### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 036, 12 pages      arXiv:1504.03921      https://doi.org/10.3842/SIGMA.2016.036

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

Sorin V. Sabau
School of Science, Department of Mathematics, Tokai University, Sapporo 005-8600, Japan

Received August 07, 2015, in final form April 06, 2016; Published online April 13, 2016

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.

Key words: Finsler manifolds; ray; co-ray (asymptotic ray); cut locus; co-points; distance function; Busemann function.

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