On Gromov's large Riemannian manifolds
type: Item
instance of: scholarly article
title: On Gromov's large Riemannian manifolds
zbMATH Open document ID: 0808.53047
DOI: 10.1007/BF01263649
author: Mingliang Cai
published in: Geometriae Dedicata
publication date: 1994-08-31T00:00:00Z
review text: The author proves that a Riemannian manifold is hyperspherical if and only if its filling radius is infinite, under suitable geometric hypotheses. The hypotheses are twofold: (i) the Ricci curvature is nonnegative (ii) the injectivity radius is positive. M. Gromov previously proved the result under the hypothesis of nonnegative sectional curvature. There are several further properties of ``largeness'' equivalent to hypersphericity (still assuming the two hypotheses), including the condition of Euclidean volume growth (for the ``biggest'' \(\rho\)-ball). The author's tools include the Anderson-Cheeger compactness theorem [\textit{M. Anderson} and \textit{J. Cheeger}, Geom. Funct. Anal. 1, No. 3, 231-252 (1991; Zbl 0764.53026)] as well as his refinement of it in [\textit{M. Cai}, Ann. Global Anal. Geom. 11, 373-385 (1993)]. A central role is played by Gromov's argument on skeletal retraction [\textit{M. Gromov}, Publ. Math., Inst. Hautes Etud. 53, 213-307 (1983), pp. 258- 260].
reviewed by: Mikhail G. Katz
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 627553
zbMATH Keywords: large Riemannian manifolds
zbMATH Keywords: nonnegative Ricci curvature
zbMATH Keywords: hyperspherical manifolds
zbMATH Keywords: filling radius
MaRDI profile type: Publication
cites work: Diffeomorphism finiteness for manifolds with Ricci curvature and \(L^{n/2}\)-norm of curvature bounded
cites work: A splitting theorem for manifolds of almost nonnegative Ricci curvature
cites work: Rigidity of manifolds with large volume
cites work: Filling Riemannian manifolds
cites work: Q3700424
description: scientific article; zbMATH DE number 627553