On the Lie curvature of Dupin hypersurfaces
type: Item
instance of: scholarly article
title: On the Lie curvature of Dupin hypersurfaces
zbMATH Open document ID: 0711.53050
author: Thomas E. Cecil
published in: Kodai Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The Lie curvature of a hypersurface with four principal curvatures is by definition their cross ratio. It turns out that the Lie curvature is invariant under Lie sphere transformations and that it is identically equal to \(\frac12\) for isoparametric hypersurfaces after ordering their principal curvatures appropriately, see the paper [ibid. 12, No. 2, 228-256 (1989; Zbl 0687.53053)] by \textit{R. Miyaoka}. In the paper under review the author gives an example of a noncompact Dupin hypersurface with four principal curvatures that is not Lie equivalent to an isoparametric hypersurface although its Lie curvature is identically \(\frac12\). The reason for this example seems rather to be that it is reducible than that it is noncompact.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4169613
zbMATH Keywords: Lie sphere geometry
zbMATH Keywords: Lie curvature
zbMATH Keywords: isoparametric hypersurfaces
zbMATH Keywords: Dupin hypersurface
reviewed by: Gudlaugur Thorbergsson
MaRDI profile type: Publication
OpenAlex ID: W2012034263
cites work: Reducible Dupin submanifolds
cites work: Tautness and Lie sphere geometry
cites work: Q3834620
cites work: Focal sets, taut embeddings and the cyclides of Dupin
cites work: Q3729588
cites work: Dupin hypersurfaces, group actions and the double mapping cylinder
cites work: Compact Dupin hypersurfaces with three principal curvatures
cites work: Dupin hypersurfaces and a Lie invariant
cites work: Dupin hypersurfaces with six principal curvatures
cites work: Q3473774
cites work: Dupin hypersurfaces
cites work: Dupin'sche Hyperflächen in \(E^ 4\). (Dupin hypersurfaces in \(E^ 4)\)
cites work: Deformations of Dupin Hypersurfaces
cites work: Dupin Hypersurfaces
DOI: 10.2996/KMJ/1138039167
Recommended article: Dupin hypersurfaces and a Lie invariant
Recommended article: Dupin Hypersurfaces in Lie Sphere Geometry
Recommended article: Dupin hypersurfaces with constant Laguerre curvatures
Recommended article: Q4466937
Recommended article: Q5289251
Recommended article: On Dupin hypersurfaces with constant Möbius curvature
Recommended article: Dupin hypersurfaces
Recommended article: Q4522315
Recommended article: Möbius curvature, Laguerre curvature and Dupin hypersurface
Recommended article: Curvatures properties of Lie hypersurfaces in the complex hyperbolic space
description: scientific article; zbMATH DE number 4169613