Nearly hypo structures and compact Nearly K\"ahler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface $N^5$ of a 6-dimensional nearly K"ahler manifold $M^6$ is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of \cite{ConS}. We show that any Sasaki-Einstein 5-manifold defines a nearly K"ahler structure on the sin-cone $N^5\times \mathbb{R}$, and a compact nearly K"ahler structure with conical singularities on $N^5\times [0,\pi ]$ when $N^5$ is compact thus providing a link between Calabi-Yau structure on the cone $N^5\times [0,\pi ]$ and the nearly K"ahler structure on the sin-cone $N^5\times [0,\pi ]$. We define the notion of {\it nearly hypo} structure that leads to a general construction of nearly K"ahler structure on $N^5\times \mathbb{R}$. We determine {\it double hypo} structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly K"ahler structure is introduced, which we refer to as {\it nearly half flat} SU(3)-structure, that leads us to generalize the construction of nearly parallel $G_2$-structures on $M^6\times \mathbb{R}$ given in \cite{BM}. For $N^5=S^5\subset S^6$ and for $N^5=S^2\times S^3\subset S^3\times S^3$, we describe explicitly a Sasaki-Einstein hypo structure as well as the corresponding nearly K"ahler structures on $N^5\times \mathbb{R}$ and $N^5\times [0,\pi ]$, and the nearly parallel $G_2$-structures on $N^5\times {\mathbb{R}}^2$ and $(N^5\times [0,\pi])\times [0,\pi]$.