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

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

1995-02-28

arxiv.org/abs/dg-ga/9502009v1

Mathematics

Differential Geometry

14 pages, AMS-LaTex

Scientific paper

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 $n+1$.

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