Rigidity of compact minimal submanifolds in a sphere
type: Item
instance of: scholarly article
title: Rigidity of compact minimal submanifolds in a sphere
zbMATH Open document ID: 1021.53037
author: Sharief Deshmukh
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: Let \(M^n\) be a compact minimal \(n\)-dimensional submanifold of the unit sphere \(S^{n+p}\). The author focusses on pinching conditions for the scalar curvature \(\tau\) of \(M\) which imly that \(M\) is a totally geodesic submanifold. First he proves that this is always the case when \(\tau> n(n-2)\) and \(p\leq 2\). To obtain a similar result for \(p>2\) he needs an additional condition on the curvature tensor of the normal connection (for example, he considers the case of a flat normal condition). Furthermore, he proves that when \(M\) is nonnegatively curved with flat normal connection, then \(M\) is totally geodesic or else \(\tau\) satisfies \(n(n-p-1)\leq\tau\leq n(n-2)\). From these results he also derives several properties for hypersurfaces \(M\), in which case the normal connection is automatically flat. For example, when \(M\) is a nonnegatively curved compact minimal hypersurface \(M\) in \(S^{n+1}\), then it is totally geodesic or it is isometric to the hypersurface \(S^m(\sqrt{{m\over n}})\times S^{n-m}(\sqrt{{n-m\over n}})\) (for which the square of the length of the second fundamental form equals \(n\)) which implies that if \(M\) is a nonnegatively curved compact minimal hypersurface in \(S^{n+1}\) which is diffeomorphic to \(S^n\), then it is a totally geodesic hypersurface.
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1463777
zbMATH Keywords: minimal
zbMATH Keywords: submanifold
zbMATH Keywords: pinching
zbMATH Keywords: scalar curvature
zbMATH Keywords: totally geodesic
MaRDI profile type: Publication
full work available at URL: pjm.2000.193.31
OpenAlex ID: W2007475506
DOI: 10.2140/PJM.2000.193.31
description: scientific article; zbMATH DE number 1463777