Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

We prove that if tau is a large positive number, then the atomic Goldberg-type space h(1)(N) and the space h(R tau)(1) (N) of all integrable functions on N of which local Riesz transform R-tau is integrable, are the same space on any complete noncompact Riemannian manifold N with Ricci curvature bounded from below and positive injectivity radius. We also relate h(1)(N) to a space of harmonic functions on the slice N x (0, delta) for delta > 0 small enough.