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ISSN 1079-6762

 

Lengths of geodesics between two points on a Riemannian manifold


Authors: Alexander Nabutovsky and Regina Rotman
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 13-20
MSC (2000): Primary 53C23, 53C22; Secondary 58E10, 53C45
Published electronically: February 9, 2007
MathSciNet review: 2285762
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ x$ and $ y$ be two (not necessarily distinct) points on a closed Riemannian manifold $ M^n$. According to a well-known theorem by J.-P. Serre, there exist infinitely many geodesics between $ x$ and $ y$. It is obvious that the length of a shortest of these geodesics cannot exceed the diameter of the manifold. But what can be said about the lengths of the other geodesics? We conjecture that for every $ k$ there are $ k$ distinct geodesics of length $ \le k\;diam(M^n)$. This conjecture is evidently true for round spheres and is not difficult to prove for all closed Riemannian manifolds with non-trivial torsion-free fundamental groups. In this paper we announce two further results in the direction of this conjecture. Our first result is that there always exists a second geodesic between $ x$ and $ y$ of length not exceeding $ 2n\;diam(M^n)$. Our second result is that if $ n=2$ and $ M^2$ is diffeomorphic to $ S^2$, then for every $ k$ every pair of points of $ M^2$ can be connected by $ k$ distinct geodesics of length less than or equal to $ (4k^2-2k-1)diam(M^2)$.


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Additional Information

Alexander Nabutovsky
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, M5S2E4, Canada, and Department of Mathematics, McAllister Bldg., The Pennsylvania State University, University Park, Pennsylvania 16802
Email: alex@math.toronto.edu

Regina Rotman
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, M5S2E4, Canada, and Department of Mathematics, McAllister Bldg., The Pennsylvania State University, University Park, Pennsylvania 16802
Email: rina@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-07-00169-2
PII: S 1079-6762(07)00169-2
Received by editor(s): September 14, 2006
Published electronically: February 9, 2007
Communicated by: Dmitri Burago
Article copyright: © Copyright 2007 American Mathematical Society