  Let $M$ be a closed oriented manifold of dimension $n$ and $\omega$ a closed
1-form on it. We discuss the question whether there exists a Riemannian metric
for which $\omega$ is co-closed. For closed 1-forms with nondegenerate zeros
the question was answered completely by Calabi in 1969. The goal of this paper
is to give an answer in the general case, i.e. not making any assumptions on
the zero set of $\omega$.
