
SIGMA 12 (2016), 036, 12 pages arXiv:1504.03921
https://doi.org/10.3842/SIGMA.2016.036
The CoPoints of Rays are Cut Points of Upper Level Sets for Busemann Functions
Sorin V. Sabau
School of Science, Department of Mathematics, Tokai University, Sapporo 0058600, Japan
Received August 07, 2015, in final form April 06, 2016; Published online April 13, 2016
Abstract
We show that the corays to a ray in a complete noncompact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the copoint set to a ray as the cut locus of such level sets. The structure theorem of the copoint 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 copoint set to the upper lever sets, are new even for Riemannian manifolds.
Key words:
Finsler manifolds; ray; coray (asymptotic ray); cut locus; copoints; distance function; Busemann function.
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