Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

On Naturally Reductive $\boldsymbol{(\alpha_1,\alpha_2)}$-Metrics

Document Type : Research Article

Authors
A. Hatami Shahi
H. R. Salimi Moghaddam
Department of Pure Mathematics, University‎ ‎of Isfahan‎, ‎P.O. Box 81746-73441, Isfahan‎, ‎Iran.
10.30504/jims.2026.578080.1328
Abstract
In this article‎, ‎we show that if $(M=G/H,g)$ is a naturally reductive homogeneous Riemannian manifold and $\frak{g}' \subset \frak{h}+\frak{m}_1$‎, ‎then the $(\alpha_1,\alpha_2)$-metric $F$ induced by $g$ and the decomposition $\frak{m}=\frak{m}_1+\frak{m}_2$ is also naturally reductive‎. ‎To establish this‎, ‎we provide an explicit formula for the Cartan tensor of homogeneous $(\alpha_1,\alpha_2)$-metrics‎. ‎We further examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and their counterparts on homogeneous $(\alpha_1,\alpha_2)$-spaces‎. ‎Finally‎, ‎we construct left-invariant $(\alpha_1,\alpha_2)$-metrics on the tangent bundle of Lie groups from left-invariant Randers metrics on the base Lie group and study their geometric relations‎.
Keywords
(\alpha_1,\alpha_2)-Metric
Randers metric
Naturally reductive
Tangent Lie group
Subjects
[1] I. Agricola and A. C. Ferreira, Tangent Lie groups are Riemannian naturally reductive spaces, Adv. Appl. Clifford Algebr. 27 (2017), no. 2, 895–911.
[2] H. An, Z. Yan, and Sh. Zhang, S-Curvature and geodesic orbit property of invariant (α1, α2)-metrics on spheres, Bull. Korean Math. Soc. 60 (2023), no. 1, 33–46.
[3] F. Asgari and H. R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo, II. Ser 67 (2018), no. 2, 185–195.
[4] P. Bahmandoust and D. Latifi, Naturally reductive homogeneous (α, β)-spaces, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 8, 2050117, 16 pp.
[5] D. Bao, S. S. Chern, and Z. Shen, An introduction to Riemannian–Finsler geometry, Grad. Texts in Math., 200, Springer, New York, 2000.
[6] S. S. Chern and Z.Shen, Riemann-Finsler Geometry, World Scientific Publishers, Singapore, 2004. [7] S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012.
[8] S. Deng and Z. Hou, Invariant Finsler metrics on homogeneous manifolds, J. Phys. A: Math. Gen. 37 (2004), no. 34, 8245–8253.
[9] S. Deng and Z. Hou, Naturally reductive homogeneous Finsler spaces, Manuscripta Math., 131 (2010), no. 1-2, 215–229.
[10] S. Deng and M. Xu, (α1, α2)-metrics and Clifford-Wolf homogeneity, J. Geom. Anal. 26 (2016), no. 3, 2282–2321. [11] S.Deng and M. Xu, Clifford–Wolf translations of Finsler spaces, Forum Math. 26 (2014), no. 5, 1413–1428.
[12] L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differential Geom. Appl. 40 (2015) 187– 208.
[13] L. Huang and X. Mo, Homogeneous Einstein metrics on (4n+3)-dimensional spheres, Canad. Math. Bull. 62 (2018), no. 3, 509–523.
[14] O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Unione Mat. Ital. 5 (1991), no. 7, 189–246.
[15] D. Latifi, Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys. 57 (2007), no. 5, 1421–1433.
[16] M. Nejadahmad and H. R. Salimi Moghaddam, The relation between automorphism group and isometry group of left invariant (α, β)-metrics, J. Finsler Geom. Appl. 4 (2023), no. 1, 81–87.
[17] M. Parhizkar and H. R. Salimi Moghaddam, Geodesic Vector fields of invariant (α, β)-metrics on Homogeneous spaces, Int. Electron. J. Geom. 6 (2013), no. 2, 39–44.
[18] J. Tan and M. Xu, Naturally reductive (α1, α2)-metrics, Acta Math. Sci. 43 (2023), no. 4, 1547–1560.
[19] M. Xu and S. Deng, Clifford–Wolf homogeneous Randers spaces, J. Lie Theory 23 (2013), no. 3, 837–845.
[20] M. Xu and S. Deng, Left invariant Clifford–Wolf homogeneous (α, β)-metrics on compact semisimple Lie groups, Transform. Groups 20 (2015), no. 2, 395–416.
[21] L. Zhang and M. Xu, Standard homogeneous (α1, α2)-metrics and geodesic orbit property, Math. Nachr. 295 (2022), no. 7, 1443–1453.
[22] S. Zhang, Z. Yan and S. Deng, Naturally reductive homogeneous (α, β) spaces, Publ. Math. Debrecen 102 (2023), no. 3-4, 415–427.
Journal of the Iranian Mathematical Society
Volume 7, Issue 2
August 2026
Pages 93-108