Totally geodesic singular orbits and symmetric singular orbits in cohomogeneity one Riemannian manifolds

Abstract

We classify cohomogeneity one G-manifolds M of a simple Lie group G such that one of the singular orbits is a simply connected symmetric space. We give a simple necessary and sufficient condition that a singular orbit P of a cohomogeneity one Riemannian G-manifold is totally geodesic. Using it, we obtain a list of cohomogeneity one G-manifolds M as above such that the singular orbit P is totally geodesic with respect to any G-invariant Riemannian metric of M. Such G-manifolds have no G-invariant metric of positive or negative curvature if the rank of the symmetric space P is greater than one.