On the Number of Geodesic Segments Connecting Two Points on Manifolds of Non-positive Curvature

1. Introduction Let M be a compact manifold Riemannian manifold of dimension n ≥ 2, with a metric of sectional curvature bounded above by χ ≤ 0 (non-positive curvature). In this paper we prove that in the case of negative curvature (χ < 0) on such manifolds there exist pairs of points connected by at least 2n + 1 geometrically distinct geodesic segments (i.e. length minimizing). A class of points which provide examples in this class are the points situated at distance equal to the diameter of the manifold. A simplified version of the method allows us to show that in the case of non-positive curvature (χ = 0) for any point there exist another point and n + 1 geometrically distinct geodesic segments connecting them. The essential ingredient in the proofs is the basic metric property of the spaces of non-positive and negative curvature to have their distance function convex and, in a sense that will be explained in the paper, even almost strictly convex. The results can also be seen as estimates for the " order " of the points in the cut locus for manifolds of non-positive curvature. In the case of positive curvature the situation changes: for the ellipsoid in IR 3 with axes of different lengths, the points at maximal distance are connected by two geodesic segments, but for the sphere by infinitely many geodesic segments. For the flat torus obtained as a quotient of IR 2 by a lattice not generated by two orthogonal vectors, the maximal " order " of the points in the cut locus is 3. Interesting is the situation for convex polyhedra in IR