A generalization of Weylʹs theorem on projectively related affine connections

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators DM and DM on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that D = D + α ⊗ 1 + 1 ⊗ α, where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weylʹs relation, a correspondence between the Yano derivatives can be detected.