In a Bianchi IX universe the metric must be invariant under the SO(3) group acting on the 3-sphere. Hence, the metric must be translation invariant in the spatial parts, where t=constant. This implies that the metric must take the form such that: ds^2 = dt^2 - g_ij(t)(x^i)(x^j), where g is a function of t alone. Am I right about all this? What concerns me is that someone told me that the metric: ds^2 = -dt^2 + a^2(t)(dx)^2 + b^2(t)(dy)^2 + (b^2sin^2y+a^2cos^2y)(dz)^2 - 2a^2cosydxdz belong to the Bianchi IX models. But this doesn't seem right?! Am I right about the Biachi IX models being homogeneous but not necesseraly isotropic?