On the pseudo-Riemann metrizability of SO(3)-symmetric Berwald-Finsler spaces

We clarify, for SO(3)-symmetric 4-dimensional Berwald-Finsler spaces of indefinite signature, the question of the existence of an affinely equivalent pseudo-Riemannian metric. It turns out that the answer depends on the holonomy distribution of the canonical affine connection and on the symmetry of its Ricci tensor. In particular, we find all classes of 4-dimensional SO(3)-invariant, symmetric affine connections which do not arise as Levi-Civita connections of any pseudo-Riemannian metric, but can still be metrized by (SO(3)-symmetric) Finsler functions; in Lorentzian signature, these will provide Berwald spacetime structures whose geodesic structure cannot be ascribed to any Lorentzian metric. Some concrete examples are also presented.