The Riemannian L2 topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L2 Riemannian metric—so-called because it induces an L2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L2 metric. We also give a user-friendly criterion for convergence (with respect to the L2 metric) in the manifold of metrics.