On cohomogeneity one flat Riemannian manifolds

We study the topological properties of cohomogeneity one flat manifolds and their orbits. Among other results we prove that principal orbits of R-n are isometric to Rn-1 or S-k(c) x Rn-k-1. We show that if M has one singular orbit, it is a totally geodesic submanifold of M and if M is orientable then there is at most one singular orbit.