Symplectization

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

Definition

Let [math]\displaystyle{ (V,\xi) }[/math] be a contact manifold, and let [math]\displaystyle{ x \in V }[/math]. Consider the set

[math]\displaystyle{ S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV }[/math]

of all nonzero 1-forms at [math]\displaystyle{ x }[/math], which have the contact plane [math]\displaystyle{ \xi_x }[/math] as their kernel. The union

[math]\displaystyle{ SV = \bigcup_{x \in V}S_xV \subset T^*V }[/math]

is a symplectic submanifold of the cotangent bundle of [math]\displaystyle{ V }[/math], and thus possesses a natural symplectic structure.

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

The coorientable case

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

[math]\displaystyle{ S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda \gt 0\} \subset T^*_xV, }[/math]

[math]\displaystyle{ S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V. }[/math]

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