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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References

  1. Bao D., Chern S.-S., Shen Z., An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, 2000.
  2. Busemann H., The geometry of geodesics, Academic Press Inc., New York, N.Y., 1955.
  3. Egloff D., Uniform Finsler Hadamard manifolds, Ann. Inst. H. Poincaré Phys. Théor. 66 (1997), 323-357.
  4. Innami N., Differentiability of Busemann functions and total excess, Math. Z. 180 (1982), 235-247.
  5. Innami N., On the terminal points of co-rays and rays, Arch. Math. (Basel) 45 (1985), 468-470.
  6. Lewis G.M., Cut loci of points at infinity, Pacific J. Math. 43 (1972), 675-690.
  7. Nasu Y., On asymptotes in a metric space with non-positive curvature, Tôhoku Math. J. 9 (1957), 68-95.
  8. Ohta S.-I., Splitting theorems for Finsler manifolds of nonnegative Ricci curvature, J. Reine Angew. Math. 700 (2015), 155-174.
  9. Shen Z., Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001.
  10. Shiohama K., Topology of complete noncompact manifolds, in Geometry of Geodesics and Related Topics (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 3, North-Holland, Amsterdam, 1984, 423-450.
  11. Shiohama K., Shioya T., Tanaka M., The geometry of total curvature on complete open surfaces, Cambridge Tracts in Mathematics, Vol. 159, Cambridge University Press, Cambridge, 2003.
  12. Tanaka M., Sabau S.V., The cut locus and distance function from a closed subset of a Finsler manifold, Houston J. Math., to appear, arXiv:1207.0918.

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