Dupin hypersurfaces with constant Laguerre curvatures

Abstract

We consider proper Dupin hypersurfaces of the Euclidean space $$\mathbb {R}^{n+1}$$ , that admit principal coordinate systems and have n distinct nonvanishing principal curvatures. We obtain explicitly all such hypersurfaces that have constant Laguerre curvatures. In particular, we show that they are determined by $$n-2$$ Laguerre curvatures and two other constants, one of them being nonzero.