|
Geodesics avoiding subsets in Hadamard manifolds
Author(s):
Albert
Borbély
Journal:
Proc. Amer. Math. Soc.
138
(2010),
1085-1092.
MSC (2000):
Primary 53C22
Posted:
October 23, 2009
MathSciNet review:
2566573
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  ,  , be an  -hyperbolic (in the sense of Gromov) Hadamard manifold. Let us assume that we are given a family of disjoint convex subsets and a point  outside these sets. It is shown that if one shrinks these sets by the constant  , then it is possible to find a complete geodesic through  that avoids the shrunk sets.
References:
-
- [B]
- A. Borbély, A note on the unclouding the sky of negatively curved manifolds, Bull. Australian Math. Soc. 77 (2008), 413-424. MR 2454972
- [BO]
- K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fundamenta Mathematicae 20 (1933), 177-190.
- [BS]
- S. Buyalo, V. Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society, Zurich (2007). MR 2327160 (2009a:53068)
- [BSW]
- S. Buyalo, V. Schroeder, M. Walz, Geodesics avoiding open subsets in surfaces of negative curvature, Ergodic Theory Dynam. Sys. 20 (2000), 991-1006. MR 1779390 (2002e:37045)
- [BT]
- R. Bott, L.W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York-Berlin (1986). MR 658304 (83i:57016)
- [LS]
- L. Lyusternik, S. Shnirel'man, Topological Methods in Variational Methods (Russian), Issledowatelskii Institut Matematiki i Mechanici, Moscow (1930).
- [M]
- Jiri Matoušek, Using the Borsuk-Ulam Theorem, Springer Universitext, 2nd corrected printing, Springer-Verlag, Berlin (2008). MR 1988723 (2004i:55001)
- [PP1]
- J. Parkkonen, F. Paulin, Unclouding the sky of negatively curved manifolds, Geom. and Funct. Anal. 15 (2005), 491-533. MR 2153908 (2006i:53048)
- [PP2]
- J. Parkkonen, F. Paulin, Prescribing the behavior of geodesics in negative curvature, arXiv:07062579v1 (2007).
- [S]
- V. Schroeder, Bounded geodesics in manifolds of negative curvature, Math. Z. 235 (2000), 817-828. MR 1801585 (2001m:53065)
- [W]
- M. Walz, Invariant subsets of the geodesic flow on negatively curved manifolds, Thesis, Zurich (1998).
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
53C22
Retrieve articles in all Journals with
MSC (2000):
53C22
Additional Information:
Albert
Borbély
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Email:
borbely.albert@gmail.com
DOI:
10.1090/S0002-9939-09-10095-3
PII:
S 0002-9939(09)10095-3
Keywords:
Convex sets,
negative curvature,
geodesics
Received by editor(s):
June 12, 2008
Posted:
October 23, 2009
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
