Möbius isoparametric hypersurfaces with three distinct principal curvatures. II
type: Item
instance of: scholarly article
title: Möbius isoparametric hypersurfaces with three distinct principal curvatures. II
zbMATH Open document ID: 1231.53011
author: Shujie Zhai
author: Zejun Hu
published in: Pacific Journal of Mathematics
publication date: 2011-03-01T00:00:00Z
review text: The authors study connected smooth and umbilic-point-free hypersurfaces \(x: M^n \to S^{n+1}\) in the \((n+1)\)-sphere \(S^{n+1}\) whose Möbius invariant 1-form vanishes and whose principal curvatures are all constant. Such hypersurfaces of \(S^{n+1}\) are called Möbius isoparametric hypersurfaces. They prove a classification theorem for Möbius isoparametric hypersurfaces which have only three distinct Möbius principal curvatures, thus extending their result in Part I [ibid. 232, No.~2, 289--311 (2007; Zbl 1154.53011)]. It is shown that there are exactly three types of such surfaces: The first type belongs to parallel Möbius second fundamental form and has three distinct Möbius principal curvatures with arbitrary multiplicities \(p\), \(q\), \(n-p-q\). The second type are the Euclidean isoparametric surfaces contained in \(S^{ n+1}\). The third type consists of certain minimal hypersurfaces where the multiplicities of the three Möbius principal curvatures are \(m\), \(m\) and \(n-2m > m\).
reviewed by: Anton Gfrerrer
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 5858901
zbMATH Keywords: Möbius isoparametric hypersurfaces
zbMATH Keywords: Möbius second fundamental form
zbMATH Keywords: Möbius metric
zbMATH Keywords: Möbius form
zbMATH Keywords: Möbius equivalence
MaRDI profile type: Publication
full work available at URL: pjm.2011.249.343
OpenAlex ID: W4230818674
DOI: 10.2140/PJM.2011.249.343
description: scientific article; zbMATH DE number 5858901