Contactomorphisms

Contactomorphisms are diffeomorphisms between contact manifolds that preserve the contact structure. A contact manifold is a (2n+1)-dimensional manifold M equipped with a completely non-integrable hyperplane field ξ. When ξ is coorientable, it can be written as the kernel of a nowhere-vanishing 1-form α with α ∧ (dα)^n ≠ 0; α is called a contact form and ξ = ker α.

A diffeomorphism φ: M → N between contact manifolds (M, ξ) and (N, η) is a contactomorphism if its differential carries ξ onto η, i.e., dφ_p(ξ_p) = η_{φ(p)} for all p. Equivalently, φ preserves the contact structure, and if α and β are contact forms defining ξ and η, then there exists a nowhere-vanishing function h with φ*β = h β. If h > 0, φ is said to preserve the orientation of the contact structure.

The set Cont(M, ξ) of all contactomorphisms forms a subgroup of the diffeomorphism group and carries the structure of an infinite-dimensional Lie group under composition. Its Lie algebra consists of vector fields X whose flows are contactomorphisms, equivalently L_X α = f α for some function f on M.

Contactomorphisms extend the idea of symmetries in contact geometry and relate to the symplectization of (M, ξ), where φ lifts to a homogeneous symplectomorphism of the symplectization. A standard example is R^{2n+1} with α = dz − ∑ y_i dx_i, where translations in the z-direction are strict contactomorphisms.