Spaces on and beyond the boundary of existence
type: Item
instance of: scholarly article
title: Spaces on and beyond the boundary of existence
zbMATH Open document ID: 0839.53032
DOI: 10.1007/BF02921805
published in: The Journal of Geometric Analysis
publication date: 1996-06-18T00:00:00Z
review text: Given an integer \(n\geq 2\) and a real number \(k\leq K\), let \(M_k (n)\) (resp. \(M^K_k (n)\)) denote the class of complete, connected, pointed Riemannian \(n\)-manifolds with sectional curvature \(\geq k\) (resp. \(\geq k\) and \(\leq K\)). Then the description of the metric spaces in the Gromov-Hausdorff closure \(\overline {M_k (n)}\) and \(\overline {M^K_k (n)}\) may be a problem worthy to pursue. \textit{K. Grove} and \textit{P. Petersen} [J. Differ. Geom. 33, No. 2, 379-394 (1991; Zbl 0729.53045)]\ have shown that \(\overline {M_k (n)}\) is an Alexandrov space with curvature \(\geq k\) and dimension \(\leq n\), while \textit{K. Fukaya} [ibid. 28, No. 1, 1-21 (1988; Zbl 0652.53031)]\ proved that \(\overline {M^K_k (n)}\) looks like \(N/O(n)\) where \(N\) is a Riemannian manifold. After establishing the theorem stated below, the authors show in Theorems 2 and 3 that the converse of the above results is not true. The theorem mentioned above is as follows: Let \(M\) be a convex Riemannian manifold with bounded curvature on which a Lie group \(G\) acts as isometry in such a way that all orbits are closed. Then \(M/G\) with the natural quotient metric can be obtained as the limit of a sequence of Riemannian manifolds with a uniform lower sectional bound and fixed dimension in the pointed Gromov-Hausdorff topology.
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 828779
zbMATH Keywords: sectional curvature
zbMATH Keywords: sequence of Riemannian manifolds
zbMATH Keywords: Gromov-Hausdorff topology
author: Peter Petersen
author: Frederick Wilhelm
author: Shun-Hui Zhu
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
cites work: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
cites work: Higher Torsion in H-Spaces
cites work: Collapsing Riemannian manifolds while keeping their curvature bounded. I
cites work: The topology of four-dimensional manifolds
cites work: Q3976707
cites work: A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters
cites work: Groups of polynomial growth and expanding maps. Appendix by Jacques Tits
cites work: Manifolds near the boundary of existence
cites work: A Pinching Theorem for Homotopy Spheres
cites work: Geometric finiteness theorems via controlled topology
cites work: A generalized sphere theorem
cites work: The fundamental equations of a submersion
cites work: Q3138486
cites work: An excess sphere theorem
cites work: Generalized Poincaré's conjecture in dimensions greater than four
cites work: Q3942069
cites work: Collapsing and pinching under a lower curvature bound
full work available at URL: https://doi.org/10.1007/bf02921805
OpenAlex ID: W2064625157
description: scientific article; zbMATH DE number 828779