Principal Curvatures of Homogeneous Hypersurfaces in a Grassmann Manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action

Abstract

We compute the principal curvatures of homogeneous hypersurfaces in a Grassmann manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action. As applications, we show that there is a unique orbit which is an austere submanifold, and that there are just two orbits which are proper biharmonic homogeneous hypersurfaces. We also show that the austere orbit is a weakly reflective submanifold.