A note on closed G2 -structures and 3-manifolds

This article shows that given any orientable 3-manifold X , the 7 -manifold T*X × R admits a closed G2 - structure φ = Re Ω − ω ∧ dt where Ω is a certain complex-valued 3-form on T*X ; next, given any 2 -dimensional submanifold S of X , the conormal bundle N*S of S is a 3-dimensional submanifold of T*X × R such that φ|N* S ≡ 0. A corollary of the proof of this result is that N*S×R is a 4 -dimensional submanifold of T*X ×R such that φ|N* S×R ≡ 0.