  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.
