Editing Contact geometry (section)

===Examples===
As a prime example, consider '''R'''<sup>3</sup>, endowed with coordinates (''x'',''y'',''z'') and the one-form {{nowrap|1=''dz'' − ''y'' ''dx''.}} The contact plane ''ξ'' at a point (''x'',''y'',''z'') is spanned by the vectors {{nowrap|1=''X''<sub>1</sub> = <big>∂</big><sub>''y''</sub>}} and {{nowrap|1=''X''<sub>2</sub> = <big>∂</big><sub>''x''</sub> + ''y'' <big>∂</big><sub>''z''</sub>.}}

<!--- Presumably, a challenge to editors to insert a picture of a contact manifold?
(Draw a picture of this!).
--->
By replacing the single variables ''x'' and ''y'' with the multivariables ''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>, ''y''<sub>1</sub>,&nbsp;...,&nbsp;''y''<sub>''n''</sub>, one can generalize this example to any '''R'''<sup>2''n''+1</sup>. By a [[Darboux theorem|theorem of Darboux]], every contact structure on a manifold looks locally like this particular contact structure on the (2''n''&nbsp;+&nbsp;1)-dimensional vector space.

The [[Sasakian manifold|Sasakian manifolds]] comprise an important class of contact manifolds.

Every [[Connected space|connected]] [[Compact space|compact]] [[Orientability|orientable]] three-dimensional manifold admits a contact structure.<ref>{{Cite journal |last=Martinet |first=J. |date=1971 |editor-last=Wall |editor-first=C. T. C. |title=Formes de Contact sur les Variétés de Dimension 3 |url=https://link.springer.com/chapter/10.1007/BFb0068901 |journal=Proceedings of Liverpool Singularities Symposium II |language=fr |location=Berlin, Heidelberg |publisher=Springer |pages=142–163 |doi=10.1007/BFb0068901 |isbn=978-3-540-36868-7|url-access=subscription }}</ref> This result generalises to any compact [[almost-contact manifold]].<ref>{{Cite journal |last=Borman |first=Matthew Strom |last2=Eliashberg |first2=Yakov |last3=Murphy |first3=Emmy |date=2015 |title=Existence and classification of overtwisted contact structures in all dimensions |url=https://projecteuclid.org/journals/acta-mathematica/volume-215/issue-2/Existence-and-classification-of-overtwisted-contact-structures-in-all-dimensions/10.1007/s11511-016-0134-4.full |journal=Acta Mathematica |volume=215 |issue=2 |pages=281–361 |doi=10.1007/s11511-016-0134-4 |issn=0001-5962|arxiv=1404.6157 }}</ref>