Symplectization

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

In mathematics, the symplectization (or symplectification) of a contact manifold is a symplectic manifold which naturally corresponds to it. S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV of all nonzero 1-forms at x , which have the contact plane \xi_x 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\} . 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 \xi 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.

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