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
: $S_xV = \ \subset T^*_xV$
of all nonzero 1-forms at $x$, which have the contact plane $\xi_x$ as their kernel. The union
:$SV = \bigcup_S_xV \subset T^*V$
is a symplectic submanifold of the cotangent bundle of $V$, and thus possesses a natural symplectic structure.

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

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:

:$S^+_xV = \ \subset T^*_xV,$

:$S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V.$

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.

Differential topology
Structures on manifolds
Symplectic geometry