Hypersurfaces of cohomogeneity one and hypersurfaces of revolution

In this paper, we study hypersurfaces f: M^n(right arrow)R^N+1, n(greater than)3, where M^n is a G-cohomogeneity one Riemannian manifold such that the principal orbits of G are umbilical submanifolds of M. In (Ann. Global Anal. Geom. 13 (1995) 169–184), n(greater than)4 and m is compact, the authors prove that such a hypersurface must be of revolution. We extend this theorem to the of complete noncompact manifolds, G compact or noncompact. We also study the case n=3.