Lengths of geodesics between two points on a Riemannian manifold
Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 13-20
Cet article a éte moissonné depuis la source American Mathematical Society
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 $(4k^2-2k-1)\operatorname{diam}(M^2)$.
Affiliations des auteurs :
Nabutovsky, Alexander 1 ; Rotman, Regina 1
@article{10_1090_S1079_6762_07_00169_2,
author = {Nabutovsky, Alexander and Rotman, Regina},
title = {Lengths of geodesics between two points on a {Riemannian} manifold},
journal = {Electronic research announcements of the American Mathematical Society},
pages = {13--20},
year = {2007},
volume = {13},
doi = {10.1090/S1079-6762-07-00169-2},
url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-07-00169-2/}
}
TY - JOUR AU - Nabutovsky, Alexander AU - Rotman, Regina TI - Lengths of geodesics between two points on a Riemannian manifold JO - Electronic research announcements of the American Mathematical Society PY - 2007 SP - 13 EP - 20 VL - 13 UR - http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-07-00169-2/ DO - 10.1090/S1079-6762-07-00169-2 ID - 10_1090_S1079_6762_07_00169_2 ER -
%0 Journal Article %A Nabutovsky, Alexander %A Rotman, Regina %T Lengths of geodesics between two points on a Riemannian manifold %J Electronic research announcements of the American Mathematical Society %D 2007 %P 13-20 %V 13 %U http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-07-00169-2/ %R 10.1090/S1079-6762-07-00169-2 %F 10_1090_S1079_6762_07_00169_2
Nabutovsky, Alexander; Rotman, Regina. Lengths of geodesics between two points on a Riemannian manifold. Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 13-20. doi: 10.1090/S1079-6762-07-00169-2
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