On the metrizability of $m$-Kropina spaces with closed null 1-form

TL;DR

This paper studies when certain Finsler spaces with $m$-Kropina metrics are locally metrizable by (pseudo-)Riemannian metrics, especially focusing on spaces with closed null 1-forms and their relation to Berwald spaces and Szabo's theorem.

Contribution

It characterizes the local metrizability of $m$-Kropina spaces with closed null 1-forms and constructs counterexamples to Szabo's metrization theorem in indefinite signatures.

Findings

Spaces are Berwald if and only if $eta$ and $ ext{α}$ have specific forms.

Metrizability is equivalent to the symmetry of the Ricci tensor.

Constructed counterexamples to Szabo's theorem for indefinite metrics.

Abstract

We investigate the local metrizability of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$, where $\beta$ is a closed null 1-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric $\alpha$ and 1-form $\beta$ have a very specific form in certain coordinates. In particular, when the signature of $\alpha$ is Lorentzian, $\alpha$ belongs to a certain subclass of the Kundt class and $\beta$ generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an $m$-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed form the affine connection is symmetric. In particular we construct all counterexamples of this type to Szabo's metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.