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
DOI:
https://doi.org/10.1090/S1079-6762-07-00169-2
Published electronically:
February 9, 2007
MathSciNet review:
2285762
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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 $\le k\operatorname{diam}(M^n)$. 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 $2n\operatorname{diam}(M^n)$. 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 $(4k^2-2k-1)\operatorname{diam}(M^2)$.
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[NR1]NR1 A. Nabutovsky, R. Rotman, The length of geodesics on a two-dimensional sphere, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-138, available at www.mpim-bonn.mpg.de/preprints/retrieve.
[NR2]NR2 A. Nabutovsky, R. Rotman, The length of a second shortest geodesic, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-113, available at www.mpim-bonn.mpg.de/preprints/retrieve.
[R]R R. Rotman, The length of a shortest geodesic loop at a point, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-105, available at www.mpim-bonn.mpg.de/preprints/retrieve.
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[NR1]NR1 A. Nabutovsky, R. Rotman, The length of geodesics on a two-dimensional sphere, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-138, available at www.mpim-bonn.mpg.de/preprints/retrieve.
[NR2]NR2 A. Nabutovsky, R. Rotman, The length of a second shortest geodesic, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-113, available at www.mpim-bonn.mpg.de/preprints/retrieve.
[R]R R. Rotman, The length of a shortest geodesic loop at a point, Preprint of the Max-Planck-Institut für Mathematik MPIM 2006-105, available at www.mpim-bonn.mpg.de/preprints/retrieve.
[S]S J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. Math. 54 (1951), 425–505.
[Schw]Schw A. S. Schwarz, Geodesic arcs on Riemannian manifolds, Uspekhi Mat. Nauk 13 (1958), no. 6, 181–184. (Russian)
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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
MR Author ID:
659650
Email:
rina@math.toronto.edu
Received by editor(s):
September 14, 2006
Published electronically:
February 9, 2007
Communicated by:
Dmitri Burago
Article copyright:
© Copyright 2007
American Mathematical Society