The Moduli Space of Local Homogeneous 3-geometries †

For a canonical formulation of quantum gravity, the superspace of all possible 3-geometries on a Cauchy hypersurface of a 3 + 1-dimensional Lorentzian manifold plays a key role. While in the analogous 2 + 1-dimensional case the superspace of all Riemannian 2-geometries is well known, the structure of the superspace of all Riemannian 3-geometries has not yet been resolved at present. In this paper, an important subspace of the latter is disentangled: The superspace of local homogenous Riemannian 3-geometries. It is finite dimensional and can be factored by conformal scale dilations, with the flat space as the center of projection. The corresponding moduli space can be represented by homothetically normalized 3-geometries. By construction, this moduli space of the local homogenous 3-geometries is an algebraic variety. An explicit parametriza-tion is given by characteristic scalar invariants of the Riemannian 3-geometry. Although the moduli space is not locally Euclidean, it is a Hausdorff space. Nevertheless, its topology is compatible with the non-Hausdorffian topology of the space of all Bianchi-Lie algebras, which characterize the moduli modulo differences in their anisotropy. † This lecture is financially supported by the DAAD (Germany) and the IPM (Iran). 1. Introduction The canonical formalism of (quantum) gravity uses a topo-logical d + 1 decomposition of the (d + 1)-dimensional Lorentzian manifold M d+1 = M 1 × M d , into a time man-ifold M 1 and a smooth topological d-space M d , which for any t ∈ M 1 is a Riemannian manifold M d (t). The action for pure Einstein gravity on M d+1 then becomes