Editing Contact geometry (section)

{{Short description|Branch of geometry}}
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[[File:Standard contact structure.svg|thumb|right|The standard contact structure on '''R'''<sup>3</sup>. Each point in '''R'''<sup>3</sup> has a plane associated to it by the contact structure, in this case as the kernel of the one-form {{nowrap|1=d''z'' − ''y'' d''x''.}} These planes appear to twist along the ''y''-axis.

It is not integrable, as can be verified by drawing an infinitesimal square in the ''x''-''y'' plane, and follow the path along the one-forms. The path would not return to the same ''z''-coordinate after one circuit.]]

In [[mathematics]],  '''contact geometry''' is the study of a geometric structure on [[smooth manifold]]s given by a hyperplane [[distribution (differential geometry)|distribution]] in the [[tangent bundle]] satisfying a condition called 'complete non-integrability'.  Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for '[[integrable system|complete integrability]]' of a hyperplane distribution, i.e. that it be tangent to a codimension one [[foliation]] on the manifold, whose equivalence is the content of the [[Frobenius theorem (differential topology)|Frobenius theorem]].

Contact geometry is in many ways an odd-dimensional counterpart of [[symplectic geometry]], a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of [[classical mechanics]], where one can consider either the even-dimensional [[phase space]] of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.