  We introduce a decreasing one-parameter family $\mathfrak{X}^{\gamma}(M)$,
$\gamma>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$
on certain nondoubling Riemannian manifolds with bounded geometry and we
investigate their properties. In particular, we prove that
$\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in
$\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the
surprising result that this space does not admit an atomic decomposition.
