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http://dx.doi.org/10.4134/BKMS.b170710

ON CONFORMAL TRANSFORMATIONS BETWEEN TWO ALMOST REGULAR (α, β)-METRICS

Chen, Guangzu (School of Science East China JiaoTong University)
Liu, Lihong (School of Science East China JiaoTong University)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.4, 2018 , pp. 1231-1240 More about this Journal

Abstract

In this paper, we characterize the conformal transformations between two almost regular ( ${\alpha},{\beta}$ )-metrics. Suppose that F is a non-Riemannian ( ${\alpha},{\beta}$ )-metric and is conformally related to ${\widetilde{F}}$ , that is, ${\widetilde{F}}=e^{{\kappa}(x)}F$ , where ${\kappa}:={\kappa}(x)$ is a scalar function on the manifold. We obtain the necessary and sufficient conditions of the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature. Further, when both F and ${\widetilde{F}}$ are regular, the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature must be a homothety.

Keywords

Finsler metric; ( ${\alpha},{\beta}$ )-metric; conformal transformation; the mean Landsberg curvature;

Citations & Related Records

Reference

1	S. Bacso and X. Cheng, Finsler conformal transformations and the curvature invariances, Publ. Math. Debrecen 70 (2007), no. 1-2, 221-231.
2	S. Bacso, X. Cheng, and Z. Shen, Curvature properties of ( ${\alpha}$ , ${\beta}$ )-metrics, in Finsler geometry, Sapporo 2005-in memory of Makoto Matsumoto, 73-110, Adv. Stud. Pure Math., 48, Math. Soc. Japan, Tokyo, 2007.
3	G. Chen and X. Cheng, An important class of conformally flat weak Einstein Finsler metrics, Internat. J. Math. 24 (2013), no. 1, 1350003, 15 pp.
4	G. Chen, X. Cheng, and Y. Zou, On conformal transformations between two ( ${\alpha}$ , ${\beta}$ )- metrics, Differential Geom. Appl. 31 (2013), no. 2, 300-307. DOI
5	G. Chen, Q. He, and Z. Shen, On conformally flat ( ${\alpha}$ , ${\beta}$ )-metrics with constant flag curvature, Publ. Math. Debrecen 86 (2015), no. 3-4, 387-400.
6	X. Cheng, H. Li, and Y. Zou, On conformally flat ( ${\alpha}$ , ${\beta}$ )-metrics with relatively isotropic mean Landsberg curvature, Publ. Math. Debrecen 85 (2014), no. 1-2, 131-144. DOI
7	X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. Math. 169 (2009), 317-340. DOI
8	X. Cheng and Z. Shen, Finsler Geometry, Science Press Beijing, Beijing, 2012.
9	X. Cheng, H. Wang, and M. Wang, ( ${\alpha}$ , ${\beta}$ )-metrics with relatively isotropic mean Landsberg curvature, Publ. Math. Debrecen 72 (2008), no. 3-4, 475-485.
10	S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
11	Y. Ichijyo and M. Hashuiguchi, On the condition that a Randers space be conformally flat, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. 22 (1989), 7-14.
12	L. Kang, On conformally flat Randers metrics (in Chinese), Sci. Sin. Math. 41 (2011), no. 5, 439-446. DOI
13	S. Kikuchi, On the condition that a Finsler space be conformally flat, Tensor (N.S.) 55 (1994), no. 1, 97-100.
14	M. S. Knebelman, Conformal geometry of generalised metric spaces, Proc. nat. Acad. Sci. USA, 15(1929), 376-379. DOI
15	B. Li and Z. Shen, On a class of weak Landsberg metrics, Sci. China Ser. A 50 (2007), no. 4, 573-589.
16	H. Rund, The Differential Geometry of Finsler Spaces, Die Grundlehren der Mathematischen Wissenschaften, Bd. 101, Springer-Verlag, Berlin, 1959.
17	Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.