LENGTH OF GEODESICS ON A TWO-DIMENSIONAL SPHERE By ALEXANDER NABUTOVSKY and REGINA ROTMAN

Let M be an arbitrary Riemannian manifold diffeomorphic to S2. Let x, y be two arbitrary points of M. We prove that for every k = 1, 2, 3, . . . there exist k distinct geodesics between x and y of length less than or equal to (4k2 − 2k − 1)d, where d denotes the diameter of M. To prove this result we demonstrate that for every Riemannian metric on S2 there are two (not mutually exclusive) possibilities: either every two points can be connected by many “short” geodesics of index 0, or the resulting Riemannian sphere can be swept-out by “short meridians”. 1. Main results. Here are the main results of the present paper: THEOREM 1. Let M be an arbitrary Riemannian manifold diffeomorphic to S2, and x, y be two arbitrary points of M. Denote the diameter of M by d. (Recall that the diameter of a compact Riemannian manifold is, by definition, the maximal distance between two points on the manifold.) For every positive integer k there exist at least k distinct geodesics starting at x and ending at y of length not exceeding (4k2 − 2k − 1)d. THEOREM 1.1. For every point x on a Riemannian manifold M diffeomorphic to S2 and any k = 1, 2, 3, . . . there exist at least k nontrivial geodesic loops based at x of length at most (4k2 + 2k)d. Theorem 1.1 deals with a particular case of Theorem 1 when y = x. In this case the shortest geodesic between x and y is trivial, and all other geodesics starting and ending at x are nontrivial geodesic loops based at x. Thus, we can apply Theorem 1 for k + 1 and obtain an upper bound for the length of the kth nontrivial geodesic loop based at x. However, the upper bound provided by Theorem 1.1 is somewhat better. Theorem 1 is a result in the direction of our conjecture made in [NR1]. Recall, that for every two points x in a closed Riemannian manifold M there exists an infinite set of distinct geodesics connecting x and y. (This is a well-known theorem of J. P. Serre, [Se].) In [NR1] we observed that if M is a nonsimply Manuscript received March 21, 2007. Research of the first author supported in part by an NSERC Discovery grant and NSF grants DMS0706803 and DMS-0405954; research of the second author supported in part by an NSERC University Faculty award, NSERC Discovery Grant and NSF grant DMS-0604113. American Journal of Mathematics 131 (2009), 0–00. c © 2009 by The Johns Hopkins University Press.