A metrical approach to Finsler geometry

TL;DR

This paper introduces a metrical formulation of Finsler geometry based on compatibility axioms, simplifying the derivation of key structures and aiding applications in Finsler gravity theories.

Contribution

It derives the metric, non-linear connection, and Chern or Cartan connections from compatibility axioms, offering a new foundational approach to Finsler geometry.

Findings

Derived metric and connections from compatibility axioms

Provided a metrical formulation suitable for field theory

Facilitated applications in Finsler gravity models

Abstract

In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the non-linear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metrical formulation of Finsler geometry which is well adapted to field theory, and which has proved useful in Einstein-Cartan-like approaches to Finsler gravity.