In this paper, we study a class of Finsler metrics called general spherically symmetric Finsler metrics which are defined by the Euclidean metric and related 1-forms. For a class of the metrics in Rn, we prove that it is projectively flat if and only if it is of scalar flag curvature.
1. D. Bao, C. Robles, and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom. 66(3) (2004), 377-435.
2. S. S. Chern and Z. Shen, Riemannian-Finsler Geometry. World Scientific Publisher, Singapore (2005).
3. X. Cheng and Z Shen, Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009), 359-370.
4. G. Cai, C. Qiu and X. Wang, On some classes of projectively flat Finsler metrics with constant flag curvature, Differ. Geom. Appl. 68 (2020), 101579.
5. J. Douglas, The general geometry of paths, Annals of Mathematics. (1927), 143-168.
6. M. Gabrani, B. Rezaei, and E. S. Sevim, General spherically symmetric Finsler metrics with constant Ricci and flag curvature, Differ. Geom. Appl. 76 (2021), 101754.
7. M. Gabrani, B. Rezaei, and E. S. Sevim, A class of Finsler metrics with almost vanishing H-and Ξ-curvatures, Results in Mathematics. 76(1) (2021), 1-17.
8. M. Gabrani and E. S. Sevim, On projectively flat Finsler warped product metrics with isotropic E-curvature, Journal of Finsler Geometry and its Applications, 1(2) (2020), 54-62.
9. L. Huang and X. Mo, On spherically symmetric Finsler metrics of scalar curvature, J. Geom. Phys. 62 (2012), 2279-2287.
10. B. Ludwig, ber dien-dimensionalen Geometrien konstanter Krmmung, in denen die Geraden die krzesten sind, Mathematische Zeitschrift. 30(1) (1929), 449-469.
11. H. Liu and X. Mo, Examples of Finsler metrics with special curvature properties, Math. Nachr. 13 (2015), 1527-1537.
12. W. Liu and B. Li, Projectively flat Finsler metrics defined by the Euclidean metric and related 1-forms, Differ. Geom. Appl. 46 (2016), 14-24.
13. X. Mo, Finsler metrics with constant (or scalar) flag curvature, Proc. Indian Acad. Sci. (Math. Sci.) 122(3) (2012), 411-427.
14. X. Mo, On some Finsler metrics of constant (or scalar) flag curvature, Houston J. Math. 38 (2012), 41-54.
15. B. Rezaei and M. Gabrani, A class of Berwaldian Finsler metrics, Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis, 107(3) (2017), 259-270.
16. B. Rezaei and M. Gabrani, A class of Finsler metrics with quadratic curvatures, Bull. Iran. Math. Soc. 46 (2020), 53-65.
17. Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Transactions of the American Mathematical Society, 355(4) (2003), 1713-1728.
18. A. Tayebi, M. Bahadori and H. Sadeghi. On spherically symmetric Finsler metrics with some non-Riemannian curvature properties, J. Geom. Phys. 163 (2021), 104125.
19. A. Tayebi and M. Shahbazi Nia. A new class of projectively flat Finsler metrics with constant flag curvature K = 1, Differ. Geom. Appl. 41 (2015), 123–133.
20. X. Wang, W. Liu and B. Li, Dually flat general spherically symmetric Finsler metrics, Sci. China. Math. 61(4) (2018), 769-782.
21. Q. Xia, On a class of Finsler metrics of scalar flag curvature, Results in Mathematics. 71(1-2) (2017), 483-507.
22. H. Zhu, A class of Finsler metrics of scalar flag curvature. Differ. Geom. Appl. 40 (2015), 321-331.
23. L. Zhou, Projective spherically symmetric Finsler metrics with constant flag curvature in Rn, Geom Dedicata. 158(1) (2012), 353-364.
24. H. Zhu and H. Zhang, Projective Ricci flat spherically symmetric Finsler metrics, Internat. J. Math. 29(11) (2018), 1850078.
Gabrani, M. (2021). On a class of Finsler metrics of scalar flag curvature defined by the Euclidean metric and related 1-forms. Journal of Finsler Geometry and its Applications, 2(1), 118-131. doi: 10.22098/jfga.2021.1269
MLA
Mehran Gabrani. "On a class of Finsler metrics of scalar flag curvature defined by the Euclidean metric and related 1-forms", Journal of Finsler Geometry and its Applications, 2, 1, 2021, 118-131. doi: 10.22098/jfga.2021.1269
HARVARD
Gabrani, M. (2021). 'On a class of Finsler metrics of scalar flag curvature defined by the Euclidean metric and related 1-forms', Journal of Finsler Geometry and its Applications, 2(1), pp. 118-131. doi: 10.22098/jfga.2021.1269
VANCOUVER
Gabrani, M. On a class of Finsler metrics of scalar flag curvature defined by the Euclidean metric and related 1-forms. Journal of Finsler Geometry and its Applications, 2021; 2(1): 118-131. doi: 10.22098/jfga.2021.1269