  Let (M, g) be a complete Riemannian manifold. Assume that the Ricci curvature
of M has quadratic decay and that the volume growth is strictly faster than
quadratic. We establish that the Hardy spaces of exact 1-differential forms on
M , introduced in [4], coincide with the closure in L p of R(d) $\cap$ L p
($\Lambda$ 1 T * M) when 1 < p < $\nu$, where $\nu$ > 2 is related to the
volume growth. The range of p is optimal. This result applies, in particular,
when M has a finite number of Euclidean ends.
