Lengths of geodesics between two points on a Riemannian manifold

Let x x and y y be two (not necessarily distinct) points on a closed Riemannian manifold M n M^n . According to a well-known theorem by J.-P. Serre, there exist infinitely many geodesics between x x and y 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 k there are k k distinct geodesics of length ≤<!-- ≤ --> k diam ⁡<!-- ⁡ --> ( M n ) \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 x and y y of length not exceeding 2 n diam ⁡<!-- ⁡ --> ( M n ) 2n\operatorname {diam}(M^n) . Our second result is that if n = 2 n=2 and M 2 M^2 is diffeomorphic to S 2 S^2 , then for every k k every pair of points of M 2 M^2 can be connected by k k distinct geodesics of length less than or equal to ( 4 k 2 −<!-- − --> 2 k −<!-- − --> 1 ) diam ⁡<!-- ⁡ --> ( M 2 ) (4k^2-2k-1)\operatorname {diam}(M^2) .

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