Morse theory of harmonic forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use a theorem of Calabi [1], characterizing 1-forms which are harmonic with respect to some metric, in an essential way. We also study some interesting examples illustrating our results.