Bulletin of the Australian Mathematical Society

Research Article

A NOTE ON THE UNCLOUDING THE SKY OF NEGATIVELY CURVED MANIFOLDS

ALBERT BORBÉLYa1

a1 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)

Abstract

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}axs2208A and for any point pxs2208Mxs22C3 axs2208ACa 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.

(Received August 15 2007)

1991 Mathematics subject classification

  • 53C22

Keywords and phrases

  • convex sets;
  • negative curvature;
  • geodesics