DOI QR코드

DOI QR Code

HARMONIC FINSLER MANIFOLDS WITH MINIMAL HOROSPHERES

  • Kim, Chang-Wan (Division of Liberal Arts and Sciences Mokpo National Maritime University)
  • Received : 2017.04.05
  • Accepted : 2018.05.16
  • Published : 2018.07.31

Abstract

In this paper we show that complete noncompact harmonic Finsler manifolds with minimal horospheres are flat.

Keywords

Finsler geometry;harmonic manifolds;minimal horospheres

References

  1. A. L. Besse, Manifolds All of Whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, 93, Springer-Verlag, Berlin, 1978.
  2. G. Besson, G. Courtois, and S. Gallot, Entropies et rigidites des espaces localement symetriques de courbure strictement negative, Geom. Funct. Anal. 5 (1995), no. 5, 731-799. https://doi.org/10.1007/BF01897050
  3. D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259-269. https://doi.org/10.1007/BF01896241
  4. D. Burago and S. Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. (2) 156 (2002), no. 3, 891-914. https://doi.org/10.2307/3597285
  5. E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 139-142. https://doi.org/10.1090/S0273-0979-1992-00293-8
  6. P. Foulon, Estimation de l'entropie des systemes lagrangiens sans points conjugues, Ann. Inst. H. Poincare Phys. Theor. 57 (1992), no. 2, 117-146.
  7. A. Freire and R. Mane, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math. 69 (1982), no. 3, 375-392. https://doi.org/10.1007/BF01389360
  8. C.-W. Kim, Finsler manifolds without conjugate points and with integral Ricci curvature, Israel J. Math. 189 (2012), 135-146. https://doi.org/10.1007/s11856-011-0129-y
  9. C.-W. Kim, Compact Finsler manifolds without focal points, Arch. Math. (Basel) 99 (2012), no. 4, 387-392. https://doi.org/10.1007/s00013-012-0435-6
  10. A. Ranjan and H. Shah, Harmonic manifolds with minimal horospheres, J. Geom. Anal. 12 (2002), no. 4, 683-694. https://doi.org/10.1007/BF02930658
  11. Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore, 2001.
  12. Z. I. Szabo, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom. 31 (1990), no. 1, 1-28. https://doi.org/10.4310/jdg/1214444087
  13. N. Zinov'ev, Examples of Finsler metrics without conjugate points: metrics of rotation, St. Petersburg Math. J. 20 (2009), 361-379. https://doi.org/10.1090/S1061-0022-09-01052-8