  We prove that if $\tau$ is a large positive number, then the atomic
Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal
R_\tau}^1(N)$ of all integrable functions on $N$ whose local Riesz transform
$\mathcal R_\tau$ 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 $\mathfrak{h}^1(N)$ to a space of harmonic
functions on the slice $N\times (0,\delta)$ for $\delta>0$ small enough.
