Anisotropic connections and parallel transport in Finsler spacetimes

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.

arXiv comment29 pages

References in corpus (7)

• On the definition and examples of cones and Finsler spacetimes
• Mathematical foundations for field theories on Finsler spacetimes
• Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes
• Foundations of Finsler spacetimes from the Observers' Viewpoint
• An equivalence of Finslerian relativistic theories
• Wind Riemannian spaceforms and Randers metrics of constant flag curvature
• Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds

Cited by in corpus (1)

• The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry