Characterization of intrinsically harmonic forms

TL;DR

This paper investigates the existence of Riemannian metrics making a given closed 1-form co-closed on closed oriented manifolds, extending previous results to forms with arbitrary zero sets.

Contribution

It provides a comprehensive answer to the existence question for co-closed metrics associated with closed 1-forms without restrictions on their zeros.

Findings

Extended Calabi's 1969 results to forms with arbitrary zeros.

Characterized conditions for the existence of co-closed metrics.

Generalized understanding of harmonic forms on manifolds.

Abstract

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 $\omega$.