Hyperpolar actions and k-flat homogeneous spaces.

Abstract

A closed, connected, k-dimensional submanifold of a compact Riemannian manifold M is called a of M if it is flat in the induced metric and totally geodesic. We call M k-flat if every geodesic lies in some of M, and if the group of isometries of M acts transitively on pairs (σ, p) consisting of a σ and a point p ∈ σ. An isometric action on M is called hyperpolar if there exists a connected, closed, flat submanifold of M that meets all orbits orthogonally. We prove that the following three properties for a compact Riemannian manifold M are equivalent: (a) M is a Riemannian homogeneous manifold and admits a cohomogeneity k hyperpolar action with a fixed-point, (b) M is homogeneous, (c) M is a rank k symmetric space. Since 1-flat homogeneous is trivially equivalent to two-point homogeneous, the equivalence of (b) and (c) generalizes the well-known fact that two-point homogeneous spaces are the same as rank 1 symmetric spaces.