Bounded geodesics in manifolds of negative curvature
type: Item
instance of: scholarly article
title: Bounded geodesics in manifolds of negative curvature
zbMATH Open document ID: 0990.53038
author: Viktor Schroeder
published in: Mathematische Zeitschrift
publication date: 2002-08-12T00:00:00Z
review text: This paper is stimulated by the following question: does the unit tangent bundle to a compact manifold \(M\) of negative sectional curvature contain a proper closed subset which projects onto \(M\) under the bundle projection and is invariant under the geodesic flow? Burns and Pollicott (unpublished) gave a positive answer if \(M\) has constant negative curvature. The present paper gives a positive answer if \(\dim(M)\geq 3\). The main theorem says that if \(M\) is a complete manifold of sectional curvature \(\leq -1\) and \(\dim(M)\geq 3\), then for any unit tangent vector \(v\) to \(M\) and any point \(x\) of \(M\), there is a complete geodesic through \(x\) whose tangent vector never comes close to \(v\) (with respect to the standard Sasaki metric). The proof also can be used to show that, if \(M\) is a finite volume complete manifold of pinched negative curvature and \(\dim(M)\geq 3\), then there is a complete geodesic through each point of \(M\) which is a bounded subset of \(M\).
reviewed by: Igor Belegradek
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 1559194
zbMATH Keywords: negative curvature
zbMATH Keywords: goedesic
MaRDI profile type: Publication
DOI: 10.1007/S002090000189
description: scientific article; zbMATH DE number 1559194