Symplectization
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
of all nonzero 1-forms at, which have the contact plane as their kernel. The union
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:
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.
