Let (M, F) be a Minkowskian product Finsler manifold of two Finsler manifolds \((M_1, F_1)\) and \((M_2, F_2)\) with \(M=M_1\times M_2\) , \(F=\sqrt{f(K, H)}\) and \(K=F_1^2\) , \(H=F_2^2\) . This paper focuses on several curvature properties of such product manifolds. We obtain a classification for the Minkowskian product Finsler metric F to be a weakly Einstein Finsler metric, and prove that (i) F has almost vanishing \({\mathcal {X}}\) -curvature if and only if it has vanishing \({\mathcal {X}}\) -curvature, (ii) F is a Douglas metric if and only if it is a Berwald metric, (iii) F is a Weyl metric if and only if it has vanishing Riemann curvature. In particular, we show that a Minkowskian product Finsler metric is a Berwald metric of scalar flag curvature if and only if it is a locally Minkowski metric.
