  In this paper we establish inclusions and noninclusions between various Hardy
type spaces on noncompact Riemannian manifolds $M$ with Ricci curvature bounded
from below, positive injectivity radius and spectral gap.
  Our first main result states that, if $\mathscr{L}$ is the positive
Laplace-Beltrami operator on $M$, then the Riesz-Hardy space
$H^1_\mathscr{R}(M)$ is the isomorphic image of the Goldberg type space
$\mathfrak{h}^1(M)$ via the map $\mathscr{L}^{1/2} (\mathscr{I} +
\mathscr{L})^{-1/2}$, a fact that is false in $\mathbb{R}^n$. Specifically,
$H^1_\mathscr{R}(M)$ agrees with the Hardy type space $\mathfrak{X}^{1/2}(M)$
recently introduced by the the first three authors; as a consequence, we prove
that $\mathfrak{h}^1(M)$ does not admit an atomic characterisation.
  Noninclusions are mostly proved in the special case where the manifold is a
Damek-Ricci space $S$. Our second main result states that $H^1_\mathscr{R}(S)$,
the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space
$H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp
contrast to the Euclidean case, where these three spaces agree.
