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When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian?

(Reposted from https://math.stackexchange.com/questions/2589600/when-is-a-divergence-free-vector-field-on-the-tangent-bundle-of-a-riemannian-man) Starting with a closed, connected Riemannian manifold $...