Contact manifold

This article describes an additional structure on a differential manifold
View other additional structures

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:

$\alpha \wedge (d\alpha )^{n}\neq 0$

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