On Finsler metrics with weakly isotropic S-curvature

Document Type : Original Article


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

2 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran


In this paper‎, ‎we focus on a class of Finsler metrics which are called general~$(alpha,beta)$-metrics‎: ‎$alpha= sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and $beta= b_{i}(x)y^{i}$ is a $1$-form. We examine the metrics as weakly isotropic $S$-curvature‎.


Main Subjects

[1] X. Cheng and Z. Shen, Randers metric with special curvature properties, Osaka. J. Math. 40 (2003), 87-101.
[2] X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. Math. 169 (2009), 317-340.
[3] X. Chun-Huan and X. Cheng, On a class of weakly-Berwald (α, β)-metrics, J. Math. Res. Expos. 29 (2009), 227-236.
[4] M. Gabrani and B. Rezaei, On general (α, β)-metric with isotropic E-curvature, J. Korean Math. Soc. 55(2) (2018), 415-424.
[5] M. Gabrani, B. Rezaei, E. S. Sevim, A Class of Finsler Metrics with Almost Vanishing H- and Ξ-curvatures, Results Math. 76(44) (2021).
[6] I. Y. Lee and M. H. Lee, On weakly-Berwald spaces of special (α, β)-metrics, Bull. Korean Math. Soc. 43 (2006), 425-441.
[7] B. Najafi and A. Tayebi, A class of Finsler metrics with isotropic mean Berwald curvature, Acad. Paedagog. Nyh´azi. 32 (2016), 113-123.
[8] Z. Shen, Nonpositively curved Finsler manifolds with constant S-curvature, Math. Z. 249 (2005), 625-639.
[9] Z. Shen, Finsler metrics with K = 0 and S = 0, Canadian J. Math. 55 (2003), 112-132.
[10] Z. Shen, Volume compasion and its applications in Riemann-Finsler geometry, Advances in Math. 128 (1997), 306-328.
[11] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001.
[12] A. Tayebi, H. Sadeghi and E. Peyghan, On Finsler metrics with vanishing S-curvature, Turkish Journal of Mathematics, 38(1) (2014), 154-165.
[13] A. Tayebi and M. Rafie-Rad, S-curvature of isotropic Berwald metrics, Science in China Series A: Mathematics, 51(12) (2008), 2198-2204.
[14] Q. Xia, Some results on the non-Riemannian quantity H of a Finsler metric, Internat. J. Math. 22(7) (2011), 925-936.
[15] C. Yu and H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl. 29(2) (2011), 244-254. [16] C. Yu and H. Zhu, Projectively flat general (α, β)-metrics with constant flag curvature, J. Math. Anal. Appl. 429(2) (2015), 1222-1239.
[17] C. Yu, On dually flat general (α, β)-metrics, Differential Geom. Appl. 40 (2015), 111-122.
[18] L. Zhou, The spherically symmetric Finsler metrics with isotropic S-curvature, J. Math. Anal. Appl. 431 (2015), 1008-1021.
[19] H. Zhu, On a class of spherically symmetric Finsler metrics with isotropic S-curvature, Differential Geom. Appl. 51 (2017), 102-108.
[20] H. Zhu, On general (α, β)-metrics with isotropic S-curvature, Journal of Mathematical Analysis and Applications, 464(2) (2018), 1127-1142.
[21] H. Zhu, On a class of Finsler metrics with isotropic Berwald curvature, Bull. Korean Math. Soc. 54(2) (2017), 399-416.
[22] M. Zohrehvand and H. Maleki, On general (α, β)-metrics of Landsberg type, Int. J. Geom. Methods Mod. Phys. 13(6) (2016), 1650085, 13 pp.