Symplectization

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 set:of all nonzero 1-forms at $x$, which have the contact plane $xi\_x$ as their kernel. The union:is a symplectic submanifold of the cotangent bundle of $V$, and thus possesses a natural symplectic structure.

The projection $pi\; :\; SV\; o\; V$ supplies the symplectization with the structure of a principal bundle over $V$ with structure group $R^*\; equiv\; R\; -\; \{0\}$.

The coorientable case

When the contact structure $xi$ 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:

:

:

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