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

TL;DR

This paper establishes lower bounds on the number of distinct geodesic segments connecting two points that maximize distance on manifolds with non-positive curvature, extending known results for negatively curved spaces.

Contribution

It provides new lower bounds for the number of connecting geodesic segments on non-positively curved manifolds, generalizing previous results from negative curvature cases.

Findings

At least 2n+1 geodesics in negatively curved manifolds for points at local distance maxima.

At least n+1 geodesics in non-positive curvature manifolds for similar points.

Results apply to complete Riemannian manifolds of dimension greater than one.

Abstract

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 $n+1$.