On Dupin hypersurfaces with constant Möbius curvature

Carlos M. C. Riveros, Luciana Avila Rodrigues and Keti Tenenblat

Vol. 236 (2008), No. 1, 89–103

DOI: 10.2140/pjm.2008.236.89

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 2000

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

Milestones

Received: 13 June 2007

Revised: 22 February 2008

Accepted: 25 February 2008

Published: 1 May 2008