标题： 伯威尔德$m$-克普林空间的任意符号：可度量化性和里奇平坦性 显示英文标题

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

摘要： 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. 事实上，唯一的可能性基本上如下两种：要么$\alpha$是平坦的且$\beta$是$\alpha$-平行的；要么$\alpha$是一个pp波且$\beta$是$\alpha$-平行的。 显示英文摘要

摘要： 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.