Anisotropic connections and parallel transport in Finsler spacetimes

TL;DR

This paper explores anisotropic connections in Finsler spacetimes, establishing their relation with Finsler metrics, introducing a new covariant derivative and parallel transport, and characterizing the Levi-Civita--Chern connection through length preservation.

Contribution

It provides a canonical interpretation of anisotropic connections, introduces a new covariant derivative and parallel transport method, and characterizes the Levi-Civita--Chern connection in Finsler geometry.

Findings

Identified vertically trivial Finsler connections with anisotropic connections.

Introduced a new covariant derivative and parallel transport along curves.

Characterized the Levi-Civita--Chern connection as length-preserving.

Abstract

The general notion of anisotropic connections $\nabla$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are identified canonically 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 ({\em 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 parallely propagated vectors.