On the existence of parallel one forms

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 TM 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 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.

Automated summary

In this paper, we study the existence of parallel one forms (or parallel vector fields) on Riemannian manifolds using Finslerian settings. We prove that a parallel one form on a Riemannian manifold is a holonomy invariant function on the tangent bundle TM with respect to the geodesic spray. We investigate the conditions for the existence of a parallel one form on Riemannian and Finsler manifolds, and show that the metrizability freedom affects the existence of proper affine Killing vector fields.