  We review some properties of the space of connections as the natural arena
for canonical (quantum) gravity, and compare to the case of the superspace of
3-metrics. We detail how a 1-parameter family of metrics on the space of
connections arises from the canonical analysis for general relativity which has
a natural interpretation in terms of invariant tensors on the algebra of the
gauge group. We also review the description of canonical GR as a geodesic
principle on the space of connections, and comment on the existence of a time
variable which could be used in the interpretation of the quantum theory.
