Symplectization Explained

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.

Definition

Let

(V,\xi) be a contact manifold, and let x \in V . Consider the setS _{xV} = \{\beta \in T ^{*} _{xV} - \{ 0 \} \mid \ker \beta = \xi _{x\}} \subset T ^{*} _{xV} of all nonzero 1-forms at x , which have the contact plane \xi _{x} as their kernel. The unionSV = c up _{x} S _{xV} \subset T ^{*V} is a symplectic submanifold of the cotangent bundle of V , and thus possesses a natural symplectic structure.\pi : SV \to V supplies the symplectization with the structure of a principal bundle over V with structure group \R ^{*} \equiv \R - \{ 0 \} .

The coorientable case

\xi is cooriented by means of a contact form \alpha , there is another version of symplectization, in which only forms giving the same coorientation to \xi as \alpha are considered:S ^{+} _{xV} = \{\beta \in T ^{*} _{xV} - \{ 0 \} | \beta = λ \alpha, λ > 0 \} \subset T ^{*} _{xV,} S ^{+V} = c up _{x} S ^{+} _{xV} \subset T ^{*V.}

Note that

\xi is coorientable if and only if the bundle \pi : SV \to V is trivial. Any section of this bundle is a coorienting form for the contact structure.