Dupin hypersurfaces with three principal curvatures
type: Item
instance of: scholarly article
title: Dupin hypersurfaces with three principal curvatures
zbMATH Open document ID: 0908.53007
author: Thomas E. Cecil
author: Gary R. Jensen
published in: Inventiones Mathematicae
publication date: 1999-03-10T00:00:00Z
review text: In this very interesting paper, the authors prove that any irreducible Dupin hypersurface in \(S^n\) with three principal curvatures is equivalent up to a Lie transformation to an isoparametric hypersurface. Note that there are no global conditions assumed in this theorem. If one replaces the word irreducible by compact in the above result, one gets a theorem of \textit{R. Miyaoka} [Math. Z. 187, 433-452 (1984; Zbl 0545.53047)]. It is a very interesting open question whether the known results on compact Dupin hypersurfaces hold true under the assumption of irreducibility instead of compactness. Here I am especially thinking of the restriction on the number of principal curvatures in my paper [\textit{G. Thorbergsson}, Bull. Lond. Math. Soc. 15, 493-498 (1983; Zbl 0529.53044)] and the recent result of \textit{S. Stolz} on the possible multiplicities of the principal curvatures [`Multiplicities of Dupin hypersurfaces' (Preprint MPI-1997-25) (Max-Planck-Institut für Mathematik, Bonn) (1997)].
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53A40
zbMATH DE Number: 1186158
zbMATH Keywords: Dupin hypersurfaces
zbMATH Keywords: Lie sphere geometry
zbMATH Keywords: irreducibility
reviewed by: Gudlaugur Thorbergsson
MaRDI profile type: Publication
full work available at URL: s002220050220
OpenAlex ID: W2035267725
DOI: 10.1007/S002220050220
description: scientific article; zbMATH DE number 1186158