On a class of Ricci-quadratic Finsler metrics

Document Type : Original Article

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran E-mail: ashirafkan@umz.ac.ir

Abstract

Let F be a (reversible) Finsler metric on a Riemannian space (M, α) of positive (or negative) sectional curvature. Suppose that the Ricci curvature of F is horizontally constant along Finslerian geodesics. Then we show that F is a Ricci-quadratic Finsler metric.

Keywords

References

  • 1. H. Akbar-Zadeh, Sur les espaces de Finsler a` courbures sectionnelles constantes, Acad.
    Roy. Belg. Bull. Cl. Sci. (5), 80(1988), 271-322.
  • 2. H. Akbar-Zadeh, Generalized Einstein manifolds, J. Geom. Phys. 17 (1995) 342-380.
  • 3. S. B´acs´o and M. Matsumoto, Finsler spaces with h-curvature tensor H dependent on
    position alone, Publ. Math. Debrecen, 55(1999), 199-210.
  • 4. S. B´acs´o and M. Matsumoto, Randers spaces with the h-curvature tensor H dependent
    on position alone, Publ. Math. Debrecen. 57(2000), 185-192.
  • 5. D. Bao, C. Robles and Z. Shen, Zermelo Navigation on Riemannian manifolds, J. Diff.
    Geom. 66(2004), 391-449.
  • 6. D. Bao and Z. Shen, Finsler metrics of constant positive curvature on the Lie group S
    3,J. London Math. Soc. 66(2002), 453-467.
  • 7. S. W. Hawking, American Astronomical Society 544 (1966).
  • 8. H. Isikawa, Note on Finslerian Relativity, J. Math. Phys. 22 No.5 (1981), 995.
  • 9. B. Li and Z. Shen, On Randers metrics of quadratic Riemann curvature, Int. Jour. Math.
    20(2009), 369-376.
  • 10. G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev.
    59(1941), 195-199.
  • 11. B. Bidabad and M. Rafie-Rad, Pure pursuit navigation on Riemannian manifolds, Nonlinear Analysis, 10(2009), 1265-1269.
  • 12. D. Latifi and A. Razavi, On homogeneous Finsler spaces, Rep. Math. Phys. 57 (2006)
    357-366.
  • 13. M. Matsumoto, A slope of a hill is a Finsler surface with respect to a time measure, J.
    Math. Kyoto. Univ. 29(1980), 17-25.
  • 14. X. Mo, On the non-Riemannian quantity H of a Finsler metric, Differential Geometry
    and its Applications 27 (2009), 7-14.
  • 15. S. Numata, On Landsberg spaces of scalar curvature, J. Korea Math. Soc. 12(1975),
    97-100.
  • 16. B. Najafi, B. Bidabad and A. Tayebi, On R-quadratic Finsler Metrics, Iranian Journal
    of Science and Technology, Trans A, Vol 31, No A4 (2007), 439-443.
  • 17. B. Najafi, Z. Shen and A. Tayebi, Finsler metrics of scalar flag curvature with special
    non-Riemannian curvature properties, Geom. Dedicata. 131(2008), 87-97.
  • 18. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
  • 19. Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001.
  • 20. Z. Shen, On R-quadratic Finsler spaces, Publ. Math. Debrecen. 58(2001), 263-274.
  • 21. H. Shimada and S.V. Sabau, An introduction to Matsumoto metric, Nonlinear Analysis:
    RWA. 63(2005) 165-168.
  • 22. G. Stephenson, La g´eometrie de Finsler et les th´eories du champ unifi´e, Annales de
    l’I.H.P, tome. 15, no.3, (1957), 205-215.
  • 23. P. C. Stravinos, On the Generalized Metric Structure of Space-Time: Finslerian
    Anisotropic Gravitational Field, Journal of Physics: Conference Series. 8 (2005), 49-57.
  • 24. P. C. Stavrinos and F. I. Diakogiannis, Gravitation and Cosmology, 10(4) (2005) 1-11.
  • 25. E.S. Sevim and M. Gabrani, On H-curvature of Finsler warped product metrics, Journal
    of Finsler Geometry and its Applications, 1(1) (2020), 37-44.
  • 26. A. Tayebi, E. Peyghan and H. Sadeghi, On Matsumoto-type Finsler metrics, Nonlinear
    Analysis: RWA, 13(2012), 2556-2561.
  • 27. S.I. Vacaru, Locally Anisotropic Kinetic Processes and Thermodynamics in Curved
    Spaces, Annals of Physics, 290 (2001), 83-123.
Journal of Finsler Geometry and its Applications
Volume 2, Issue 1
July 2021
Pages 96-107

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