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

Definition

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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

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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.