Bulletin of the
Korean Mathematical Society BKMS

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