In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Definition
Let
be a contact manifold, and let
. Consider the set
![{\displaystyle S_{x}V=\{\beta \in T_{x}^{*}V-\{0\}\mid \ker \beta =\xi _{x}\}\subset T_{x}^{*}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d4cd31abeff6d28240a329c323c326fae2d98)
of all nonzero 1-forms at
, which have the contact plane
as their kernel. The union
![{\displaystyle SV=\bigcup _{x\in V}S_{x}V\subset T^{*}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/618c9592302781c3b1c4da5f815d571ac0ce1457)
is a symplectic submanifold of the cotangent bundle of
, and thus possesses a natural symplectic structure.
The projection
supplies the symplectization with the structure of a principal bundle over
with structure group
.
The coorientable case
When the contact structure
is cooriented by means of a contact form
, there is another version of symplectization, in which only forms giving the same coorientation to
as
are considered:
![{\displaystyle S_{x}^{+}V=\{\beta \in T_{x}^{*}V-\{0\}\,|\,\beta =\lambda \alpha ,\,\lambda >0\}\subset T_{x}^{*}V,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee14bed83f5591c20e92eb3554f5715d887036f)
![{\displaystyle S^{+}V=\bigcup _{x\in V}S_{x}^{+}V\subset T^{*}V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88c0378e01f73c141bfff18cd699980578f18317)
Note that
is coorientable if and only if the bundle
is trivial. Any section of this bundle is a coorienting form for the contact structure.