Nearly half-flat $\rm{SU}(3)$-structures on $S^3\times S^3$

Abstract

We study the $\rm{SU}(3)$-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel $\rm{G}_2$-structure. We call such $\rm{SU}(3)$-structures nearly half-flat. We characterise the left invariant nearly half-flat structures on $S^3\times S^3$. This characterisation then help us to systematically analyse nearly parallel $\rm{G}_2$-structures on an interval times $ S^3\times S^3$.