contact manifold

Let M be a smooth manifoldMathworldPlanetmath and α a one form on M. Then α is a contact form on M if

  1. 1.

    for each point mM, αm0 and

  2. 2.

    the restriction dαm|kerαm of the differentialMathworldPlanetmath of α is nondegenerate.

Condition 1 ensures that ξ=kerα is a subbundle of the vector bundle TM. Condition 2 equivalently says dα is a symplectic structure on the vector bundle ξM. A contact structure ξ on a manifold M is a subbundle of TM so that for each mM, there is a contact form α defined on some neighborhood of m so that ξ=kerα. A co-oriented contact structure is a subbundle of TM of the form ξ=kerα for some globally defined contact form α.

A (co-oriented) contact manifold is a pair (M,ξ) where M is a manifold and ξ is a (co-oriented) contact structure. Note, symplectic linear algebra implies that dimM is odd. If dimM=2n+1 for some positive integer n, then a one form α is a contact form if and only if α(dα)n is everywhere nonzero.

Suppose now that (M1,ξ1=kerα1) and (M2,ξ2=kerα2) are co-oriented contact manifolds. A diffeomorphism ϕ:M1M2 is called a contactomorphism if the pullback along ϕ of α2 differs from α1 by some positive smooth functionMathworldPlanetmath f:M1, that is, ϕ*α2=fα1.


  1. 1.

    3 is a contact manifold with the contact structure induced by the one form α=dz+xdy.

  2. 2.

    Denote by 𝕋2 the two-torus 𝕋2=S1×S1. Then, ×𝕋2 (with coordinates t,θ1,θ2) is a contact manifold with the contact structure induced by α=costθ1+sintθ2.

Title contact manifold
Canonical name ContactManifold
Date of creation 2013-03-22 13:43:27
Last modified on 2013-03-22 13:43:27
Owner RevBobo (4)
Last modified by RevBobo (4)
Numerical id 4
Author RevBobo (4)
Entry type Definition
Classification msc 53D10
Related topic SymplecticManifold
Defines contact structure
Defines contact form
Defines contactomorphism