Ricci curvature and the topology of open manifolds
type: Item
instance of: scholarly article
title: Ricci curvature and the topology of open manifolds
zbMATH Open document ID: 0958.53025
published in: Mathematische Annalen
publication date: 2001-04-02T00:00:00Z
review text: Let \((M, g)\) be an \(n\)-dimensional complete Riemannian manifold. The authors consider a global metric invariant \(A\) which expresses the \(r\)-limiting value of the ratio \(\text{Vol}(B(p,r))/E(n,r)\), where \(B(p,r)\) denotes the geodesic ball with radius \(r\), centered in \(p\in M\); \(E(n,r)\) represents the volume of the Euclidean \(n\)-ball with radius \(r\). The space \((M, g)\) has large volume growth if \(A\) is positive. In this paper, the analogous of the theorem of \textit{V. B. Marenich} and \textit{V. T. Toponogov} [Sib. Mat. Zh. 26, 191-194 (1985; Zbl 0578.53030)] for the case of spaces with large volume growth and with nonnegative Ricci curvature is obtained: the theorem of Marenich-Toponogov states that an \(n\)-dimensional complete manifold with nonnegative sectional curvature and large volume growth is diffeomorphic to a Euclidean \(n\)-space.
reviewed by: Valentin Boju
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1423137
zbMATH Keywords: Ricci curvature
zbMATH Keywords: sectional curvature
zbMATH Keywords: geodesic ball
zbMATH Keywords: large volume growth
zbMATH Keywords: comparison theorem
author: Manfredo P. do Carmo
author: Changyu Xia
MaRDI profile type: Publication
full work available at URL: s002080050018
OpenAlex ID: W4249335751
DOI: 10.1007/S002080050018
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description: scientific article; zbMATH DE number 1423137