  In this paper we show that on a complete Riemannian manifold of negative
curvature and dimension $n>1$ every two points which realize a local maximum
for the distance function are connected by at least $2n+1$ geometrically
distinct geodesic segments (i.e. length minimizing). Using a similar method, we
obtain that in the case of non-positive curvature, for every two points with
the same property as above the number of connecting distinct geodesic segments
is at least $n+1$.
