The Co-Points of Rays are Cut Points of Upper Level Sets for Busemann Functions

TL;DR

This paper explores the relationship between co-rays, co-points, and upper level sets of Busemann functions in Finsler manifolds, revealing new insights into their structure and properties, including for Riemannian cases.

Contribution

It characterizes the co-point set as the cut locus of upper level sets of Busemann functions and establishes structural properties, some of which are new even for Riemannian manifolds.

Findings

Co-rays contain geodesic segments to upper level sets.

Co-point set is characterized as the cut locus of these level sets.

On surfaces, the co-point set forms a local tree.

Abstract

We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut locus. We point out that some of our findings, in special the relation of co-point set to the upper lever sets, are new even for Riemannian manifolds.