Towards a dual formulation of quantum gravity via metric-curvature bijections

We prove that Riemannian metrics in General Relativity in the \emph{`normal-coordinates’} gauge are in one-to-one correspondence with curvature 2-forms. We discuss how this can be used as a change of variables in the operator formalism to construct a dual formulation of quantum gravity pertinent in the context of asymptotic safety-like approaches to quantum gravity.