A contact structure on a
(2n + 1)-dimensional manifold M is a completely non-integrable hyperplane
distribution in the tangent bundle TM, i.e. a distribution which is (at least locally)
defined by a 1-form α satisfying α ∧ (dα)n≠0. An almost contact structure is a
reduction of the structure group of TM to U(n) × 1. Every contact structure induces
an almost contact structure.
Applying results of Eliashberg and Weinstein on contact surgery, we show that an
(n− 1)-connected (2n + 1)-manifold is contact (to be precise: almost diffeomorphic to
a contact manifold) if and only if it is almost contact.
