contact manifold

Let $M$ be a smooth manifold and $\alpha$ a one form on $M$. Then $\alpha$ is acontact form on $M$ if

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

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 $dimM$ 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 $(M_1,{\xi }_1=\ker {\alpha }_1)$ and $(M_2,{\xi }_2=\ker {\alpha }_2)$ are co-oriented 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 $f:M_1\to ℝ$, that is, ${\varphi }^{*}{\alpha }_2=f{\alpha }_1$.

Examples:

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

2. Denote by ${𝕋}^2$ the two-torus ${𝕋}^2=S^1\times S^1$. Then, $ℝ\times {𝕋}^2$ (with coordinates $t,{\theta }_1,{\theta }_2$) is a contact manifold with the contact structure induced by$\alpha =\cos t{\theta }_1+\sin t{\theta }_2$.