Let M be a closed oriented manifold of dimension n and ω a closed 1-form on M. We discuss
the question of whether there exists a Riemannian metric for which ω is co-closed. For closed 1-forms with nondegenerate zeros, the question was answered completely by Calabi in 1969; (cf.Calabi, Global analysis, Papers in honour of K. Kodaira). The goal of this paper is to give an answer in the general case, that is, without making any assumptions on the zero set of ω.