Contact manifold

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This article describes an additional structure on a differential manifold
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Definition

A contact manifold is the following data:

  • A differential manifold M of dimension (2n + 1)
  • A hyperplane field (viz a smooth association of a tangent hyperplane at every point) that occurs as the kernel of a smooth 1-form \alpha on M satisfying:

\alpha \wedge (d\alpha)^n \ne 0

Note that \alpha is uniquely determined upto a scalar multiple.

Terminology

  • A smooth 1-form \alpha described as above is termed a contact form
  • A hyperplane field described as above is termed a contact field
  • The overall datum is termed a contact structure