In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the nonlinear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metrical formulation of Finsler geometry which is well adapted to field theory, and which has proved useful in Einstein–Cartan-like approaches to Finsler gravity.

中文翻译：

---

芬斯勒几何的度量方法

在芬斯勒几何的标准方法中，度量被定义为垂直的 Hessian 矩阵，而 Chern 或 Cartan 连接仅显示为回拉切线丛上许多可能的自然线性连接中的两个。这里表明，度量的 Hessian 性质、非线性连接以及 Chern 或 Cartan 连接可以从度量和 Finsler 连接之间的一些兼容性公理导出。这一结果提供了芬斯勒几何的度量公式，该公式非常适合场论，并且已被证明在类似于爱因斯坦-嘉当的芬斯勒引力方法中很有用。