CANONICAL STRUCTURE OF LOCALLY HOMOGENEOUS SYSTEMS ON COMPACT CLOSED 3-MANIFOLDS OF TYPES E-3, NIL AND SOL

In this paper we investigate the canonical structure of diffeomorphism -invariant phase spaces for spatially locally homogeneous spacetimes w ith 3-dimensional compact closed spaces. After giving a general algori thm to express the diffeomorphism-invariant phase space and the canoni cal structure of a locally homogeneous system in terms of those of a h omogeneous system on a covering space and a moduli space, we determine completely the symplectic structures and the Hamiltonians of locally homogeneous pure gravity systems on orientable, compact closed 3-space s of the Thurston-type E-3, Nil and Sol for all possible space topolog ies and invariance groups. We point out that in many cases the symplec tic structure of the phase space becomes degenerate in the moduli sect or. This implies that locally homogeneous systems are not canonically closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces.