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

1995/02/28 by Paul Horja, Horja, Paul
Engineering · Mathematics · #3D Shape Modeling and Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #dg-ga #math.DG

paper · pdf · doi:10.48550/arxiv.dg-ga/9502009

14 pages, AMS-LaTex

arxiv created 1995/02/28 · openalex publication_date 1995/02/28 · arxiv updated 2016/08/31 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

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.

Related

• Lengths of geodesics between two points on a Riemannian manifold
• Minimizing geodesic nets and critical points of distance
• Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below
• A quadratic lower bound for the number of minimal geodesics
• Length spectrum of geodesic loops in manifolds of nonpositive curvature