Symplectization

{{Unreferenced|date=December 2020}} In mathematics, the '''symplectization''' (or '''symplectification''') of a contact manifold is a symplectic manifold which naturally corresponds to it.

== Definition ==

Let <math>(V,\xi)</math> be a contact manifold, and let <math>x \in V</math>. Consider the set : <math>S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV</math> of all nonzero 1-forms at <math>x</math>, which have the contact plane <math>\xi_x</math> as their kernel. The union :<math>SV = \bigcup_{x \in V}S_xV \subset T^*V</math> is a symplectic submanifold of the cotangent bundle of <math>V</math>, and thus possesses a natural symplectic structure.

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

== The coorientable case ==

When the contact structure <math>\xi</math> is cooriented by means of a contact form <math>\alpha</math>, there is another version of symplectization, in which only forms giving the same coorientation to <math>\xi</math> as <math>\alpha</math> are considered:

:<math>S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda > 0\} \subset T^*_xV,</math>

:<math>S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V.</math>

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

Category:Differential topology Category:Structures on manifolds Category:Symplectic geometry