Weakly reflective submanifolds and austere submanifolds

Abstract

An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by $-1$. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.