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 x \in V. Consider the set

S_xV = \{\beta \in T^*_xV - \{0\} \,|\, \ker \beta = \xi_x\} \subset T^*_xV

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

SV = \bigcup_{x \in V}S_xV \subset T^*V

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

The projection \pi : SV \to V supplies the symplectization with the structure of a principal bundle over V with structure group \R^* \equiv \R - \{0\}.

[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:

S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda > 0\} \subset T^*_xV,
S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V.

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