A characterization of quadric constant scalar curvature hypersurfaces of spheres
type: Item
instance of: scholarly article
title: A characterization of quadric constant scalar curvature hypersurfaces of spheres
zbMATH Open document ID: 1307.53048
author: Oscar M. Perdomo
author: Guoxin Wei
published in: The Journal of Geometric Analysis
publication date: 2014-11-18T00:00:00Z
review text: The authors prove the following result (Theorem 1.1) : Let \(M \subset S^4\) be a complete orientable hypersurface with constant scalar curvature \(R\), and let \(l_v, f_v :M\rightarrow \mathbb{R}\) be the real functions defined by \(l_v(x)=\langle x,v\rangle, f_v(x)=\langle \nu (x),v\rangle\), where \(\nu :M\rightarrow S^4\) is a Gauss map of \(M\). If \(l_v=\lambda f_v\) for some nonzero vector \(v \in \mathbb{R}^5\) and some real number \(\lambda\), then \(M\) is either totally umbilical (a Euclidean sphere) or \(M\) is a Cartesian product of Euclidean spheres. In Section 4 of the paper, an example illustrates that the completeness condition is necessary.
reviewed by: Dorin Andrica
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 6370449
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: principal curvature
zbMATH Keywords: sectional curvature
zbMATH Keywords: Gauss map
zbMATH Keywords: orientable hypersurface
MaRDI profile type: Publication
full work available at URL: s12220-013-9399-4
OpenAlex ID: W4251416216
cites work: A characterization of quadric constant mean curvature hypersurfaces of spheres
cites work: On some rigidity results of hypersurfaces in a sphere
cites work: Global rigidity theorems of hypersurfaces
DOI: 10.1007/S12220-013-9399-4
description: scientific article; zbMATH DE number 6370449