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