On general $(\alpha,\beta)$-metrics with Cartan torsion, mean Cartan torsion and Landsberg curvature

Document Type : Original Article


1 Department of Mathematics and Statistics Faculty of Science, Dr. Rammanohar Lohia Avadh University, Ayodhya-224001, India

2 Department of Mathematics Institute of Applied Sciences and Humanities GLA University Mathura-281406, India


In this paper, we derive a formula for the (mean) Cartan torsion of a class of general $(\alpha,\beta)$-metrics. Also, we study weak Landsberg general $(\alpha,\beta)$-metrics under a certain condition.


Main Subjects

[1] A. Ala, A. Behzadi, and M. Rafiei-Rad, On general (α, β)-metrics of weak Landsberg type, 2017. preprint.
[2] R. Bryant, Finsler structures on the 2-sphere satisfying K = 1, in Finsler Geometry (Seattle, WA, 1995), vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 27–41.
[3] , Projectively flat finsler 2-spheres of constant curvature, Sel. math. (New ser.), 3 (1997), pp. 161–203.
[4] S.-S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc., 43 (1996), pp. 959–963.
[5] S.-S. Chern and Z. Shen, Riemann-Finsler geometry, vol. 6 of Nankai Tracts in Mathematics, World Sci[1]entific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
[6] M. Matsumoto, On C-reducible Finsler spaces, Tensor (N.S.), 24 (1972), pp. 29–37.
[7] G. Randers, On an asymmetrical metric in the fourspace of general relativity, Phys. Rev. (2), 59 (1941), pp. 195–199.
[8] S. F. Rutz, Symmetry in Finsler spaces, in Finsler geometry (Seattle, WA, 1995), vol. 196 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1996, pp. 289–300.
 [9] E. S. Sevim, Z. Shen, and S. Ulgen ¨ , Spherically symmetric Finsler metrics with constant Ricci and flag curvature, Publ. Math. Debrecen, 87 (2015), pp. 463–472.
[10] Z. Shen, Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[11] A. Tayebi, E. Peyghan, and H. Sadeghi, On locally dually flat (α, β)-metrics with isotropic S-curvature, Indian J. Pure Appl. Math., 43 (2012), pp. 521–534.
[12] A. Tayebi, H. Sadeghi, and E. Peyghan, On a class of locally dually flat (α, β)-metrics, Math. Slovaca, 65 (2015), pp. 191–198.
[13] C. Yu and H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl., 29 (2011), pp. 244–254.
[14] , Projectively flat general (α, β)-metrics with constant flag curvature, J. Math. Anal. Appl., 429 (2015), pp. 1222–1239.
[15] L. Zhou, Spherically symmetric Finsler metrics in Rn, Publ. Math. Debrecen, 80 (2012), pp. 67–77.
[16] , The spherically symmetric Finsler metrics with isotropic S-curvature, J. Math. Anal. Appl., 431 (2015), pp. 1008–1021.
[17] M. Zohrehvand and H. Maleki, On general (α, β)-metrics of Landsberg type, Int. J. Geom. Methods Mod. Phys., 13 (2016), pp. 1650085, 13.