On conformally flat square-root (α,β)-metrics

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Victoriei 76 North University, Center of Baia Mare, Technical University of Cluj Napoca, 430122 Baia Mare, Romania. Laurian.PISCORAN@mi.utcluj.ro

2 Department of Mathematics, Faculty of science, University of Qom, Iran. marzeia.amini@gmail.com

Abstract

Let F = √α(α + β) be a conformally flat square-root (α; β)-metric on a manifold M of dimension n ≥ 3, where α = √aij(x)yiyj is a Riemannian metric and β = bi(x)yi is a 1-form on M. Suppose that F has relatively isotropic mean Landsberg curvature. We show that F reduces to a Riemannian
metric or a locally Minkowski metric.

Keywords

References

  • 1. M. Amini, On conformally flat cubic (α, β)-metrics, Journal of Finsler Geometry and its
    Applications, 2(1) (2021), 75-85.
  • 2. G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rve.
    59 (1941), 195-199.
  • 3. P. Ginsparg, Applied conformal field theory, Ecole dEte de Physique Theorique, 1989.
  • 4. H. Shimada and V.S. Sabau, An introduction to Matsumoto metric, Nonlinear Analysis:
    RWA. 63(2005), 165-168.
  • 5. R.S. Ingarden, On the geometrically absolute optical representation in the electron microscope, Trav. Soc. Sci. Lett. Wrochlaw. Ser. B. 3 (1957), 60 pp.
  • 6. M. Matsumoto, Theory of Finsler spaces with m-th root metric, Publ. Math. Debrecen.
    49(1996), 135-155.
  • 7. P.L. Antonelli, R.S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler
    Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993.
  • 8. G.S. Asanov, Finslerian metric functions over product R × M and their potential applications, Rep. Math. Phys. 41(1998), 117-132.
  • 9. X. Cheng, H. Wang and M. Wang, (α, β)-metrics with relatively isotropic mean Landsberg
    curvature, Publ. Math. Debrecen, 72(2008), 475-485.
  • 10. S. S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientiflc, Singapore, 2005.
  • 11. M.S. Knebelman, Conformal geometry of generalized metric spaces, Proc. Nat Acad. Sci.
    USA. 15(1929), 376-379.
  • 12. B. Li and Z. Shen, On a class of weak Landsberg metrics, Science in China Series A,
    50(2007), 75-85.
  • 13. Z. Shen, On R-quadratic Finsler spaces, Publ. Math. Debrecen, 58(2001), 263-274.
  • 14. T. Tabatabaeifar, On generalized 4-th root Finsler metrics, Journal of Finsler Geometry
    and its Applications, 1(1) (2020), 54-59.
  • 15. A. Tayebi and M. Amini, Conformally flat 4-th root (α, β)-metrics with relatively
    isotropic mean Landsberg curvature, Mathematical Analysis and Convex Optimization,
    1(2) (2020), 25-34.
  • 16. A. Tayebi and B. Najafi, On m-th root Finsler metrics, J. Geom. Phys. 61(2011), 1479-
    1484.
  • 17. A. Tayebi and B. Najafi, On m-th root metrics with special curvature properties, C. R.
    Acad. Sci. Paris, Ser. I. 349(2011), 691-693.
  • 18. A. Tayebi and M. Razgordani, On conformally flat fourth root (α, β)-metrics, Differ.
    Geom. Appl. 62(2019), 253-266.
  • 19. A. Tayebi and M. Shahbazi Nia, A new class of projectively flat Finsler metrics with
    constant flag curvature K = 1, Differ. Geom. Appl. 41(2015), 123-133.
  • 20. G. Yang, On a class of Einstein-reversible Finsler metrics, Differ. Geom. Appl. 60(2018),
    80-103.
Journal of Finsler Geometry and its Applications
Volume 2, Issue 2
December 2021
Pages 89-102

Files

Share

How to cite

Statistics