Berwald m-Kropina spaces of arbitrary signature: Metrizability and Ricci-flatness

Sjors Heefer (Corresponding author)

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Abstract

The (pseudo-)Riemann-metrizability and Ricci-flatness of Finsler spaces with m-Kropina metric F = α 1+mβ −m of Berwald type are investigated. We prove that the affine connection of F can locally be understood as the Levi-Civita connection of some (pseudo-)Riemannian metric if and only if the Ricci tensor of the canonical affine connection is symmetric. We also obtain a third equivalent characterization in terms of the covariant derivative of the 1-form β. We use these results to classify all locally metrizable m-Kropina spaces whose 1-forms have a constant causal character. In the special case where the first de Rham cohomology group of the underlying manifold is trivial (which is true of simply connected manifolds, for instance), we show that global metrizability is equivalent to local metrizability and hence, in that case, our necessary and sufficient conditions also characterize global metrizability. In addition, we further obtain explicitly all Ricci-flat, locally metrizable m-Kropina metrics in (3 + 1)D whose 1-forms have a constant causal character. In fact, the only possibilities are essentially the following two: either α is flat and β is α-parallel, or α is a pp-wave and β is α-parallel.

Original languageEnglish
Article number122502
Number of pages11
JournalJournal of Mathematical Physics
Volume65
Issue number12
DOIs
Publication statusPublished - 1 Dec 2024

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