Spatial geometry of the electric field representation of non-abelian gauge theories

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.