A characterization of Laguerre isoparametric hypersurfaces of the Euclidian space

Abstract

We consider Mn,n≥3, umbilic-free hypersurfaces in the Euclidean space, with nonvanishing principal curvatures. We prove that M is a Laguerre isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Laguerre curvatures.