  A unitary transformation $\Ps [E]=\exp (i\O [E]/g) F[E]$ is used to simplify
the Gauss law constraint of non-abelian gauge theories in the electric field
representation. This leads to an unexpected geometrization because
$\o^a_i\equiv -\d\O [E]/\d E^{ai}$ transforms as a (composite) connection. The
geometric information in $\o^a_i$ is transferred to a gauge invariant spatial
connection $\G^i_{jk}$ and torsion by a suitable choice of basis vectors for
the adjoint representation which are constructed from the electric field
$E^{ai}$. A metric is also constructed from $E^{ai}$. For gauge group $SU(2)$,
the spatial geometry is the standard Riemannian geometry of a 3-manifold, and
for $SU(3)$ it is a metric preserving geometry with both conventional and
unconventional torsion. The transformed Hamiltonian is local. For a broad class
of physical states, it can be expressed entirely in terms of spatial geometric,
gauge invariant variables.
