Cohomogeneity one hypersurfaces of Euclidean spaces

We study isometric immersions f : M--> E into Euclidean space E of dimension n+1 of a complete Riemannian manifold M of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either n ≥ 3 and M is compact or if n ≥ 5 and the connected components of the flat part of M are bounded. We also provide several sufficient conditions for f to be a hypersurface of revolution.