Effective universal coverings and local minima of the length functional on loop spaces
type: Item
instance of: scholarly article
title: Effective universal coverings and local minima of the length functional on loop spaces
zbMATH Open document ID: 1203.53039
author: Alexander Nabutovsky
published in: Geometric and Functional Analysis. GAFA
publication date: 2010-10-14T00:00:00Z
review text: The author describes a construction of effective universal coverings of Riemannian manifolds and of path metric spaces. Moreover, he gives two applications of this construction. In the first application the author demonstrates that if the length functional on the space \(\Omega_xM^n\) of all loops based at a point \(x\in M^n\) has a non-trivial very deep local minimum, then it has many deep local minima. The second application is a version of a theorem proved by M. Anderson. If \(M^n\) is a closed \(n\)-dimensional manifold with volume \(\geq v> 0\), diameter \(d\) and Ricci curvature \(\geq -(n-1)\) then there exist explicitly given positive \(\varepsilon = \varepsilon(n,v,d)\) and \(N=N(n,v,d)\) such that for each closed curve \(\gamma\) of length smaller than \(\varepsilon\) one of its first \(N\) iterates is contractible. The author finds an explicit upper bound for the length of curves in homotopy contracting one of the first \(N\) iterates of \(\gamma\).
reviewed by: Jan Kurek
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 58D15
zbMATH DE Number: 5800300
zbMATH Keywords: effective universal coverings
zbMATH Keywords: geodesic loops
zbMATH Keywords: loop spaces
zbMATH Keywords: Ricci curvature
MaRDI profile type: Publication
full work available at URL: s00039-010-0067-6
OpenAlex ID: W1988882890
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DOI: 10.1007/S00039-010-0067-6
description: scientific article; zbMATH DE number 5800300