On conformal transformations between two almost regular $(\alpha,\beta)$-metrics
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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 $\tF$, 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 $\tF$ preserving the mean Landsberg curvature. Further, when both $F$ and $\tF$ are regular, the conformal transformation between $F$ and $\tF$ preserving the mean Landsberg curvature must be a homothety.
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 $\tF$, 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 $\tF$ preserving the mean Landsberg curvature. Further, when both $F$ and $\tF$ are regular, the conformal transformation between $F$ and $\tF$ preserving the mean Landsberg curvature must be a homothety.
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