A topological rigidity theorem on open manifolds with nonnegative Ricci curvature
type: Item
instance of: scholarly article
title: A topological rigidity theorem on open manifolds with nonnegative Ricci curvature
zbMATH Open document ID: 1060.53049
DOI: 10.1007/BF02385488
author: Qiaoling Wang
published in: Arkiv för Matematik
publication date: 2004-09-28T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature, \(B(p,r)\) the open geodesic ball of radius r at \(p \in M\), and vol\(B(p,r)\) its volume. Let \(\omega_n\) be the volume of the unit ball in the Euclidean space \(\mathbb R^n\), and denote by \(\alpha_M\) the \(\lim_{r\rightarrow + \infty}\frac{\operatorname{vol}B(p,r)}{\omega_n r^n}\). It is well-known that \(\alpha_M\) is well-defined, i.e. the limit exists and it does not depend on the choice of the point \(p\). The manifold \(M\) is said to have large volume growth if \(\alpha_M > 0.\) Such manifolds have been studied extensively. In [Fields Inst. Monogr. 4, 87--115 (1996; Zbl 0843.53031)], \textit{P. Petersen} put forward the following conjecture: An \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature and \(\alpha_M > \frac{1}{2}\) is diffeomorphic to \(\mathbb{R}^n\). In the paper, the author proves the following topological rigidity theorem which supports Petersen's conjecture and contains no condition on the sectional curvature. Theorem. Given \(\alpha\in (\frac{1}{2},1), \rho_0 >0\) and an integer \(n\geq 2\), there exist positive constants \(r_0 = r_0(\alpha, \rho_0, n)\) and \(\varepsilon = \varepsilon (\alpha, \rho_0, n)\) such that any complete Riemannian \(n\)-manifold \(M\) with Ricci curvature \(\text{Ricci}_M \geq 0\), \(\alpha_M \geq \alpha\), conjugate radius \(\text{conj}_M \geq \rho_0\) and \[ \frac{\operatorname{vol}B(p,r)}{\omega_n r^n} - \alpha_M \leq \frac{\varepsilon}{r^{n-2+1/n}} \] for some \(p \in M\) and all \(r\geq r_0,\) is diffeomorphic to \(\mathbb R^n\).
reviewed by: Robert A. Wolak
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 2103857
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: nonnegative Ricci curvature
zbMATH Keywords: topological rigidity
MaRDI profile type: Publication
cites work: On Complete Manifolds With Nonnegative Ricci Curvature
cites work: On the topology of complete manifolds of non-negative Ricci curvature
cites work: Ricci curvature and the topology of open manifolds
cites work: Q4287762
cites work: On the structure of spaces with Ricci curvature bounded below. I
cites work: A comparison-estimate of Toponogov type for Ricci curvature
cites work: Curvature, diameter and Betti numbers
cites work: Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
cites work: Large time behavior of the heat equation on complete manifolds with non- negative Ricci curvature
cites work: Large volume growth and finite topological type
cites work: Manifolds of Positive Ricci Curvature with Almost Maximal Volume
cites work: Q4872653
cites work: Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity
cites work: On Complete Manifolds of Nonnegative kth-Ricci Curvature
cites work: Complete manifolds with nonnegative Ricci curvature and large volume growth
cites work: Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups.
cites work: Open manifolds with nonnegative Ricci curvature and large volume growth
cites work: Large volume growth and the topology of open manifolds
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description: scientific article; zbMATH DE number 2103857