Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L²$L^2$ metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.