Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(\alpha ,\beta )$-Finsler spaces or spacetimes, respectively, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the geometry defining 1-form. We illustrate our results with novel examples of $(\alpha ,\beta )$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries.

Berwald 几何是接近（伪）-黎曼几何的 Finsler 几何。我们建立了一个简单的一阶偏微分方程作为充要条件，一个给定的芬斯勒拉格朗日量必须满足它是 Berwald 类型的。应用于$(\alpha ,\beta )$-Finsler 空间或时空，这分别简化为定义 1-形式的几何的 Levi-Civita 协变导数的必要和充分条件。我们用新的例子来说明我们的结果$(\alpha ,\beta )$- 代表昆特（恒定标量不变）时空的 Finslerian 版本的 Berwald 几何。结果概括了 Tavakol 和 van den Bergh 的早期发现，以及 Randers 和 m-Kropina 的 Berwald 条件。非常特殊/广义相对论几何。