1. On the metrizability of m-Kropina spaces with closed null 1-form
- Author
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Heefer, Sjors, Pfeifer, Christian, van Voorthuizen, Jorn, Fuster, Andrea, Applied Differential Geometry, Center for Analysis, Scientific Computing & Appl., Mathematics and Computer Science, Applied Physics and Science Education, and Mathematical Image Analysis
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,differential geometry ,Finsler geometry ,General Relativity and Quantum Cosmology - 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.
- Published
- 2023