Jump to content

Symplectization

From Wikipedia, the free encyclopedia

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

Definition

[edit]

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

[edit]

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.