Lengths of geodesics between two points on a Riemannian manifold

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