Rigidity of compact submanifolds in a unit sphere
type: Item
instance of: scholarly article
title: Rigidity of compact submanifolds in a unit sphere
zbMATH Open document ID: 0835.53070
author: Guanghua Chen
author: Xiaorong Zou
published in: Kodai Mathematical Journal
publication date: 1996-03-31T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold of dimension \(n\) immersed in a unit sphere \(S^{n + p}\) of dimension \(n + p\). Let \(S\) be the square of the length of the second fundamental form. The study of upper bounds for \(S\) in connection with some properties of \(M\) is an interesting problem. The aim of the present paper is to give some results in this direction, which can be stated as follows. (I) Let \(M^n\) be an \(n\)-dimensional compact submanifold in \(S^{n + p}\) with non-zero parallel mean curvature. If either of the following conditions is satisfied, then \(M^n\) lies in a totally geodesic \(S^{n+1}\): (i) \(S \leq \min \left \{ {2 \over 3} n,\;{2n \over 1 + \sqrt{n \over 2}}\right\}\), \(p > 2\) and \(n \neq 8\), (ii) \(S \leq \min \left \{ {n \over 2 - {1\over p - 1}} , {2n \over 1 + \sqrt {n\over 2}} \right \}\), \(p > 1\) and \((n,p) \neq (8, 3)\). (II) Let \(M^n\) be an \(n\)-dimensional \((2 \leq n \leq 7)\) compact submanifold in \(S^{n + p}\) with \(p > 2\) which has non-zero parallel mean curvature. If \(S \leq (2/3)n\), then \(M^n\) is totally umbilical. (III) Let \(M^n\) be a compact minimal submanifold in the sphere \(S^{n + p}\) with \(p > 1\). Suppose the sectional curvatures of \(M^n\) are everywhere not less than \((1/2 - 1/3p)\). Then either \(M^n\) is a totally geodesic sphere or the Veronese surface in \(S^4\).
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 815800
zbMATH Keywords: immersed manifold
zbMATH Keywords: unit sphere
zbMATH Keywords: second fundamental form
zbMATH Keywords: compact submanifold
zbMATH Keywords: totally geodesic sphere
zbMATH Keywords: Veronese surface
reviewed by: Grigorios Tsagas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043355
OpenAlex ID: W2033645292
cites work: Q4711212
cites work: Hypersurfaces and a Pinching Problem on the Second Fundamental Tensor
cites work: Minimal varieties in Riemannian manifolds
cites work: Submanifolds with Constant Mean Curvature II
DOI: 10.2996/KMJ/1138043355
description: scientific article; zbMATH DE number 815800