The group-theoretical classification of the 11-dimensional classical homogeneous Kaluza–Klein cosmologies

Abstract

In the context of the classical Kaluza–Klein cosmology the generalized Bianchi models in 11 dimensions are considered. These are space-times whose spacelike ten-dimensional sections are the hypersurfaces of transitivity for a ten-dimensional isometry group of the total space-time. Such a space-time is a trivial principal fiber bundle P(M,G7), where M is a four-dimensional physical space-time with an isometry group G3 (of a Bianchi type) and G7 is a compact isometry group of the compact internal space. The isometry group of P is G10=G3⊗G7, hence all the generalized Bianchi models are classified by enumerating the relevant groups G7. Due to the compactness of G7 the result is astonishingly simple: there are three distinct homogeneous internal spaces in addition to the 11 ordinary Bianchi types for M.