Trisha Shetty (Editor)

Symplectization

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

Contents

Definition

Let ( V , ξ ) be a contact manifold, and let x V . Consider the set

S x V = { β T x V { 0 } ker β = ξ x } T x V

of all nonzero 1-forms at x , which have the contact plane ξ x as their kernel. The union

S V = x V S x V T V

is a symplectic submanifold of the cotangent bundle of V , and thus possesses a natural symplectic structure.

The projection π : S V V supplies the symplectization with the structure of a principal bundle over V with structure group R R { 0 } .

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:

S x + V = { β T x V { 0 } | β = λ α , λ > 0 } T x V , S + V = x V S x + V T V .

Note that ξ is coorientable if and only if the bundle π : S V V is trivial. Any section of this bundle is a coorienting form for the contact structure.

References

Symplectization Wikipedia


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