Symplectization

In mathematics, the symplectization (or symplectification) 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

$S_{x}V=\{\beta \in T_{x}^{*}V-\{0\}\mid \ker \beta =\xi _{x}\}\subset T_{x}^{*}V$

of all nonzero 1-forms at $x$, which have the contact plane $\xi _{x}$ as their kernel. The union

$SV=\bigcup _{x\in V}S_{x}V\subset T^{*}V$

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

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:

$S_{x}^{+}V=\{\beta \in T_{x}^{*}V-\{0\}\,|\,\beta =\lambda \alpha ,\,\lambda >0\}\subset T_{x}^{*}V,$

$S^{+}V=\bigcup _{x\in V}S_{x}^{+}V\subset T^{*}V.$

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.