Contact distribution symplectic form ($d\alpha {|}_{\xi }$)

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.