Abstract:

Given a closed differential form on an oriented smooth manifold M, does there exist a Riemannian metric on M with respect to which the form becomes harmonic? Do forms allowing such metrics have some intrinsic description/properties? This problem was first set in the literature in 1969 by Calabi, who tried to tackle it for 1-forms. We will discuss this problem and see how it gives rise to a lot of beautiful geometry. The talk will be fairly elementary and all the necessary prerequisites will be given.