A rigidity theorem for submanifolds with parallel mean curvature in a sphere
type: Item
instance of: scholarly article
title: A rigidity theorem for submanifolds with parallel mean curvature in a sphere
zbMATH Open document ID: 0785.53049
DOI: 10.1007/BF01207549
published in: Archiv der Mathematik
publication date: 1994-04-21T00:00:00Z
review text: It is proved that if \(M^ n\) is an \(n\)-dimensional compact submanifold with parallel mean curvature in a unit sphere \(S^{n+p}(1)\), and \(S\leq C(n,p,H)\), where \(H\) and \(S\) are the mean curvature and the square norm of the second fundamental form of \(M\) respectively, and \(C(n,p,H)\) is a positive constant depending on \(n\), \(p\) and \(H\), then either \(M\) is the totally umbilical sphere \(S^ n({1\over \sqrt{1+H^ 2}})\), the standard immersion of the product of two spheres or the Veronese surface in \(S^ 4({1\over \sqrt{1+H^ 2}})\).
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 493370
zbMATH Keywords: totally umbilical
zbMATH Keywords: product of two spheres
zbMATH Keywords: Veronese surface
author: Hongwei Xu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 493370