Berwald $m$-Kropina Spaces of Arbitrary Signature: Metrizability and Ricci-Flatness

TL;DR

This paper investigates the conditions under which certain Finsler spaces with $m$-Kropina metrics are metrizable and Ricci-flat, providing classifications and explicit examples in four-dimensional cases.

Contribution

It establishes necessary and sufficient conditions for local and global metrizability of $m$-Kropina spaces and classifies Ricci-flat examples with constant causal character in four dimensions.

Findings

Affine connection is Levi-Civita iff Ricci tensor is symmetric.

Classification of locally metrizable $m$-Kropina spaces with constant causal forms.

Explicit Ricci-flat, locally metrizable $m$-Kropina metrics in 4D with specific geometric structures.

Abstract

The (pseudo-)Riemann-metrizability and Ricci-flatness of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$ of Berwald type are investigated. We prove that the affine connection on $F$ 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 $\beta$. 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 Rahm 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 $\alpha$ is flat and $\beta$ is $\alpha$-parallel, or $\alpha$ is a pp-wave and $\beta$ is $\alpha$-parallel.