Stability of \(e^\epsilon\)-Lipschitz and co-Lipschitz maps in Gromov-Hausdorff topology
type: Item
instance of: scholarly article
title: Stability of \(e^\epsilon\)-Lipschitz and co-Lipschitz maps in Gromov-Hausdorff topology
zbMATH Open document ID: 1267.53045
author: Xiaochun Rong
author: Shicheng Xu
published in: Advances in Mathematics
publication date: 2012-09-19T00:00:00Z
review text: The notion of Gromov-Hausdorff distance between compact metric spaces and the related Gromov-Hausdorff convergence is a basic one in Riemannian geometry and geometry of more general metric spaces. Since limits of Riemannian manifolds are often not smooth anymore, many concepts and results for Riemannian manifolds have been generalized to suitable metric spaces, for example, Alexandrov spaces. One important problem is concerned with stability of maps between these more general spaces. One result of Perelman states that sufficiently small Gromov-Hausdorff approximations between Alexandrov spaces can be approximated by homeomorphisms. This was reformulated by V. Kapovitch as a special case of a stability result for a submetry from an Alexandrov space to a Riemannian manifold, where the notion of submetry between metric spaces is an analog of the Riemannian submersion between Riemannian manifolds. In this paper, the author proves the stability of \(e^\epsilon\)-Lipschitz and co-Lipschitz maps between Alexandrov spaces and Lipschitz manifolds. One of the motivation for this generalization is that submetries is rare but \(e^\epsilon\)-Lipschitz and co-Lipschitz maps occur naturally.
reviewed by: Lizhen Ji
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 6084081
zbMATH Keywords: Gromov-Hausdorff topology
zbMATH Keywords: Alexandrov spaces
zbMATH Keywords: stability of fiber bundles
MaRDI profile type: Publication
full work available at URL: j.aim.2012.05.018
OpenAlex ID: W2065575205
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DOI: 10.1016/J.AIM.2012.05.018
description: scientific article; zbMATH DE number 6084081