In a spatially homogeneous model, spacetime is filled with a one-parameter set of invariant hypersurfaces H(t). Spatial homogeneity means that the metric on each H(t) is described in terms of constants. Meaning that the metric becomes a function of time only. I guess that this means that given an isometry group (belonging to the Bianchi classes) one have to choose a set of three 1-forms such that the metric depends on time only? That is, all the Bianchi models can be written in the form where ds^2 is given by: ds^2 = -dt^2 + g_ij(t)w^ïw^j, where w^i is the set of forms determined by the isometry group such that that the metric becomes a function of t alone. Could someone clearify this? What do the forms really mean? In Biachi I the forms are given by dx,dy,dz which makes sence. For then the metric will only depend on time since in Biachi I, the isometry group is the group of translations along the spatial coordinate axes.