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 L (2) Riemannian metric-so-called because it induces an L (2) topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L (1)-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L (2) metric. We also give a user-friendly criterion for convergence (with respect to the L (2) metric) in the manifold of metrics.