On quintic (α,β)-metrics in Finsler geometry

Document Type : Original Article

Authors

1 Department of Mathematics Basic Sciences Faculty University of Bonab, Bonab, Iran

2 Department of Mathematics, Basic Sciences Faculty, University of Bonab, Bonab 5551395133, Iran.

Abstract

 In this paper, we study the class of quintic (α,β)-metrics. We show that every weakly Landsberg 5-th root (α,β)-metrics has vanishing S-curvature. Using it, we prove that a quintic (α,β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. Then, we show that a quintic (α,β)-metric satisfies Ξ = 0 if and only if S = 0.

Keywords


  • 1. P. L. Antonelli, R. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler
    Spaces with Applications in Physics and Biology, Springer Science and Busines Media. 48 (1993).
  • 2. D. Bao and S.S. Chern, A note on the Gauss-Bonnet theorem for Finsler spaces,
    Ann. Math. 143(1996), 233-252.
  • 3. D. Bao and Z. Shen, On the volume of unite tangent spheres in a Finsler space,
    Results in Math. 26(1994), 1-17.
  • 4. M. Matsumoto, On Finsler spaces with a cubic metric, Tensor, ns. 33 (1979), 153–
    162.
  • 5. M. Matsumoto and S. Numata, On Finsler spaces with a cubic metric, Tensor (N.S.)
    33(2) (1979), 153-162.
  • 6. B. Najafi and A. Tayebi, Some curvature properties of (α, β)-metrics, Bull. Math.
    Soc. Sci. Math. Roumanie. (2017), 277–291.
  • 7. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001.
  • 8. Z. Shen, Finsler manifolds with nonpositive flag curvature and constant S-curvature,
    Math. Z. 249(2005), 625-639.
  • 9. Z. Shen, On some non-Riemannian quantities in Finsler geometry, Canadian Math.
    Bull. 56(1) (2013), 184-193.
  • 10. H. Shimada, On Finsler spaces with roots metric L = mp
    ai1...im(x)yi1 yi2 ...yim,Tensor. ns. 33(1979), 365–372.
  • 11. A. Tayebi and B. Najafi, On m-th root Finsler metrics, J. Geom. Phys, 61(2011),
    1479–1484.
  • 12. A. Tayebi and B. Najafi, On m-th root metrics with special curvature properties,
    Comptes Rendus Mathematique. 349(11-12)(2011), 691–693.
  • 13. A. Tayebi, A. Nankali and E. Peyghan, Some properties of m-th root Finsler metrics,
    J. Contemporary. Math Analysis. 49(4) (2014), 184–193.
  • 14. Y. Yu and Y. You, On Einstein m-th root metrics, Differ. Geom. Appl. 28 (2010),
    290–294.