6533b7d1fe1ef96bd125cc21
RESEARCH PRODUCT
Prescribing the behaviour of geodesics in negative curvature
Jouni ParkkonenFrédéric Paulinsubject
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 functiongeodesicgeodesics11J06description
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 described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2010-01-01 |