Symplectization
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In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
[edit] Definition
Let (V,ξ) be a contact manifold, and let . Consider the set
of all nonzero 1-forms at x, which have the contact plane ξx as their kernel. The union
is a symplectic submanifold of the cotangent bundle of V, and thus possesses a natural symplectic structure.
The projection supplies the symplectization with the structure of a principal bundle over V with structure group .
[edit] The coorientable case
When the contact structure ξ is cooriented by means of a contact form α, there is another version of symplectization, in which only forms giving the same coorientation to ξ as α are considered:
Note that ξ is coorientable if and only if the bundle is trivial. Any section of this bundle is a coorienting form for the contact structure.