Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem
type: Item
instance of: scholarly article
title: Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem
zbMATH Open document ID: 1039.53042
author: Alexander Nabutovsky
author: Regina Rotman
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2003-10-16T00:00:00Z
review text: Summary: In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. Related questions that will also be studied are the following: given a contractible \(k\)-dimensional sphere in \(M^n\), how ''fast'' can this sphere be contracted to a point, if \(\pi_i(M^n)=\{0\}\) for \(1\leq i<k\). That is, what is the maximal length of the trajectory described by a point of a sphere under an ''optimal'' homotopy? Also, what is the ''size'' of the smallest non-contractible \(k\)-dimensional sphere in a \((k-1)\)-connected manifold \(M^n\) providing that \(M ^n\) is not \(k\)-connected?
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1993526
zbMATH Keywords: shortest closed geodesic
zbMATH Keywords: sectional curvature
zbMATH Keywords: volume
zbMATH Keywords: diameter
zbMATH Keywords: smallest non-contractible \(k\)-dimensional sphere
MaRDI profile type: Publication
full work available at URL: s10097-003-0051-7
OpenAlex ID: W2042537709
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DOI: 10.1007/S10097-003-0051-7
description: scientific article; zbMATH DE number 1993526