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

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 m...

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Bibliographic Details
Published in:arXiv.org 1995-02
Main Author: Horja, Paul
Format: Article
Language:English
Subjects:
Curvature
Riemann manifold
Segments
Online Access:Get full text
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Summary: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 \(n+1\).
ISSN:2331-8422