Totally geodesic submanifolds and polar actions on Stiefel manifolds

Abstract

We classify totally geodesic submanifolds of the real Stiefel manifolds of orthogonal two-frames. We also classify polar actions on these Stiefel manifolds, specifically, we prove that the orbits of polar actions are lifts of polar actions on the corresponding Grassmannian. In the case of cohomogeneity-one actions we are able to obtain a classification for all real, complex and quaternionic Stiefel manifolds of $k$-frames.