  A four dimensional generally covariant field theory is presented which
describes non-dynamical three geometries coupled to scalar fields. The theory
has an infinite number of physical observables (or constants of the motion)
which are constructed from loops made from scalar field configurations. The
Poisson algebra of these observables is closed and is the same as that for the
3+1 gravity loop variables in the Ashtekar formalism. The theory also has
observables that give the areas of open surfaces and the volumes of finite
regions. Solutions to all the Hamilton-Jacobi equations for the theory and the
Dirac quantization conditions in the coordinate representation are given. These
solutions are holonomies based on matter loops. A brief discussion of the loop
space representation for the quantum theory is also given together with some
implications for the quantization of 3+1 gravity.
