Contact Manifolds

Abstract

By a contact manifold we mean a C ^∞ manifold M ²ⁿ⁺¹ together with a 1-form η such that η ∧ (dη)ⁿ ≠ 0. In particular η ∧ (dη)ⁿ ≠ 0 is a volume element on M so that a contact manifold is orientable. Also dη has rank 2n on the Grassmann algebra ∧ T ^* _m M at each point m ∈ M and thus we have a 1-dimensional subspace, {X ∈ T _m M|dη(X, T _m M) = 0}, on which η ≠ 0 and which is complementary to the subspace on which η = 0. Therefore choosing ξ _m in this subspace normalized by η(ξ _m ) = 1 we have a global vector field ξ satisfying

$$ d\eta \left( {\xi ,X} \right) = 0,\;\eta \left( \xi \right) = 1 $$

.