Finsler spaces with Riemannian geodesics

In Finsler spaces the intervalds=F(x ⁱ ,dx ⁱ ) is an arbitrary function of the coordinatesx ⁱ and coordinate incrementsdx ⁱ withF homogeneous of degree one in thedx ⁱ . It is shown that for Riemannian spacesds _R ²=g _ij dx ⁱ dx ⁱ 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.