On Homogeneous Randers Metrics

Yıl 2021, Cilt: 14 Sayı: 1, 217 - 225, 15.04.2021

Akbar Sadighi Megerdich Toomanian Behzad Najafi

https://doi.org/10.36890/iejg.797112

Öz

Anahtar Kelimeler

Homogeneous metric

On Homogeneous Randers Metrics

Öz

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.

Anahtar Kelimeler

Homogeneous metric, Ricci tensor, Randers metric

Kaynakça

• [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.