Weakly Douglas Finsler warped product metrics

Document Type : Original Article

Authors

1 Department of Mathematics, Faculty of Science, Urmia University Urmia, Iran.

2 Department of Mathematics, Istanbul Bilgi University, 34060, Eski Silahtaraga Elektrik Santrali, Kazim Karabekir Cad. No: 2/13 Eyupsultan, Istanbul, Turkey.

Abstract

Recent studies show that warped product manifolds are useful in differential geometry as well as in physics. The goal of this paper is to study on some projective invariants of a special product manifold with Finsler metrics arising from warped products. Firstly, we consider the class of weakly Douglas metrics, weaker notion of Douglas metrics, introduced by Atashafrouz, Najafi and Tayebi in [4]. We prove that every Finsler warped product manifold Mn (n ≥ 3) is weakly Douglas if and only if it is Douglas. Finally, under a certian condition, we show that a class of Finsler warped product metric is locally projectively flat if and only if it is of scalar flag curvature.

Keywords

References

  • 1. B. Chen, Z. Shen and L. Zhao, Constructions of Einstein Finsler metrics by warped
    product, International Journal of Mathematics, 29 (11), (2018) 1850081.
  • 2. P. Antonelli, R. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces
    with applications in physics and biology, Kluwer Academic Publishers. (1993).
  • 3. G. S. Asanov, Finslerian metric functions over the product R × M and their potential
    applications, Rep. Math. Phys. 41 (1998), 117–132.
  • 4. M. Atashafrouz, B. Najafi and A. Tayebi, Weakly Douglas Finsler metrics, Period. Math.
    Hung. (2020), 1–7.
  • 5. R. L. Bishop and B. ONeill, Manifolds of negative curvature, Trans. Amer. Math. Soc.
    145 (1969), 1–49.
  • 6. J. K. Beem, Indefinite Finsler spaces and timelike spaces, Canad. J. Math. 22(5) (1970),
    1035–1039.
  • 7. S.S. Chern, Z. Shen, Riemannian-Finsler Geometry. World Scientific Publisher, Singapore (2005).
  • 8. B. Chen, Z. Shen and L. Zhao, Constructions of Einstein Finsler metrics by warped
    product, Internat. J. Math. 29 (2018), 1850081.
  • 9. J. Douglas, The general geometry of paths, Ann. Math. (1927), 143–168.
  • 10. M. Gabrani and E. S. Sevim, On a class of Finsler metrics of quadratic Weyl curvature,
    Tohoku Math. J. 75 (2) (2023), 283-298.
  • 11. M. Gabrani, B. Rezaei and E. S. Sevim, On warped product Finsler metrics of isotropic
    E-curvature, Mathematical Researches. 7 (4) (2021), 878-888.
  • 12. M. Gabrani, B. Rezaei and E. S. Sevim, On Landsberg warped product metrics, Journal
    Math. Phys. Aalysis. Geometry, 17 (4) (2021), 468-483.
  • 13. L. Kozma, I. R. Peter and V. Varga, Warped product of Finsler manifolds, Ann. Univ.
    Sci. Budapest. Etvs Sect. Math. 44 (2001), 157–170.
  • 14. H. Liu and X. Mo, Finsler warped product metrics of Douglas type, Canad. Math. Bull.
    62 (2019), 119–130.
  • 15. H. Liu, X. Mo and H. Zhang, Finsler warped product metrics with special Riemannian
    curvature properties, Science China Math. (2019), 1–18.
  • 16. H. Liu and X. Mo, On projectively flat Finsler warped product metrics of constant flag
    curvature, J. Geom. Anal. (2021), 1–22.
  • 17. M. Matsumoto, Projective changes of Finsler metrics and projectively flat Finsler spaces,
    Tensor. NS. 34 (1980), 303–315.
  • 18. B Najafi, Z. Shen and A. Tayebi, On a projective class of Finsler metrics, Publ. Math.
    Debrecen. 70 (2007), 211–219.
  • 19. B. Najafi and A. Tayebi, A new quantity in Finsler geometry, Comptes Rendus Math.
    349 (2011), 81–83.
  • 20. J. Parvaneh, Warped product Finsler manifolds from Hamiltonian point of view. Int. J.
    Geom. Methods Mod. Phys. 14(02) (2017), 1750029.
  • 21. E. Peyghan, A. Tayebi and B. Najafi, Doubly warped product of Finsler manifolds with
    some non-Riemannian curvature properties, Ann. Polon. Math. 105 (2012), 293–311.
  • 22. C. Pfeifer. Finsler spacetime geometry in Physics, Int. J. Geom. Methods Mod. Phys.
    16(02) (2019), 1941004.
  • 23. E. S. Sevim and M. Gabrani, On H-curvature of Finsler warped product metrics, Journal.
    Finsler. Geometry. Appl. 1 (2020), 37–44.
  • 24. Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans Am Math
    Soc. 355 (2003), 1713–1728.
  • 25. A. Tayebi, H. Sadeghi and E. Peyghan, On generalized Douglas-Weyl spaces. Bull.
    Malays. Math. Sci. Soc. 36 (2013), 587–594.
  • 26. H. Zhu, Finsler warped product metrics with almost vanishing H-curvature. Int. J. Geom.
    Methods Mod. Phys. 17 (2020), 2050188-49. 
Journal of Finsler Geometry and its Applications
Volume 4, Issue 1
July 2023
Pages 102-113

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