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

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.

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• From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology
• Cosmological Landsberg Finsler spacetimes
• Raychaudhuri equations, Tidal forces and Weak field Limit in Schwarzshild-Finsler-Randers spacetime
• Berwald $m$-Kropina Spaces of Arbitrary Signature: Metrizability and Ricci-Flatness
• Metrizability of SO(3)-invariant connections: Riemann versus Finsler