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