Closed hypersurfaces of \(S^4\) with two constant symmetric curvatures
type: Item
instance of: scholarly article
title: Closed hypersurfaces of \(S^4\) with two constant symmetric curvatures
zbMATH Open document ID: 0905.53041
author: Sebastião Carneiro de Almeida
author: Fabiano G. B. Brito
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1999-01-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73415
review text: This paper deals with closed hypersurfaces of the 4-dimensional Euclidean sphere \(S^4\) with given geometric properties. The second fundamental form of this kind of immersions is determined by three symmetric curvatures: the mean curvature \(H\), the scalar curvature \(k\) and the Gauss-Kronecker curvature \(K\). A hypersurface \(M\) is said to be isoparametric if all of them are constant. Combining results of the authors [\textit{S. C. de Almeida} and \textit{F. G. B. Brito}, Duke Math. J. 61, 195-206 (1990; Zbl 0721.53056)] and \textit{S.-P. Chang} [Commun. Anal. Geom. 1, 71-100 (1993; Zbl 0791.53058)], a closed hypersurface in \(S^4\) with constant mean curvature and constant scalar curvature is isoparametric. In this paper, the authors prove in their first theorem that a closed oriented hypersurface in \(S^4\) with constant Gauss-Kronecker curvature and constant nonnegative scalar curvature is isoparametric; in their second theorem, they suppose that \(H\) and \(K\neq 0\) are constant and, in addition, \(-3 \leq H/K\), and obtain the same conclusion (\(M\) is isoparametric).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1139492
zbMATH Keywords: isoparametric hypersurfaces of the 4-sphere
zbMATH Keywords: constant curvature
MaRDI profile type: Publication
OpenAlex ID: W2327074874
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DOI: 10.5802/AFST.860
description: scientific article; zbMATH DE number 1139492