On the number of geodesic segments connecting two points on manifolds of non-positive curvature

Abstract

We prove that on a complete Riemannian manifold M M of dimension n n with sectional curvature K M > 0 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 2 n + 1 2n+1 geodesic segments. A simpler version of the argument shows that if one of the points is fixed and K M ≤ 0 K_M \leq 0 then the two points are connected by at least n + 1 n+1 geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.