On a class of Dupin hypersurfaces in $$R^5$$ with nonconstant Lie curvature

Abstract

We consider a class of Dupin hypersurface in \(R^5\) parametrized by lines of curvature, with four distinct principal curvatures. We provide a characterization of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable. We prove that these functions describe plane curves. The Lie curvature of these hypersurfaces is not constant but some Moebius curvatures are constant along certain lines of curvature. We give explicit examples of such Dupin hypersurfaces.