TY - JOUR
T1 - Berwald m-Kropina spaces of arbitrary signature
T2 - Metrizability and Ricci-flatness
AU - Heefer, Sjors
PY - 2024/12/1
Y1 - 2024/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85212184072&partnerID=8YFLogxK
U2 - 10.1063/5.0227285
DO - 10.1063/5.0227285
M3 - Article
SN - 0022-2488
VL - 65
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 12
M1 - 122502
ER -