On conformal transformations between two almost regular $(\alpha,\beta)$-metrics
Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1231-1240
https://doi.org/10.4134/BKMS.b170710
Published online July 1, 2018
Guangzu Chen, Lihong Liu
East China JiaoTong University, East China JiaoTong University
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 $\tilde{F}$, that is, $\tilde{F}=e^{\kappa(x)}F$, 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 $\tilde{F}$ preserving the mean Landsberg curvature. Further, when both $F$ and $\tilde{F}$ are regular, the conformal transformation between $F$ and $\tilde{F}$ preserving the mean Landsberg curvature must be a homothety.
Keywords : Finsler metric, $(\alpha,\beta)$-metric, conformal transformation, the mean Landsberg curvature
MSC numbers : 53B40, 53C60
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