PlanetMath: contact manifold

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contact manifold

(Definition)

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

for each point $m \in M$ , $\alpha_{m} \neq 0$ and
the restriction $d\alpha_{m}\vert _{\ker{\alpha_{m}}}$ of the differential of $\alpha$ is nondegenerate.

Condition 1 ensures that $\xi=\ker{\alpha}$ is a subbundle of the vector bundle . Condition 2 equivalently says $d\alpha$ is a symplectic structure on the vector bundle $\xi \to M$ . A contact structure $\xi$ on a manifold is a subbundle of so that for each $m \in M$ , there is a contact form $\alpha$ defined on some neighborhood of so that $\xi = \ker{\alpha}$ . A co-oriented contact structure is a subbundle of of the form $\xi=\ker{\alpha}$ for some globally defined contact form $\alpha$ .

A (co-oriented) contact manifold is a pair $(M,\xi)$ where 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 , 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:

$\mathbb{R}^3$ is a contact manifold with the contact structure induced by the one form $\alpha = dz + xdy$ .
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 $\alpha = \cos{t\theta_1}+\sin{t\theta_2}$ .

"contact manifold" is owned by RevBobo.

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