Nearly hypo structures and compact Nearly Kähler 6-manifolds with conical singularities

We consider 6-dimensional nearly Kähler manifolds M and prove that any totally geodesic hypersurface N of M is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of [8]. We show that such a hypo structure defines a nearly Kähler structure on N × R, and a compact nearly Kähler structure with conical singularities on N 5 × S 1 when N is compact. Moreover, an extension of the concept of nearly Kähler structure is introduced, which we refer to as nearly half flat SU(3)-structure, that leads us to generalize the construction of nearly parallel G2-structures on M 6 × R given in [2]. For N 5 = S ⊂ S and for N = S × S ⊂ S × S, we describe explicitly a Sasaki-Einstein hypo structure as well as the corresponding nearly Kähler structures on N×R and N×S, and the nearly parallel G2-structures on N 5 × R 2 and (N × S)× S.