ON THE NUMBER OF GEODESIC SEGMENTS CONNECTING 2 POINTS ON MANIFOLDS OF NONPOSITIVE CURVATURE

We prove that on a complete Riemannian manifold M of dimension n with sectional curvature K-M < 0, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least 2n + 1 geodesic segments. A simpler version of the argument shows that if one of the points is fixed and K-M less than or equal to 0 then the two points are connected by at least n + 1 geodesic segments. The proof uses mainly the convexity properties of the distance unction for metrics of negative curvature.