  We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes
equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence
of the Hodge decomposition. We show that this phenomenon does not occur on the
hyperbolic spaces of higher dimension. We also describe the corresponding
general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds,
which includes the hyperbolic setting.
