Search results for "Closed geodesic"

showing 2 items of 2 documents

Prescribing the behaviour of geodesics in negative curvature

2010

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods 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 in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], 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 desc…

Mathematics - Differential GeometryhoroballsPure mathematicsGeodesicDisjoint setsLagrange spectrum52A5501 natural sciences53C22Mathematics - Metric Geometry0103 physical sciences0101 mathematicshoroball[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]MathematicsFinite volume methodHall rayAMS : 53 C 22 11 J 06 52 A 55 53 D 25Mathematics - Number Theory010102 general mathematicsnegative curvatureRiemannian manifold[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]Closed geodesic53D25[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Totally geodesic010307 mathematical physicsGeometry and TopologyNegative curvatureMathematics::Differential GeometryConvex functiongeodesicgeodesics11J06
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Appendix: Diophantine Approximation on Hyperbolic Surfaces

2002

In this (independent) appendix, we study the Diophantine approximation properties for the particular case of the cusped hyperbolic surfaces, in the spirit of Sect. 2 (or [11]), and the many still open questions that arise for them. We refer to [9], [10]for fundamental results and further developments. We study in particular the distance to a cusp of closed geodesics on a hyperbolic surface.

Surface (mathematics)Cusp (singularity)Pure mathematicsGeodesicDiophantine setMathematics::Number TheoryDiophantine equationMathematical analysisHyperbolic manifoldDiophantine approximationMathematics::Geometric TopologyMathematicsClosed geodesic
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