P. Horja, ON THE NUMBER OF GEODESIC SEGMENTS CONNECTING 2 POINTS ON MANIFOLDS OF NONPOSITIVE CURVATURE, Transactions of the American Mathematical Society, 349(12), 1997, pp. 5021-5030
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.