In the present paper, we prove that a general $(α, β)$ -metric $F$ is of isotropic $S$ -curvature if and only if it is of isotropic $E$ -curvature under an extra condition on $α$ and $β$ .

Keywords:

AMSC: 53B40, 53C60

References

1. D. Bao and C. Robles , Ricci and Flag Curvature in Finsler Geometry, Riemann-Finsler Geometry, Vol. 50 (MSRI Publications, Cambridge University Press, 2004), pp. 197–260. Google Scholar
2. D. Bao, C. Robbles and Z. Shen , Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004) 391–449. Web of Science, Google Scholar
3. S. Bacso, X. Cheng and Z. Shen , Curvature properties of $(α, β)$ -metrics, Adv. Studies in Pure Math. 48 (2007) 73–110. Google Scholar
4. Y. Chen and W. Song , Spherically symmetric Finsler metrics with isotropic $E$ -curvature, J. Math. Res. Appl. 35(5) (2015) 561–567. Google Scholar
5. X. Cheng and Z. Shen , A class of Finsler metrics with isotropic $S$ -curvature, Israel J. Math. 169 (2009) 317–340. Web of Science, Google Scholar
6. X. Cheng and Z. Shen , Randers metric with special curvature properties, Osaka J. Math. 40 (2003) 87–101. Web of Science, Google Scholar
7. S. S. Chern and Z. Shen , Riemannian-Finsler Geometry (World Scientific, Singapore, 2005). Link, Google Scholar
8. M. Gabrani and B. Rezaei , On general $(α, β)$ -metrics with isotropic $E$ -curvature, J. Korean Math. Soc. 55(2) (2018) 415–424. Web of Science, Google Scholar
9. X. Mo and H. Zhu , On a class of locally projectively flat general $(α, β)$ -metrics, Bull. Korean Math. Soc. 54(4) (2017) 1293–1307. Web of Science, Google Scholar
10. S. F. Rutz , Symmetry in Finsler spaces, Contemp. Math. 196 (1996) 289–300. Google Scholar
11. Z. Shen , Finsler metrics with $K = 0$ and $S = 0$ , Canadian J. Math. 55 (2003) 112–132. Web of Science, Google Scholar
12. Z. Shen and C. Yu , On a class of Einstein Finsler metrics, Int. J. Math. 25(4) (2014) 1450030-1–18. Link, Web of Science, Google Scholar
13. Z. Shen , Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128(2) (1997) 306–328. Web of Science, Google Scholar
14. Q. Xia , On a class of Finsler metrics with scalar curvature properties, Results in Math. 71 (2017) 483–507. Web of Science, Google Scholar
15. C. Yu, Deformations and Hilberts Fourth Problem, http://arxiv.org/abs/1209.0845. Google Scholar
16. C. Yu and H. Zhu , On a new class of Finsler metrics, Differential Geom. Appl. 29 (2011) 244–254. Web of Science, Google Scholar
17. L. Zhou , Spherically symmetric Finsler metrics in $ℝ^{n}$ , Publ. Math. Debrecen 80 (2010) 48–70. Web of Science, Google Scholar
18. H. Zhu , On general $(α, β)$ -metrics with isotropic $S$ -curvature, J. Math. Anal. Appl. 464 (2018) 1127–1142. Web of Science, Google Scholar