Transactions of the American Mathematical Society, Vol. 238 (Apr., 1978), pp. 45-55 (11 pages) Any analytic symplectic diffeomorphism $\Phi$ of a symplectic manifold ...

In 1966, Chern asked which compact orientable manifolds carry a Riemannian metric and a vector field which is parallel under this metric. An earlier paper answered this without assuming orientability.