On homogeneous hypersurfaces in the manifold of quaternionic subalgebras of the Cayley algebra

Abstract

In this work  we consider a class of real hypersurfaces of $N=G_2 / SU(2) \times SU(2)$, the manifold of quaternionic subalgebras of the Cayley algebra. They are the  family of tubes centered at the maximal totally geodesic submanifolds of maximal rank of $N$, which are (up to isomorphism) $\mathbb{CP}^2$ and $S^2 \times S^2$. We determine which of those tubes are homogeneous and for them we obtain the spectral decomposition of the shape operator. Moreover we show that the universal covering space of the focal set of $\mathbb{CP}^2$ is the sphere $S^5$.