The fundamental group of compact manifolds without conjugate points
type: Item
instance of: scholarly article
title: The fundamental group of compact manifolds without conjugate points
zbMATH Open document ID: 0608.53038
DOI: 10.1007/BF02621908
author: Christopher B. Croke
author: Viktor Schroeder
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140046
review text: A complete Riemannian manifold M has no conjugate points if every Jacobi vector field Y along a geodesic \(\gamma\) of M vanishes at most once on \(\gamma\). Equivalently, a manifold M has no conjugate points if the universal Riemannian cover \(\tilde M\) admits a unique geodesic joining any two given distinct points. Manifolds with nonpositive sectional curvature are the most common examples of manifolds without conjugate points. Consequently, it is challenging to extend geometric properties of nonpositively curved manifolds to the larger class of manifolds without conjugate points. These extensions are often difficult since most of the convexity properties of nonpositively curved manifolds are absent in a general manifold without conjugate points. The outstanding problem of this type is the Hopf problem to show that any n-torus without conjugate points is flat. A closely related direction of research is to determine algebraic properties of the fundamental group of a manifold M without conjugate points; such a manifold M is a K(\(\pi\),1)- space and the fundamental group contains the homotopy of M. In particular one may consider properties expressible in terms of the word length metric. If M is compact then its fundamental group \(\Gamma\) is finitely generated. If \(S=\{\gamma_ 1,...,\gamma_ p\}\) is a fixed set of generators for \(\Gamma\), then one may define a word length metric \(| |\) by setting \(| \phi |\), for any element \(\phi\) of \(\Gamma\), to be the minimum length of a word in the generators \(\{\gamma_ i\}\) and their inverses that represents \(\phi\). Different generating sets \(S_ 1,S_ 2\) define equivalent word length metrics \(| |_ 1\) and \(| |_ 2\); that is, 1/c\(\leq | \phi |_ 1/| \phi |_ 2\leq c\) for some \(c>0\) and all \(\phi\in \Gamma\). A finitely generated subgroup \(\Gamma_ 0\) in \(\Gamma\) is called straight if the word length metric on \(\Gamma_ 0\) is equivalent to the restriction of the word length metric on \(\Gamma\) to \(\Gamm…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 3984055
zbMATH Keywords: no conjugate points
zbMATH Keywords: Hopf problem
zbMATH Keywords: fundamental group
zbMATH Keywords: K(\(\pi \) ,1)-space
zbMATH Keywords: word length metric
zbMATH Keywords: geodesics
zbMATH Keywords: Busemann functions
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3984055