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

We investigate the local metrizability of Finsler spaces with m-Kropina metric F = alpha(1+m)beta(-m), where beta is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric alpha and one-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 from 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.

Automated summary

This paper investigates the local metrizability of Finsler spaces with m-Kropina metric F = alpha(1+m)beta(-m), where beta is a closed null one-form. It is shown that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric alpha and one-form beta have a very specific form in certain coordinates. Furthermore, the affine connection on such an m-Kropina space is proven to be locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. The paper also provides counterexamples to Szabo's metrization theorem.