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We systematically investigate the possible transitions between classicalhomogeneous Reimannian 3-hypersurfaces of cosmological models on one hand, and between homogeneous Lorentzian minisuperspace 3-geometries on the other hand.\\ For the Riemannian case, in contrast to our earlier approaches where we only evaluated the three scalar invariants from the Ricci tensor, here we additionally evaluate a scalar invariant constructed from the firstcovariant derivative of the Ricci tensor. Coincidence of these four invariants implies local isometry in the set of homogeneous Riemannian 3-manifolds, whereas the three eigenvalues of hte Ricci tensor do ot suffice.Possible deformations between homogeneous 3-dimensional factor spaces can be classified this way.\\ The analogous classification in the Lorentzian case gives a different result for each signature of the metric. This can be applied to investigate homogeneous deformations of conformal classes of 3-dimensional minisuperspaces. The latter are fundamental ingredient fo the canonical quantization of a factor space cosmology.
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