The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space-time. In this paper, we first obtain the PDE characterization of Finsler warped product metrics with a vanishing Riemannian curvature. Moreover, we obtain equivalent conditions for locally Minkowski Finsler warped product spaces. Finally, we explicitly construct two types of non-Riemannian examples.
The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space–time. In this paper, we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we find differential equations of Finsler warped product metrics with vanishing χ-curvature or vanishing H-curvature. Furthermore, we show that, for Finsler warped product metrics, the χ-curvature vanishes if and only if the H-curvature vanishes.
In this paper, we study conformal transformations between two almost regular general (α,β)-metrics. By using the method of special coordinate system, the necessary and sufficient conditions for conformal transformations preserving the mean Landsberg curvature are obtained. Further, a rigidity theorem for regular general (α,β)-metrics is proved.
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