Dupin hypersurfaces and a Lie invariant
type: Item
instance of: scholarly article
title: Dupin hypersurfaces and a Lie invariant
zbMATH Open document ID: 0687.53053
published in: Kodai Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The author first shows that the cross ratio of four distinct principal curvatures of a hypersurface in a space form is invariant under Lie transformations. Let us call this invariant Lie curvature. Consequently, a necessary condition for a Dupin hypersurface (i.e., a hypersurface each of which principal curvatures has constant multiplicity and vanishing derivative in corresponding principal direction) with exactly four principal curvatures to be Lie equivalent to an isoparametric hypersurface is that its Lie curvature is constant. The author shows that this is also a sufficient condition if one adds a certain condition on how leaves of curvature distributions can intersect. The method of proof is similar in spirit, although much more difficult, to the paper of the author in [Math. Z. 187, 433-452 (1984; Zbl 0529.53045)]. The Lie curvature introduced in this paper has been used to prove that there are Dupin hypersurfaces that are not Lie equivalent to isoparametric hypersurfaces, see the author and \textit{T. Ozawa} [Proc. Symp. Differ. Geom. 1988, Perspect Math. 8, 181-189 (1989)] and \textit{U. Pinkall} and the reviewer [Proc. Am. Math. Soc. 107, No.4, 1037-1043 (1989)].
reviewed by: Gudlaugur Thorbergsson
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 4126243
zbMATH Keywords: principal curvatures
zbMATH Keywords: Lie transformations
zbMATH Keywords: Dupin hypersurface
zbMATH Keywords: isoparametric hypersurface
zbMATH Keywords: Lie curvature
author: Reiko Miyaoka
MaRDI profile type: Publication
OpenAlex ID: W1985052947
cites work: Tautness and Lie sphere geometry
cites work: Focal sets, taut embeddings and the cyclides of Dupin
cites work: Q3729588
cites work: WAVE EQUATION AND DUPIN HYPERSURFACE
cites work: Compact Dupin hypersurfaces with three principal curvatures
cites work: Q3473774
cites work: Dupin Hypersurfaces
cites work: Dupin hypersurfaces
cites work: Deformations of Dupin Hypersurfaces
DOI: 10.2996/KMJ/1138039036
description: scientific article; zbMATH DE number 4126243