A note on closed G2 -structures and 3-manifolds

Abstract

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.