Finsler spaces with Riemannian geodesics

In Finsler spaces the intervalds=F(xⁱ,dxⁱ) is an arbitrary function of the coordinatesxⁱ and coordinate incrementsdxⁱ withF homogeneous of degree one in thedxⁱ. It is shown that for Riemannian spacesds_R²=g_ijdxⁱdxⁱ which admit a non trivial covariantly constant tensorH_i.(x^k) there is an associated Finsler space with the same geodesic structure. The subset of such Finsler spaces withH_i.(x^k) a vector or second rank decomposable tensor is determined.