Symplectization

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

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.