Classes of hypersurfaces with vanishing Laplace invariants

Abstract

Consider a hypersurface M n in R n+1 with n distinct princi- pal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hyper- surfaces for n ≥ 4, and for n = 3, they are, up to Möbius transformations, Dupin hypersurfaces with constant Möbius curvature. (2) If the principal curvatures are given by a sum of functions of sepa- rated variables, there are no such hypersurfaces for n ≥ 4, and for n = 3, they are, up to Möbius transformations, Dupin hypersurfaces with con- stant Möbius curvature.