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On Homogeneous Randers Metrics

Year 2021, , 217 - 225, 15.04.2021
https://doi.org/10.36890/iejg.797112

Abstract

References

  • [1] H. Akbar-Zadeh, Generalized Einstein manifolds, Jour. Geom. Phys. 17(1995), 342- 380.
  • [2] D. Bao, On two curvature-driven problems in Finsler geometry, Adv. Study Pure Math, 48(2007), Finsler Geometry, Sapporo 2005 - In Memery of Makoto Mat- sumoto, 19-71.
  • [3] D. Bao and C. Robles, Ricci and Flag curvatures in Finsler geometry, A Sampler of Riemann-Finlser Geometry, MSRIP. 50(2004), 198-256.
  • [4] L. Berwald, Über Parallelübertragung in Raumen mit allgemeiner Massbestimmung,Jber. Deutsch. Math.-Verein, 34(1925), 213-220.
  • [5] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific. J. Math, 207(2002), 149-155.
  • [6] S. Deng and Z. Hou, Homogeneous Einstein-Randers spaces of negative Ricci curva- ture, C. R. Math. Acad. Sci. Paris. 347(2009), 1169-1172.
  • [7] S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240(2013), 194-226.
  • [8] S. Deng and Z. Hu, On flag curvature of homogeneous Randers spaces, Canad. J. Math. 65(2013), 66-81.
  • [9] E. Esrafilian and H.R. Salimi Moghaddam, Flag curvature of invariant Randers metrics on homogeneous manifolds, J. Phys. A: Math. Gen. 39(2006), 3319-3324.
  • [10] Z. Hu and S. Deng, Ricci-quadratic homogeneous Randers spaces, Nonlinear Analysis. 92(2013), 130-137.
  • [11] B. Li and Z. Shen, Ricci curvature tensor and non-Riemannian quantities, Canadian Mathematical Bulletin, 58(2015), 530-537.
  • [12] M. Matsumoto, Foundations of Finsler Geometry and special Finsler Spaces, Kai- seisha Press, Japan 1986.
  • [13] M. Matsumoto, An improvment proof of Numata and Shibata’s theorem on Finsler spaces of scalar curvature, Publ. Math. Debrecen, 64(2004), 489-500.
  • [14] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59(1941), 195-199.
  • [15] C. Robles, Einstein metrics of Randers type, Doctoral Dissertation, University of British Columbia, 2003.
  • [16] Z. Shen, Finsler metrics with K = 0 and S = 0, Canadian J. Math. 55(2003), 112-132.
  • [17] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Pub- lishers, 2001.
  • [18] C. Shibata, On Finsler spaces with Kropina metrics, Report. Math., 13(1978), 117- 128.
  • [19] H. Shimada, On the Ricci tensors of particular Finsler spaces, J. Korean Math. Soc. 14(1977), 41-63.
  • [20] A. Tayebi and E. Peyghan, On Ricci tensors of Randers metrics, J. Geom. Phys. 60(2010), 1665-1670.
  • [21] M. Xu and S. Deng, Homogeneous (α,β)-spaces with positive flag curvature and vanishing S-curvature, Nonlinear Analysis. 127(2015), 45-54.

On Homogeneous Randers Metrics

Year 2021, , 217 - 225, 15.04.2021
https://doi.org/10.36890/iejg.797112

Abstract

In this paper, we study the curvature features of the class of homogeneous Randers metrics. For these metrics, we first find a reduction criterion to be a Berwald metric based on a mild restriction on their Ricci tensors. Then, we prove that every homogeneous Randers metric with relatively isotropic (or weak) Landsberg curvature must be Riemannian. This provides an extension of well-known Deng-Hu theorem that proves the same result for a homogeneous Berwald-Randers metric of non-zero flag curvature.

