  Let x and y be two (not necessarily distinct) points on a closed Riemannian
manifold M of dimension n. According to a celebrated theorem by J.P. Serre
there exist infinitely many geodesics between x and y. The length of the
shortest of these geodesics is obviously less than the diameter of M. But what
can be said about the length of the other geodesics? We conjecture that for
every k there are k geodesics between x and y of length not exceeding kd, where
d denotes the diameter of M.This conjecture is obviously true for round spheres
and it is not difficult to prove it 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
the length of the second shortest geodesic between x and y does not exceed 2nd.
Our second result is that if n=2 and M is diffeomorphic to the two-dimensional
sphere, then for every k every two points on M can be connected by k geodesics
of length not exceeding $(k^2/2 + 3k/2 +2)d$.
