Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$. 


**Question**: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic, i.e., $\Delta\_g \omega = 0$? 


Here $\Delta\_g$ is the Laplace-deRham operator, defined as usual by
$\Delta\_g = d \delta + \delta d$, where $\delta$ is
the $g$-codifferential. Note that non-exactness is important, since if $\omega$ were to be exact and harmonic, then by the Hodge decomposition theorem $\omega = 0$.


For instance, if $\omega$ is a 1-form on the unit circle, then it is not hard to see that $\omega$ is harmonic with respect to some metric $g$ if and only if it is a volume form (i.e., it doesn't vanish). This observation generalizes to forms of top degree on any $M$.


What can be said in general for forms which are not of top degree?



When is a closed differential form harmonic relative to some metric?
A closed $k$-form is called *intrinsically harmonic* if there is some Riemannian metric with respect to which it is harmonic. E. Calabi (*Calabi, Eugenio*, An intrinsic characterization of harmonic one-forms, Global Analysis, Papers in Honor of K. Kodaira 101-117 (1969). [ZBL0194.24701](https://zbmath.org/?q=an:0194.24701).)
showed that a one-form having non-degenerate zeros on a compact manifold without boundary is intrinsically harmonic if and only if it satisfies a property called *transitivity*. The precise statement and proof can be found in chapter 9 of M. Farber's book "Topology of closed one-forms". In what comes I am following Farber. That a closed one-form $\omega$ have non-degenerate zeros means that near each zero it can be written in the form $\omega = df$ with $f$ a Morse function. For such a one-form, the additional assumption of harmonicity means that the Morse index of a zero cannot be $0$ or $n$ (write $\omega = df$ near the zero; because $\omega$ is co-closed, $f$ is harmonic, so by the maximum principle cannot have a max or min at the zero). That $\omega$ be *transitive* means that for any point $p$ of $M$ which is not a zero of $\omega$ there is a smooth $\omega$-positive loop $\gamma: [0, 1] \to M$; that is, $\gamma(0) = p = \gamma(1)$, and $\omega(\dot{\gamma}(t)) > 0$ for $t \in [0, 1]$. Then Calabi's theorem states that a closed one-form with non-degenerate zeros is intrinsically harmonic if and only if it is transitive.
Near a non-degenerate index $0$ zero of a closed one-form the one-form can be written in the form $\delta\_{ij}x^{i}dx^{j}$, for which it can be checked there are no positive loops beginning sufficiently near the origin.


(If one can handle $k$-forms then by Hodge duality one expects to be able to get somewhere with $(n-k)$-forms. The intrinsic harmonicity of $(n-1)$-forms was characterized in terms of transitivity in the thesis of Ko Honda, available on his web page). Evgeny Volkov (*Volkov, Evgeny*, [**Characterization of intrinsically harmonic forms**](http://dx.doi.org/10.1112/jtopol/jtn014), J. Topol. 1, No. 3, 643-650 (2008). [ZBL1148.57036](https://zbmath.org/?q=an:1148.57036).)
weakens the non-degeneracy condition, replacing it with the condition that the closed one-form be *locally intrinsically harmonic* - that is, the restriction of the form to a suitable open neighborhood of its zero set is intrinsically harmonic.


As far as I know, for higher degree forms nothing much is known at all, though for some special cases, like $2$-forms on $4$-manifolds, something more has been said. One imagines that with further assumptions on the form, perhaps more can be said - for example a symplectic form is always intrinsically harmonic (use the metric determined by a compatible almost complex structure). On the other hand, Volkov's paper exhibits a closed $2$-form of rank $2$ on a $4$-manifold which is transitive but not intrinsically harmonic.


