Extending Bianchi's Classification of Homogeneous Three-Manifolds

A few of the Bianchi types of homogeneous cosmologies described by three-dimensional groups of motions G(3) with transitive actions on three- manifolds were extended to those involving transitive groups of motions G(r) with additional dimensions of four-manifolds. In order to extend a three-manifold metric which admits the Bianchi type N (N=I II V IX) to that of four-manifold the group G(r) was chosen.