  Consider a real-analytic orientable connected complete Riemannian manifold
$M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le
n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the
manifold and the metric on it can be reconstructed, up to an isometry, from the
set of the Cauchy data for harmonic $k$-forms, given on an open subset of the
boundary. This extends a result of [13] when $k=0$. In the two-dimensional
case, the same conclusion is obtained when considering the set of the Cauchy
data for harmonic $1$-forms. Under additional assumptions on the curvature of
the manifold, we carry out the same program when $M$ is complete non-compact.
In the case $n\ge 3$, this generalizes the results of [12] when $k=0$. In the
two-dimensional case, we are able to reconstruct the manifold from the set of
the Cauchy data for harmonic $1$-forms.
