Frobenius integrability and Finsler metrizability for $2$-dimensional sprays

TL;DR

This paper investigates the conditions under which a 2-dimensional spray can be metrized by a Finsler function, using Frobenius integrability of a specific distribution and properties of associated differential forms.

Contribution

It introduces the Berwald distribution and characterizes Finsler metrizability through its Frobenius integrability, providing explicit criteria and formulas.

Findings

Finsler metrizability is equivalent to the Frobenius integrability of the Berwald distribution.

A closed, homogeneous 1-form from the annihilator of the Berwald distribution yields the Finsler function.

Regularity conditions are expressed via a 2-form associated with the spray.

Abstract

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous $1$-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a $2$-form, canonically associated to the given spray. The integrability of the Berwald distribution and the regularity of the $2$-form have simple and useful expressions in terms of the Berwald frame.