  In this paper, using the Finslerian settings, we study the existence of
parallel one forms (or, equivalently parallel vector fields) on a Riemannian
manifold. We show that a parallel one form on a Riemannian manifold M is a
holonomy invariant function on the tangent bundle T M with respect to the
geodesic spray. We prove that if the metrizability freedom of the geodesic
spray of (M, F) is 1, then the (M, F) does not admit a parallel one form. We
investigate a sufficient condition on a Riemannian manifold to admit a parallel
one form. As a by-product, we relate the existence of a proper affine Killing
vector field by the metrizability freedom. We establish sufficient conditions
for the existence of a parallel one form on a Finsler manifold. By
counter-examples, we show that if the metrizability freedom is greater than 1,
then the manifold (Riemannian or Finslerian) does not necessarily admit a
parallel one form. Various special cases and examples are studied and
discussed.
