A Characterization of Dupin Hypersurfaces in $\mathbb R^4$

Abstract

In this paper we study Dupin hypersurfaces in $\mathbb R^4$ parametrized by lines of curvature, with three distinct principal curvatures and $m_{jik}= 0$. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, which are invariant under inversions and homotheties.