Vanishing of homology groups, Ricci estimate for submanifolds and applications
type: Item
instance of: scholarly article
title: Vanishing of homology groups, Ricci estimate for submanifolds and applications
zbMATH Open document ID: 1007.53044
published in: Kodai Mathematical Journal
publication date: 2002-06-30T00:00:00Z
review text: Let \(x: M^n\to Q_c^{n+m}\) be an isometric immersion of a Riemannian manifold \(M^n\) (superscripts denote dimensions) into a complete simply-connected space \(Q_c^{n+m}\) of constant sectional curvature \(c\). The authors obtain, among other results, an estimate for the Ricci curvature of \(M^n\) and, when \(M^n\) is compact, a criterion for the vanishing of its homology groups. Here is an interesting corollary of the Ricci curvature estimate. Let \(M^n\), \(n\geq 3\), be a compact, minimal submanifold of the unit sphere \(S^{n+m}\). Assume that the norm squared \(|A|^2\) of the second fundamental form \(A\) satisfies \(|A|^2\leq n\) and that the fundamental group of \(M^n\) is infinite. Then \(M^n\) is a Clifford torus \(S^1\times S^{n-1}\) in a totally geodesic submanifold \(S^{n+1}\subset S^{n+m}\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1742079
zbMATH Keywords: isometric immersion
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Ricci curvature
zbMATH Keywords: minimal submanifold
zbMATH Keywords: totally geodesic submanifold
author: Antonio C. Asperti
author: Ézio de Araujo Costa
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1106168806
OpenAlex ID: W2055599945
DOI: 10.2996/KMJ/1106168806
description: scientific article; zbMATH DE number 1742079