Short geodesic segments between two points on a closed Riemannian manifold
type: Item
instance of: scholarly article
title: Short geodesic segments between two points on a closed Riemannian manifold
zbMATH Open document ID: 1180.53043
author: Alexander Nabutovsky
author: Regina Rotman
published in: Geometric and Functional Analysis. GAFA
publication date: 2009-11-06T00:00:00Z
review text: Inspired by a nice theorem of J.-P. Serre (`any two points of a closed Riemannian manifold can be connected by infinitely many geodesics'), in an earlier paper [Electron. Res. Announc. Am. Math. Soc. 13, 13--20, electronic only (2007; Zbl 1113.53026)], the authors formulated the following Conjecture. If \(M\) is an \(n\)-dimensional closed Riemannian manifold of diameter \(d\), then there exists a function \(f(k,n)\) such that any two points of \(M\) can be connected by at least \(k\) geodesics of length at most \(f(k,n)d\). Concerning this conjecture, in the present paper it is shown that if the closed Riemannian manifold in question is homotopy equivalent to the product of \(S^2\) and an arbitrary \((n-2)\)-dimensional manifold, then there exist at least \(k\) distinct geodesics between any two points of \(M\) of length at most \(20k!d\), for every positive integer \(k\).
reviewed by: József Szilasi
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 5627020
zbMATH Keywords: closed Riemannian manifolds
zbMATH Keywords: length of geodesics
MaRDI profile type: Publication
full work available at URL: s00039-009-0004-8
OpenAlex ID: W1968407298
Wikidata QID: Q115389246
cites work: A Zoll counterexample to a geodesic length conjecture
cites work: Q4039749
cites work: Area and the length of the shortest closed geodesic
cites work: Q4463916
cites work: The Morse landscape of a Riemannian disk
cites work: Filling Riemannian manifolds
cites work: Q4181002
cites work: Lengths of geodesics between two points on a Riemannian manifold
cites work: The length of the second shortest geodesic
cites work: Length of geodesics on a two-dimensional sphere
cites work: The length of a shortest geodesic loop at a point
cites work: Global and local volume bounds and the shortest geodesic loops
DOI: 10.1007/S00039-009-0004-8
description: scientific article; zbMATH DE number 5627020