Symplectization

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Definition.
Let formula_1 be a contact manifold, and let formula_2. Consider the set
of all nonzero 1-forms at formula_4, which have the contact plane formula_5 as their kernel. The union
is a symplectic submanifold of the cotangent bundle of formula_7, and thus possesses a natural symplectic structure.
The projection formula_8 supplies the symplectization with the structure of a principal bundle over formula_7 with structure group formula_10.
The coorientable case.
When the contact structure formula_11 is cooriented by means of a contact form formula_12, there is another version of symplectization, in which only forms giving the same coorientation to formula_11 as formula_12 are considered:
Note that formula_11 is coorientable if and only if the bundle formula_8 is trivial. Any section of this bundle is a coorienting form for the contact structure.