On radius, systole, and positive Ricci curvature
type: Item
instance of: scholarly article
title: On radius, systole, and positive Ricci curvature
zbMATH Open document ID: 0823.53033
DOI: 10.1007/BF02571926
published in: Mathematische Zeitschrift
publication date: 1995-10-17T00:00:00Z
full work available at URL: https://eudml.org/doc/174751
review text: Recall that the radius of a compact metric space, \((X, \text{dist})\) is defined by \[ \text{Radius } X = \min_{x \in X} \max_{y \in X} \text{dist} (x,y). \] In this note we prove that a closed Riemannian \(n\)- manifold with \(\text{Ricci }M \geq n -1\) is diffeomorphic and almost isometric to \(S^ n\), if its radius is sufficiently close to \(\pi\) compared to its lower bound for sectional curvature. We also show that the same conclusion can be obtained if we replace the hypothesis on diameter, provided also that the \(d\)-excess of \(M\) is sufficiently small. As a corollary of the radius theorem we get that \(M\) is diffeomorphic and almost isometric to \(\mathbb{R} P^ n\), if \(\text{Ricci }M \geq n - 1\) and \(\text{Sys}_ 1M\) is sufficiently close to \(\pi\) compared to the minimum of the sectional curvatures of \(M\). Here \(\text{Sys}_ 1 M\) denotes the length of the shortest closed noncontractible curve in \(M\).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 784092
zbMATH Keywords: almost isometric
zbMATH Keywords: sectional curvature
zbMATH Keywords: excess
author: Frederick Wilhelm
MaRDI profile type: Publication
cites work: Metrics of positive Ricci curvature with large diameter
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cites work: Eigenvalue comparison theorems and its geometric applications
cites work: Filling Riemannian manifolds
cites work: A Pinching Theorem for Homotopy Spheres
cites work: Volume comparison à la Aleksandrov
cites work: The filling radius of two-point homogeneous spaces
cites work: Riemannian manifolds with positive mean curvature
cites work: On manifold of positive Ricci curvature with large diameter
cites work: On Manifolds of Small Excess
cites work: A diameter sphere theorem for manifolds of positive Ricci curvature
cites work: Manifolds with small excess and bounded curvature
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cites work: Lipschitz convergence of manifolds of positive Ricci curvature with large volume
cites work: Collapsing and pinching under a lower curvature bound
Wikidata QID: Q125791637
OpenAlex ID: W2087103523
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description: scientific article; zbMATH DE number 784092