Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-29T01:13:42.688Z Has data issue: false hasContentIssue false

A NOTE ON THE UNCLOUDING THE SKY OF NEGATIVELY CURVED MANIFOLDS

Published online by Cambridge University Press:  01 June 2008

ALBERT BORBÉLY*
Affiliation:
Kuwait University, Faculty of Science, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 13060, Kuwait (email: borbely@mcs.sci.kuniv.edu.kw)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of finding geodesics that avoid certain obstacles in negatively curved manifolds has been studied in different situations. In this note we give a generalization of the unclouding theorem of J. Parkkonen and F. Paulin: there is a constant s0=1.534 such that for any Hadamard manifold M with curvature ≤−1 and for any family of disjoint balls or horoballs {Ca}aA and for any point pM−⋃ aACa if we shrink these balls uniformly by s0 one can always find a geodesic ray emanating from p that avoids the shrunk balls. It will be shown that in the theorem above one can replace the balls by arbitrary convex sets.

MSC classification

Type
Research Article
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Buyalo, S., Schroeder, V. and Walz, M., ‘Geodesics avoiding subsets in surfaces of negative curvature’, Ergod. Th. & Dynam. Sys. 20 (2000), 9911006.Google Scholar
[2]Chavel, I., Riemannian Geometry — A Modern Introduction (Cambridge University Press, Cambridge, 1993).Google Scholar
[3]Parkkonen, J. and Paulin, F., Geom. Funct. Anal. 15 (2005), 491533.Google Scholar
[4]Parkkonen, J. and Paulin, F., ‘Prescribing the behavior of geodesics in negative curvature’, Preprint, 2007, arXiv:07062579v1.Google Scholar
[5]Schroeder, V., ‘Bounded geodesics in manifolds of negative curvature’, Math. Z. 235 (2000), 817828.Google Scholar
[6]Walz, M., ‘Invariant subsets of the geodesic flow on negatively curved manifolds’, Thesis, Zurich, 1998.Google Scholar