On D-Recurrent Finsler Metrics

Abstract

In this paper, we study D-recurrent Finsler metrics and characterize D-recurrent Randers metrics. Indeed, we show that a Randers metric \(F=\alpha +\beta \) is D-recurrent if and only if \(\mathrm{d}\beta \) is nearly recurrent. We prove that a GDW-Randers metric is a Douglas metric provided that it is D-recurrent. Then, we extend this fact to all the Finsler metrics: GDW-metrics are Douglas metric if they are D-recurrent. Then, we show that in the class of spherically symmetric Finsler metrics, Douglas metrics coincide with D-recurrent ones.

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References

  1. 1.

    Akbar-Zadeh, H.: Sur les espaces de Finsler à courbures sectionnelles constantes. Acad. R. Belg. Bull. Cl. Sci. 80(5), 271–322 (1988)

    MATH  Google Scholar 

  2. 2.

    Bácsó, S., Papp, I.: A note on generalized Douglas space. Period. Math. Hung. 48(1–2), 181–184 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bácsó, S., Matsumoto, M.: On Finsler spaces of Douglas type, a generalization of notion of Berwald space. Publ. Math. Debr. 51, 385–406 (1997)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Ceyhan, S., Glçin, Ç.: D-Recurrent Kropina spaces with generalized Douglas metric. Bilecik Seyh Edebali Univ. Fen Bilimleri Dergisi 1(1), 17–21 (2014)

    Google Scholar 

  5. 5.

    Emamian, M.H., Tayebi, A.: Generalized Douglas–Weyl Finsler spaces. Iran. J. Math. Sci. Inform. 10(2), 67–75 (2015)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Guo, E., Liu, H., Mo, X.: On spherically symmetric Finsler metrics with isotropic Berwald curvature. Int. J. Geom. Methods Mod. Phys. 10(10), 1350054 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Huang, L., Mo, X.: On spherically symmetric Finsler metrics of scalar curvature. J. Geom. Phys. 62(11), 2279–2287 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kumar, A., Shulka, H.S., Tripathi, R.P.: On the existence of projective affine motion in a \( W \)-recurrent Finsler space. Tamkang J. Math. 31, 71–78 (2000)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Huang, L., Mo, X.: On spherically symmetric Finsler metrics of scalar curvature. J. Geom. Phys. 6211, 2279–2287 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mishra, R.S., Pande, H.D.: Recurrent Finsler spaces. J. Indian Math. Soc. 32, 17–22 (1968)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Mo, X., Newton, M.S., Keti, T.: On spherically symmetric Finsler metrics with vanishing Douglas curvature. Differ. Geom. Appl. 31, 746–758 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Mo, X., Zhu, H.: On a projective class of Finsler metrics with orthogonal invariance. Differ. Geom. Appl. 52, 167–180 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Najafi, B., Shen, Z., Tayebi, A.: On a projective class of Finsler metrics. Publ. Math. Debr. 70, 211–219 (2007)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Najafi, B., Bidabad, B., Tayebi, A.: On R-quadratic Finsler metrics. Iran. J. Sci. Technol. (Sci.) 32, 439–443 (2008)

    MATH  Google Scholar 

  15. 15.

    Nasehi, M.: On 5-dimensional 2-step homogeneous Randers nilmanifolds of Douglas type. Bull. Iran. Math. Soc. 4(3), 695–706 (2017)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Randers, G.: On an asymmetric metric in the four-space of general relativity. Phys. Rev. 59, 195–199 (1941)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, London (2001)

    Google Scholar 

  18. 18.

    Tayebi, A., Peyghan, E.: On Douglas surfaces. Bull. Math. Soc. Sci. Math. (Roumanie, Tome) 55(103), 327–335 (2012)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Tayebi, A., Barzegari, M.: Generalized Berwald spaces with \((\alpha, \beta )\)-metrics. Indag. Math. 27, 670–683 (2016)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Tayebi, A., Najafi, B.: On a class of homogeneous Finsler metrics. J. Geom. Phys. 140, 265–270 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Tayebi, A., Sadeghi, H.: On generalized Douglas–Weyl \((\alpha, \beta )\)-metrics. Acta Math. Sin. Engl. Ser. 31(10), 1611–1620 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tayebi, A., Sadeghi, H., Peyghan, E.: On generalized Douglas–Weyl spaces. Bull. Malays. Math. Sci. Soc. (2) 36(3), 587–594 (2013)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Tayebi, A., Peyghan, E.: On Douglas surfaces. Bull. Math. Soc. Sci. Math. Roum. (55) 103, 327–335 (2012)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Wong, Y.C.: Recurrent tensors on a linearly connected differentiable manifold. Trans. Am. Math. Soc. 99, 325–341 (1961)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Youssef, N.L., Soleiman, A.: On concircularly recurrent Finsler manifolds. Balkan J. Geom. Appl. 18(1), 101–113 (2013)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Youssef, N.L., Soleiman, A.: On horizontal recurrent Finsler connections. Rend. Circ. Mat. Palermo II. Ser. 68(1), 1–9 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Zhou, L.: Spherically symmetric Finsler metrics in \(R^n\). Publ. Math. Debr. 80, 1–11 (2012)

    Google Scholar 

  28. 28.

    Zhou, L.: Projective spherically symmetric Finsler metrics with constant flag curvature in \(R^n\). Geom. Dedic. 158, 353–364 (2012)

    Article  Google Scholar 

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Acknowledgements

We thank the referee for his/her valuable suggestions and comments.

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Correspondence to Behzad Najafi.

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Communicated by Mohammad Koushesh .

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Atashafrouz, M., Najafi, B. On D-Recurrent Finsler Metrics. Bull. Iran. Math. Soc. 47, 143–156 (2021). https://doi.org/10.1007/s41980-020-00372-y

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Keywords

  • D-recurrent
  • Douglas metrics
  • Spherically symmetric metrics

Mathematics Subject Classification

  • 53B40
  • 53C60