Cohomogeneity one hypersurfaces of Euclidean Spaces

Abstract

We study isometric immersions $f: M^{n}\to \mathbb{R}^{n+1}$ into Euclidean space of dimension $n+1$ of a complete Riemannian manifold 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 $n\geq 3$ and $M^n$ is compact or if $n\geq 5$ and the connected components of the flat part of $M^n$ are bounded. We also provide several sufficient conditions for $f$ to be a hypersurface of revolution.