  The space of all Riemannian metrics is infinite-dimensional. Nevertheless a
great deal of usual Riemannian geometry can be carried over. The superspace of
all Riemannian metrics shall be endowed with a class of Riemannian metrics;
their curvature and invariance properties are discussed. Just one of this class
has the property to bring the lagrangian of General Relativity into the form of
a classical particle's motion. The signature of the superspace metric depends
in a non-trivial manner on the signature of the original metric, we derive the
corresponding formula. Our approach is a local one: the essence is a metric in
the space of all symmetric rank-two tensors, and then the space becomes a
warped product of the real line with an Einstein space.
