On cohomogeneity one hyperpolar actions related to $G_2$

Abstract

Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and exceptional actions. In this paper, we consider exceptional actions related to the exceptional compact Lie group $G_2$ and investigate some properties of their orbits as Riemannian submanifolds. In particular, we examine the principal curvatures of principal orbits and classify principal orbits that are minimal, austere, weakly reflective, and proper biharmonic.