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

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.