Lengths of geodesics between two points on a Riemannian manifold

Nabutovsky, Alexander¹ ; Rotman, Regina¹

¹ Department of Mathematics, University of Toronto, Toronto, Ontario, M5S2E4, Canada, and Department of Mathematics, McAllister Bldg., The Pennsylvania State University, University Park, Pennsylvania 16802

Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 13-20.

Résumé

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 $(4k^2-2k-1)\operatorname{diam}(M^2)$.

Détail

DOI : 10.1090/S1079-6762-07-00169-2

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