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A NOTE ON THE UNCLOUDING THE SKY OF NEGATIVELY CURVED MANIFOLDS

Part of: Global differential geometry

Published online by Cambridge University Press:  01 June 2008

ALBERT BORBÉLY
ALBERT BORBÉLY*
Affiliation:
Kuwait University, Faculty of Science, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 13060, Kuwait (email: [email protected])
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Abstract

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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}a∈A and for any point p∈M−⋃ a∈ACa 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.

Keywords

convex setsnegative curvaturegeodesics

MSC classification

Secondary: 53C22: Geodesics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 3 , June 2008 , pp. 413 - 424
Copyright
Copyright © 2008 Australian Mathematical Society

References

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