mathrm{d}x_1 \wedge \mathrm{d}y_1 + \ldots + \mathrm{d}x_m \wedge \mathrm{d}y_m.</math>
The chart ''<math>U </math>'' is said to be a '''Darboux chart''' around ''<math>p </math>''.<ref name=":32">{{Cite book |last=McDuff |first=Dusa |url=https://academic.oup.com/book/43512 |title=Introduction to Symplectic Topology |last2=Salamon |first2=Dietmar |date=2017-06-22 |publisher=[[Oxford University Press]] |isbn=978-0-19-879489-9 |volume=1 |language=en |doi=10.1093/oso/9780198794899.001.0001 |author-link=Dusa McDuff |author-link2=Dietmar Salamon}}</ref> The manifold ''<math>M </math>'' can be [[cover (topology)|covered]] by such charts.

To state this differently, identify <math>\mathbb{R}^{2m}</math> with <math>\mathbb{C}^{m}</math> by letting <math>z_j=x_j+\textit{i}\,y_j</math>. If <math>\varphi: U \to \mathbb{C}^n</math> is a Darboux chart, then <math> \omega </math> can be written as the [[pullback (differential geometry)|pullback]] of the standard symplectic form <math>\omega_0</math> on <math>\mathbb{C}^{n}</math>:
:<math>\omega = \varphi^{*}\omega_0.\,</math>
A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using [[Moser's trick]].<ref name=":32" /><ref name=":02">{{Cite book |last=Cannas Silva |first=Ana |url=https://link.springer.com/book/10.1007/978-3-540-45330-7 |title=Lectures on Symplectic Geometry |publisher=[[Springer Science+Business Media|Springer]] |year=2008 |isbn=978-3-540-42195-5 |language=en |doi=10.1007/978-3-540-45330-7 |author-link=Ana Cannas da Silva}}</ref>

=== Comparison with Riemannian geometry ===
Darboux's theorem for symplectic manifolds implies that there are no local invariants in symplectic geometry: a [[Darboux frame|Darboux basis]] can always be taken, valid near any given point. This is in marked contrast to the situation in [[Riemannian geometry]] where the [[curvature of Riemannian manifolds|curvature]] is a local invariant, an obstruction to the [[metric tensor|metric]] being locally a sum of squares of coordinate differentials.

The difference is that Darboux's theorem states that <math> \omega </math> can be made to take the standard form in an ''entire neighborhood'' around ''<math>p </math>''. In Riemannian geometry, the metric can always be made to take the standard form ''at'' any given point, but not always in a neighborhood around that point.

=== Darboux's theorem for contact manifolds ===
Another particular case is recovered when <math> n=2p+1 </math>; if <math> \theta \wedge \left( \mathrm{d} \theta \right)^p \ne 0 </math> everywhere, then <math> \theta </math> is a [[Contact geometry#Contact forms and structures|contact form]]. A simpler proof can be given, as in the case of symplectic structures, by using [[Moser's trick]].<ref>{{Cite book |last=Geiges |first=Hansjörg |url=https://www.cambridge.org/core/books/an-introduction-to-contact-topology/F851B2A2E7E78C6B9967A18A6641B40C |title=An Introduction to Contact Topology |date=2008 |publisher=[[Cambridge University Press]] |isbn=978-0-521-86585-2 |series=Cambridge Studies in Advanced Mathematics |location=Cambridge |pages=67-68 |doi=10.1017/cbo9780511611438}}</ref>