The connections of pseudo-Finsler spaces

TL;DR

This paper provides a comprehensive introduction to pseudo-Finsler geometry and its various connections, comparing their properties and implications for Finslerian generalizations of gravity.

Contribution

It introduces a unified framework for Finsler connections, characterizes notable connections via torsion, and explores implications for Finslerian gravity theories.

Findings

Comparison of Finsler connections highlights their advantages.

Curvature symmetries suggest mean Cartan torsion vanishes in Finsler gravity.

New Finsler connection proposed with potential benefits over existing ones.

Abstract

We give an introduction to (pseudo-)Finsler geometry and its connections. For most results we provide short and self contained proofs. Our study of the Berwald non-linear connection is framed into the theory of connections over general fibered spaces pioneered by Mangiarotti, Modugno and other scholars. The main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald's, Cartan's or Chern-Rund's. In this way it becomes easy to compare them and see the advantages of one connection over the other. Since we introduce two soldering forms we are able to characterize the notable Finsler connections in terms of their torsion properties. As an application, the curvature symmetries implied by the compatibility with a metric suggest that in Finslerian generalizations of general relativity the mean Cartan torsion vanishes. This observation allows us to obtain dynamical equations which imply a satisfactory conservation law. The work ends with a discussion of yet another Finsler connection which has some advantages over Cartan's and Chern-Rund's.