In a Finsler spaces, we consider a special (α,β)-metric L satisfying L2(α,β) = c1 α 2 +2c2 αβ+c3β2, where ci are constant. In this paper, the existence of invariant vector elds on a special homogeneous (α,β)-space with L metric is proved. Then we study geodesic vectors and investigate the set of all homogeneous geodesics of invariant (α,β)-metric L on homogeneous spaces and simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure.
1. H. An, S. Deng, Invariant (α, β)-metrics on homogeneous manifolds, Monatsh. Math, 154(2008), 89-102.
2. D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer, New York, (2000).
3. M. L. Barberis, Hypercomplex structures on four-dimensional Lie groups, Proc. Am. Math. Soc. 125 (1997), 1043-1054.
4. M. L. Barberis, Hyper-Kahler Metrics Conformal to Left Invariant Metrics on FourDimensional Lie Groups, Mathematical Physics, Analysis and Geometry 6(2003), 1-8.
5. S. Deng, Homogeneous Finsler Spaces, Springer Monographs in Mathematics, New York, 2012.
6. S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207(2002), 149-155.
7. S. Deng and Z. Hou, Invariant Randers metrics on Homogeneous Riemannian manifolds, J. Phys. A: Math. General 37 (2004) 4353-4360; Corrigendum, ibid, 39(2006), 5249-5250.
8. M. Ebrahimi and D. Latifi,On Flag Curvature and Homogeneous Geodesics of Left Invariant Randers Metrics on the Semi-Direct Product a ⊕p r , Journal of Lie Theory 29(2019), 619-627.
9. P. Habibi, Homogeneous geodesics in Homogeneous Randers spaces-examples, Journal of Finsler Geometry and its Applications, 1(1) (2020), 89-95.
10. S. Homolya and O. Kowalski, Simply connected two-step homogeneous nilmanifolds of dimension 5, Note Mat. 26(2006), 69-77.
11. K. Kaur and G. Shanker, On the geodesics of a homogeneous Finsler space with a special (α, β)-metric, Journal of Finsler Geometry and its Applications, 1(1) (2020), 26-36.
12. O. Kowalski and L. Vanhecke, Riemannian-manifolds with homogeneous geodesics, Boll. Unione. Mat. Ital. 5(1991), 189-246.
13. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, 1969.
14. D. Latifi, Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys. 57(2007), 1421-1433.
15. D. Latifi, A. Razavi, Bi-invariant Finsler metrics on Lie groups, Aust. J. Basic Appl. Sci. 5(2011), 507-511.
16. M. Matsumoto, Theory of Finsler spaces with (α, β)-metric, Rep. Math. Phys. 31(1992), 43-83.
17. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76(1954), 33-65.
18. G. Randers, On an asymmetrical metric in the four-space of general relativity, Phys. Rev. 59(1941), 195-199.
19. G. Shanker and S. Rani, On S-curvature of a homogeneous Finsler space with square metric, Int. J. Geom. Meth. Mod. Physics, 17(2) (2020), 2050019.
20. Z. Shen, Lectures on Finsler Geometry. World Scientific, 2001.
Ebrahimi, M. (2020). Invariant vector field on a homogeneous Finsler space with special (α,β)-metric. Journal of Finsler Geometry and its Applications, 1(2), 1-14. doi: 10.22098/jfga.2020.1235
MLA
Mahnaz Ebrahimi. "Invariant vector field on a homogeneous Finsler space with special (α,β)-metric", Journal of Finsler Geometry and its Applications, 1, 2, 2020, 1-14. doi: 10.22098/jfga.2020.1235
HARVARD
Ebrahimi, M. (2020). 'Invariant vector field on a homogeneous Finsler space with special (α,β)-metric', Journal of Finsler Geometry and its Applications, 1(2), pp. 1-14. doi: 10.22098/jfga.2020.1235
VANCOUVER
Ebrahimi, M. Invariant vector field on a homogeneous Finsler space with special (α,β)-metric. Journal of Finsler Geometry and its Applications, 2020; 1(2): 1-14. doi: 10.22098/jfga.2020.1235
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