References

  • [1] H. Akbar-Zadeh, Generalized Einstein manifolds, Jour. Geom. Phys. 17(1995), 342- 380.
  • [2] D. Bao, On two curvature-driven problems in Finsler geometry, Adv. Study Pure Math, 48(2007), Finsler Geometry, Sapporo 2005 - In Memery of Makoto Mat- sumoto, 19-71.
  • [3] D. Bao and C. Robles, Ricci and Flag curvatures in Finsler geometry, A Sampler of Riemann-Finlser Geometry, MSRIP. 50(2004), 198-256.
  • [4] L. Berwald, Über Parallelübertragung in Raumen mit allgemeiner Massbestimmung,Jber. Deutsch. Math.-Verein, 34(1925), 213-220.
  • [5] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific. J. Math, 207(2002), 149-155.
  • [6] S. Deng and Z. Hou, Homogeneous Einstein-Randers spaces of negative Ricci curva- ture, C. R. Math. Acad. Sci. Paris. 347(2009), 1169-1172.
  • [7] S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240(2013), 194-226.
  • [8] S. Deng and Z. Hu, On flag curvature of homogeneous Randers spaces, Canad. J. Math. 65(2013), 66-81.
  • [9] E. Esrafilian and H.R. Salimi Moghaddam, Flag curvature of invariant Randers metrics on homogeneous manifolds, J. Phys. A: Math. Gen. 39(2006), 3319-3324.
  • [10] Z. Hu and S. Deng, Ricci-quadratic homogeneous Randers spaces, Nonlinear Analysis. 92(2013), 130-137.
  • [11] B. Li and Z. Shen, Ricci curvature tensor and non-Riemannian quantities, Canadian Mathematical Bulletin, 58(2015), 530-537.
  • [12] M. Matsumoto, Foundations of Finsler Geometry and special Finsler Spaces, Kai- seisha Press, Japan 1986.
  • [13] M. Matsumoto, An improvment proof of Numata and Shibata’s theorem on Finsler spaces of scalar curvature, Publ. Math. Debrecen, 64(2004), 489-500.
  • [14] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59(1941), 195-199.
  • [15] C. Robles, Einstein metrics of Randers type, Doctoral Dissertation, University of British Columbia, 2003.
  • [16] Z. Shen, Finsler metrics with K = 0 and S = 0, Canadian J. Math. 55(2003), 112-132.
  • [17] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Pub- lishers, 2001.
  • [18] C. Shibata, On Finsler spaces with Kropina metrics, Report. Math., 13(1978), 117- 128.
  • [19] H. Shimada, On the Ricci tensors of particular Finsler spaces, J. Korean Math. Soc. 14(1977), 41-63.
  • [20] A. Tayebi and E. Peyghan, On Ricci tensors of Randers metrics, J. Geom. Phys. 60(2010), 1665-1670.
  • [21] M. Xu and S. Deng, Homogeneous (α,β)-spaces with positive flag curvature and vanishing S-curvature, Nonlinear Analysis. 127(2015), 45-54.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Akbar Sadighi This is me 0000-0002-9992-2958

Megerdich Toomanian 0000-0003-1263-2120

Behzad Najafi 0000-0003-2788-3360

Publication Date April 15, 2021
Acceptance Date March 22, 2021
Published in Issue Year 2021

Cite

APA Sadighi, A., Toomanian, M., & Najafi, B. (2021). On Homogeneous Randers Metrics. International Electronic Journal of Geometry, 14(1), 217-225. https://doi.org/10.36890/iejg.797112
AMA Sadighi A, Toomanian M, Najafi B. On Homogeneous Randers Metrics. Int. Electron. J. Geom. April 2021;14(1):217-225. doi:10.36890/iejg.797112
Chicago Sadighi, Akbar, Megerdich Toomanian, and Behzad Najafi. “On Homogeneous Randers Metrics”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 217-25. https://doi.org/10.36890/iejg.797112.
EndNote Sadighi A, Toomanian M, Najafi B (April 1, 2021) On Homogeneous Randers Metrics. International Electronic Journal of Geometry 14 1 217–225.
IEEE A. Sadighi, M. Toomanian, and B. Najafi, “On Homogeneous Randers Metrics”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 217–225, 2021, doi: 10.36890/iejg.797112.
ISNAD Sadighi, Akbar et al. “On Homogeneous Randers Metrics”. International Electronic Journal of Geometry 14/1 (April 2021), 217-225. https://doi.org/10.36890/iejg.797112.
JAMA Sadighi A, Toomanian M, Najafi B. On Homogeneous Randers Metrics. Int. Electron. J. Geom. 2021;14:217–225.
MLA Sadighi, Akbar et al. “On Homogeneous Randers Metrics”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 217-25, doi:10.36890/iejg.797112.
Vancouver Sadighi A, Toomanian M, Najafi B. On Homogeneous Randers Metrics. Int. Electron. J. Geom. 2021;14(1):217-25.