Contact distribution ($\xi =\ker \alpha$)

A contact form $\alpha$ defines the hyperplane distribution $\xi =\ker \alpha$. Multiplying $\alpha$ by a nowhere-zero function leaves this distribution unchanged. The contact distribution symplectic form distinguishes it from an integrable hyperplane distribution.

For a contact form on a $(2r+1)$-manifold, $d\alpha {|}_{\ker \alpha }$ is nondegenerate: the contact condition says its top exterior power is nowhere zero on the hyperplanes. This fiberwise symplectic form identifies contact-plane vectors with covectors on that plane and determines the horizontal generator of contact stability.