Average of geometric structures in Finsler spaces with Lorentzian signature

TL;DR

This paper explores the averaging process of geometric structures in Finsler spaces with Lorentzian signature, analyzing associated metrics and connections, and highlighting the independence of the average connection from a timelike vector field.

Contribution

It introduces a method to average geometric structures in Lorentzian Finsler spaces and examines the relationship between the averaged metric and connection.

Findings

A pseudo-Riemannian metric of signature n-1 is associated to the Finsler structure.

An affine, torsion-free connection is derived from the Chern connection.

The average connection is defined without using the timelike vector field.

Abstract

Given the class of Finsler spaces with Lorentzian signature $(M,L)$ on a manifold $M$ endowed with a timelike vector field $\mathcal{X}$ satisfying $g_{(x,y)}(\mathcal{X},\mathcal{X})<0$ at any point $(x,y)$ of the slit tangent bundle, a pseudo-Riemannian metric defined on $M$ of signature $n-1$ is associated to the fundamental tensor $g$. Furthermore, an affine, torsion free connection is associated to the Chern connection determined by $L$. The definition of the average connection does not make use of $\mathcal{X}$. Therefore, there is no direct relation between these two averaged objects.