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.
