About the S³ Group-manifold Reduction of Einstein Gravity

We exhibit a new consistent group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S^3. The novel feature in the reduction is to exploit the two 3-dimensional Lie algebras that S^3 admits. The first algebra is introduced into the group-manifold reduction in the standard way through the Maurer-Cartan 1-forms associated to the symmetry of the general coordinate transformations. The second algebra is associated to the linear adjoint group and it is introduced into the group-manifold reduction through a local transformation in the internal tangent space. We discuss the characteristics of the resulting lower-dimensional theory and we emphasize the novel results generated by the new group-manifold reduction. As an application of the reduction we show that the lower-dimensional theory admits a domain wall solution which upon uplifting to the higher-dimension results to be the self-dual (in the non-vanishing components of both curvature and spin connection) Kaluza-Klein monopole. This discussion may be relevant in the dimensional reductions of M-theory, string theory and also in the Bianchi cosmologies in four dimensions.