  The Hamiltonians of $SU(2)$ and $SU(3)$ gauge theories in 3+1 dimensions can
be expressed in terms of gauge invariant spatial geometric variables, i.e.,
metrics, connections and curvature tensors which are simple local functions of
the non-Abelian electric field. The transformed Hamiltonians are local. New
results from the same procedure applied to the $SU(2)$ gauge theory in 2+1
dimensions are also given.
