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
 * $$S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV$$

of all nonzero 1-forms at $$x$$, which have the contact plane $$\xi_x$$ as their kernel. The union
 * $$SV = \bigcup_{x \in V}S_xV \subset T^*V$$

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 $$\R^* \equiv \R - \{0\}$$.

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:


 * $$S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda > 0\} \subset T^*_xV,$$


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

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.