Anisotropic Connections and Parallel Transport in Finsler Spacetimes

The general notion of anisotropic connections $$\nabla $$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the metric nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are canonically identified with anisotropic connections. So, these connections provide a simple intrinsic interpretation of a part of any Finsler connection closer to the Koszul formulation in M. Moreover, a new covariant derivative and parallel transport along curves is introduced, taking first a self-propagated vector (instantaneous observer) so that it serves as a reference for the propagation of the others. The covariant derivative of any anisotropic tensor is given by the natural derivative of a curve of tensors obtained by parallel transport along a curve and, in the case of pseudo-Finsler metrics, this is used to characterize the Levi-Civita–Chern anisotropic connection as the one that preserves the length of parallelly propagated vectors.