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2010 Prescribing the behaviour of geodesics in negative curvature
Jouni Parkkonen, Frédéric Paulin
Geom. Topol. 14(1): 277-392 (2010). DOI: 10.2140/gt.2010.14.277

Abstract

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighbourhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much into) them. Several applications are given, including a definite improvement of the unclouding problem of our paper [Geom. Func. Anal. 15 (2005) 491–533], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt–Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.

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Jouni Parkkonen. Frédéric Paulin. "Prescribing the behaviour of geodesics in negative curvature." Geom. Topol. 14 (1) 277 - 392, 2010. https://doi.org/10.2140/gt.2010.14.277

Information

Received: 1 June 2007; Revised: 21 July 2009; Accepted: 15 April 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1191.53026
MathSciNet: MR2578306
Digital Object Identifier: 10.2140/gt.2010.14.277

Subjects:
Primary: 11J06 , 52A55 , 53C22
Secondary: 53D25

Keywords: geodesics , Hall ray , horoballs , Lagrange spectrum , negative curvature

Rights: Copyright © 2010 Mathematical Sciences Publishers

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