  In the geometrodynamical setting of general relativity in Lagrangian form,
the objects of study are the {\it Riemannian} metrics (and their time
derivatives) over a given 3-manifold $M$. It is our aim in this paper to study
the gauge properties that the space Riem(M) of all metrics over $M$ possesses,
specially as they relate to the constraints of geometrodynamics. For instance,
the Hamiltonian constraint does not generate a group, and it is thus hard to
view its action in Riem(M) in a gauge setting. However, in view of the recent
results representing GR as a dual theory, invariant under foliation preserving
3--diffeomorphisms and 3D conformal transformations, but not under
refoliations, we are justified in considering the gauge structure pertaining
only to the groups $\mathcal{D}$ of diffeomorphisms of $M$, and $\mathcal{C}$,
of conformal diffeomorphisms on $M$. For these infinite-dimensional symmetry
groups, Riem(M) has a natural principal fiber bundle (PFB) structure, which
renders the gravitational field amenable to the full range of gauge-theoretic
treatment. We discuss some of these structures and construct explicit formulae
for supermetric-induced gauge connections. To apply the formalism, we compute
general properties for a specific connection bearing strong resemblance to the
one naturally induced by the deWitt supermetric, showing it has desirable
relationalist properties. Finally, we find that the group of conformal
diffeomorphisms solves the pathologies inherent in the $\DD$ group and also
brings it closer to Horava gravity and the dual conformal theory called Shape
Dynamics.
