Dupin hypersurfaces with four principal curvatures in ℝ5 with principal coordinates

Abstract

In this paper we study Dupin hypersurfaces in R 5 parametrized by lines of curvature, with four distinct principal curvatures and T ij kl = 0. We characterize locally a family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. Moreover, we show that these vector valued functions are invariant under inversions and homotheties.