  The space of all Riemannian metrics on a smooth second countable finite
dimensional manifold is itself a smooth manifold modeled on the space of
symmetric (0,2)-tensor fields with compact support. It carries a canonical
Riemannian metric which is invariant under the action of the diffeomorphism
group. We determine its geodesics, exponential mapping, curvature, and Jacobi
fields in a very explicit manner.
