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 $(N^5\times [0,\pi])\times [0,\pi]$.