Abstract
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold , such as balls, horoballs, tubular neighbourhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in 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 , or of spiraling times around a closed geodesic in a closed such . 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.
Citation
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
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