Cohomogeneity one manifolds and hypersurfaces of the Euclidean space (Q1894608)

Let \((M,g)\) be a compact Riemannian manifold of dimension \(n > 3\) with an isometry group \(G\) which has a codimension one orbit and \(i : M \to \mathbb{R}^{n + 1}\) be an isometric immersion of \(M\) into the Euclidean space as a hypersurface. The authors prove that \(i(M)\) is a revolution hypersurface, that is it is invariant under the group of rotations around a line through the origin, if and only if all principal orbits of \(G\) on \(M\) are umbilical.