The Metric Geometry of the Manifold of Riemannian Metrics over a Closed\n Manifold

We prove that the L^2 Riemannian metric on the manifold of all smooth\nRiemannian metrics on a fixed closed, finite-dimensional manifold induces a\nmetric space structure. As the L^2 metric is a weak Riemannian metric, this\nfact does not follow from general results. In addition, we prove several\nresults on the exponential mapping and distance function of a weak Riemannian\nmetric on a Hilbert/Frechet manifold. The statements are analogous to, but\nweaker than, what is known in the case of a Riemannian metric on a\nfinite-dimensional manifold or a strong Riemannian metric on a Hilbert\nmanifold.\n