Metrizability of SO(3)-invariant connections: Riemann versus Finsler

For a torsion-free affine connection on a given manifold, which does not necessarily arise as the Levi–Civita connection of any pseudo-Riemannian metric, it is still possible that it corresponds in a canonical way to a Finsler structure; this property is known as Finsler (or Berwald–Finsler) metrizability. In the present paper, we clarify, for four-dimensional SO (3)-invariant, Berwald–Finsler metrizable connections, the issue of the existence of an affinely equivalent pseudo-Riemannian structure. In particular, we find all classes of S O ( 3 ) -invariant connections which are not Levi–Civita connections for any pseudo-Riemannian metric—hence, are non-metric in a conventional way—but can still be metrized by SO (3)-invariant Finsler functions. The implications for physics, together with some examples are briefly discussed.