Contact Geometry - Contact Forms and Structures

Contact Forms and Structures

Given an n-dimensional smooth manifold M, and a point p ∈ M, a contact element of M with contact point p is an (n − 1)-dimensional linear subspace of the tangent space to M at p. A contact element can be given by the zeros of a 1-form on the tangent space to M at p. However, if a contact element is given by the zeros of a 1-form ω, then it will also be given by the zeros of λω where λ ≠ 0. Thus, { λω : λ ≠ 0 } all give the same contact element. It follows that the space of all contact elements of M can be identified with a quotient of the cotangent bundle T*M, namely:

A contact structure on an odd dimensional manifold M, of dimension 2k+1, is a smooth distribution of contact elements, denoted by ξ, which is generic at each point. The genericity condition is that ξ is non-integrable.

Assume that we have a smooth distribution of contact elements, ξ, given locally by a differential 1-form α; i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly as:

Notice that if ξ is given by the differential 1-form α, then the same distribution is given locally by β = ƒ⋅α, where ƒ is a non-zero smooth function. If ξ is co-orientable then α is defined globally.