Short geodesic segments on closed Riemannian manifolds

A well-known result of J. P. Serre states that for an arbitrary pair of points on a closed Riemannian manifold there exist infinitely many geodesics connecting these points. A natural question is whether one can estimate the length of the k-th geodesic in terms of the diameter of a manifold. We will demonstrate that given a pair of points p, q on a closed Riemannian manifold of dimension n and diameter d, there always exist at least k geodesics of length at most 4nk^2d connecting them. We will also demonstrate that for any two points of a manifold that is diffeomorphic to the 2-sphere there always exist at least k geodesics between them of length at most 24kd. (Joint with A. Nabutovsky)