Global pinching theorems of submanifolds in spheres
type: Item
instance of: scholarly article
title: Global pinching theorems of submanifolds in spheres
zbMATH Open document ID: 1014.53031
DOI: 10.1155/S0161171202106247
author: Kairen Cai
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/50102
full work available at URL: S0161171202106247.html
review text: Let \(M\) be an \(m\)-dimensional (\(m \geq 3\)) compact submanifold of the unit sphere \(S^n\) having parallel mean curvature and positive Ricci curvature. Denote by \(\sigma\) the length of the second fundamental form. In the present paper the author shows that there is a constant \(C\), depending only on \(m\), the mean curvature and \(k\), where \((m-1)k\) is the lower bound of the Ricci curvature, such that if \((\int_M \sigma^{m/2})^{2/m} < C\), then \(M\) is a totally umbilical hypersurface in a sphere \(S^{m+1}\). A similar result for surfaces, with Gaussian curvature playing the role of Ricci curvature, is also stated.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1780415
zbMATH Keywords: submanifolds
zbMATH Keywords: length of the second fundamental form
zbMATH Keywords: pinching
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1780415