Symplectization

Symplectization is a standard construction in contact and symplectic geometry that associates a symplectic manifold to a given contact manifold. If (M, ξ) is a (2n−1)-dimensional contact manifold and α is a contact form with ξ = ker α, then the symplectization is the product M × R equipped with the symplectic form ω = d(e^t α), where t ∈ R is the coordinate on the extra factor. The choice of α is not unique, but different choices yield symplectomorphic results that depend only on the contact structure ξ.

Equivalently, the symplectization carries the Liouville form λ = e^t α, so ω = dλ. The Liouville vector field Z is defined by i_Z ω = λ, and in this standard model Z = ∂/∂t. The hyperplanes ξ on M are recovered as ker α, and each slice M × {t} is naturally a copy of M with the same underlying contact structure, with the contact form e^t α on that slice.

A common variant uses the positive ray M × R_+ with ω = d(r α) for r > 0, which is isomorphic to the previous formulation by the change of variables r = e^t. The symplectization is an exact, complete Liouville manifold and serves as a canonical way to realize a contact manifold as a hypersurface of a symplectic manifold. It provides a natural setting for studying holomorphic curves and invariants in symplectic field theory, where the symplectization plays the role of a noncompact ambient space in which contact dynamics are analyzed.