We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics.

中文翻译：

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Finsler几何中使用一类各向异性连接的曲率计算

我们展示了如何使用仿射连接来计算张量导数和曲率张量。这允许在不使用坐标系的情况下以与经典黎曼几何中出现的计算平行的方式获得所有计算。特别是，我们获得了任何各向异性连接的曲率张量的Bianchi身份，我们比较了任何两个各向异性连接的曲率张量，并且找到了一个非常适合研究Finsler度量几何的各向异性连接族。