Abstract

We show that proper Dupin hypersurfaces Mⁿ for n ≥ 4 in Rⁿ⁺¹ with n distinct principal curvatures and constant Möbius curvature cannot be parametrized by lines of curvature. For n = 3, up to Möbius transformations, there is a unique proper Dupin hypersurface, parametrized by lines of curvature, with three distinct principal curvatures and constant Möbius curvature. Moreover, these hypersurfaces are the only conformally flat proper Dupin hypersurfaces M³ ⊂ R⁴ with three distinct principal curvatures and constant Möbius curvature.

Keywords

Dupin hypersurfaces, constant Möbius curvature, conformally flat hypersurfaces

Mathematical Subject Classification

Primary: 53C42, 53A30, 53C40, 53A07

Authors