Symplectization

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

Definition

Let ${\displaystyle (V,\xi )}$ be a contact manifold, and let ${\displaystyle x\in V}$. Consider the set

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

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

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

is a symplectic submanifold of the cotangent bundle of ${\displaystyle V}$, and thus possesses a natural symplectic structure.

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

The coorientable case

When the contact structure ${\displaystyle \xi }$ is cooriented by means of a contact form ${\displaystyle \alpha }$, there is another version of symplectization, in which only forms giving the same coorientation to ${\displaystyle \xi }$ as ${\displaystyle \alpha }$ are considered:

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

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

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