Editing Differential geometry (section)

===Contact geometry===
{{main|Contact geometry}}

[[Contact geometry]] deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A ''contact structure'' on a {{nowrap|(2''n'' + 1)}}-dimensional manifold ''M'' is given by a smooth hyperplane field ''H'' in the [[tangent bundle]] that is as far as possible from being associated with the level sets of a differentiable function on ''M'' (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point ''p'', a hyperplane distribution is determined by a nowhere vanishing [[Differential form|1-form]] <math>\alpha</math>, which is unique up to multiplication by a nowhere vanishing function:

: <math> H_p = \ker\alpha_p\subset T_{p}M.</math>

A local 1-form on ''M'' is a ''contact form'' if the restriction of its [[exterior derivative]] to ''H'' is a non-degenerate two-form and thus induces a symplectic structure on ''H''<sub>''p''</sub> at each point. If the distribution ''H'' can be defined by a global one-form <math>\alpha</math> then this form is contact if and only if the top-dimensional form

: <math>\alpha\wedge (d\alpha)^n</math>

is a [[volume form]] on ''M'', i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.