contact manifold
Let $M$ be a smooth manifold^{} and $\alpha $ a one form on $M$. Then $\alpha $ is a contact form on $M$ if

1.
for each point $m\in M$, ${\alpha}_{m}\ne 0$ and

2.
the restriction ${d{\alpha}_{m}}_{\mathrm{ker}{\alpha}_{m}}$ of the differential^{} of $\alpha $ is nondegenerate.
Condition 1 ensures that $\xi =\mathrm{ker}\alpha $ is a subbundle of the vector bundle $TM$. Condition 2 equivalently says $d\alpha $ is a symplectic structure on the vector bundle $\xi \to M$. A contact structure $\xi $ on a manifold $M$ is a subbundle of $TM$ so that for each $m\in M$, there is a contact form $\alpha $ defined on some neighborhood of $m$ so that $\xi =\mathrm{ker}\alpha $. A cooriented contact structure is a subbundle of $TM$ of the form $\xi =\mathrm{ker}\alpha $ for some globally defined contact form $\alpha $.
A (cooriented) contact manifold is a pair $(M,\xi )$ where $M$ is a manifold and $\xi $ is a (cooriented) contact structure. Note, symplectic linear algebra implies that $dimM$ is odd. If $dimM=2n+1$ for some positive integer $n$, then a one form $\alpha $ is a contact form if and only if $\alpha \wedge {(d\alpha )}^{n}$ is everywhere nonzero.
Suppose now that $({M}_{1},{\xi}_{1}=\mathrm{ker}{\alpha}_{1})$ and $({M}_{2},{\xi}_{2}=\mathrm{ker}{\alpha}_{2})$ are cooriented contact manifolds. A diffeomorphism $\varphi :{M}_{1}\to {M}_{2}$ is called a contactomorphism if the pullback along $\varphi $ of ${\alpha}_{2}$ differs from ${\alpha}_{1}$ by some positive smooth function^{} $f:{M}_{1}\to \mathbb{R}$, that is, ${\varphi}^{*}{\alpha}_{2}=f{\alpha}_{1}$.
Examples:

1.
${\mathbb{R}}^{3}$ is a contact manifold with the contact structure induced by the one form $\alpha =dz+xdy$.

2.
Denote by ${\mathbb{T}}^{2}$ the twotorus ${\mathbb{T}}^{2}={S}^{1}\times {S}^{1}$. Then, $\mathbb{R}\times {\mathbb{T}}^{2}$ (with coordinates $t,{\theta}_{1},{\theta}_{2}$) is a contact manifold with the contact structure induced by $\alpha =\mathrm{cos}t{\theta}_{1}+\mathrm{sin}t{\theta}_{2}$.
Title  contact manifold 

Canonical name  ContactManifold 
Date of creation  20130322 13:43:27 
Last modified on  20130322 13:43:27 
Owner  RevBobo (4) 
Last modified by  RevBobo (4) 
Numerical id  4 
Author  RevBobo (4) 
Entry type  Definition 
Classification  msc 53D10 
Related topic  SymplecticManifold 
Defines  contact structure 
Defines  contact form 
Defines  contactomorphism 