# 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. 1.

for each point $m\in M$, $\alpha_{m}\neq 0$ and

2. 2.

the restriction $d\alpha_{m}|_{\ker{\alpha_{m}}}$ of the differential of $\alpha$ is nondegenerate.

Condition 1 ensures that $\xi=\ker{\alpha}$ 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 $\xi=\ker{\alpha}$. A co-oriented contact structure is a subbundle of $TM$ of the form $\xi=\ker{\alpha}$ for some globally defined contact form $\alpha$.

A (co-oriented) contact manifold is a pair $(M,\xi)$ where $M$ is a manifold and $\xi$ is a (co-oriented) contact structure. Note, symplectic linear algebra implies that $\dim{M}$ is odd. If $\dim{M}=2n+1$ 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 $(M_{1},\xi_{1}=\ker{\alpha_{1}})$ and $(M_{2},\xi_{2}=\ker{\alpha_{2}})$ are co-oriented contact manifolds. A diffeomorphism $\phi:M_{1}\to M_{2}$ is called a contactomorphism if the pullback along $\phi$ of $\alpha_{2}$ differs from $\alpha_{1}$ by some positive smooth function $f:M_{1}\to\mathbb{R}$, that is, $\phi^{*}\alpha_{2}=f\alpha_{1}$.

Examples:

1. 1.

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

2. 2.

Denote by $\mathbb{T}^{2}$ the two-torus $\mathbb{T}^{2}=S^{1}\times S^{1}$. Then, $\mathbb{R}\times\mathbb{T}^{2}$ (with coordinates $t,\theta_{1},\theta_{2}$) is a contact manifold with the contact structure induced by $\alpha=\cos{t\theta_{1}}+\sin{t\theta_{2}}$.

Title contact manifold ContactManifold 2013-03-22 13:43:27 2013-03-22 13:43:27 RevBobo (4) RevBobo (4) 4 RevBobo (4) Definition msc 53D10 SymplecticManifold contact structure contact form contactomorphism