Nearly half-flat SU(3) structures on S3 × S3

Abstract

We study the $\mathrm{SU}\left(3\right)$-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel $G_2$-structure. Such $\mathrm{SU}\left(3\right)$-structures are called nearly half-flat. We characterise the left invariant nearly half-flat structures on $S^3\times S^3$. This characterisation then helps us to systematically analyse nearly parallel $G_2$-structures on an interval times $S^3\times S^3$.