Anisotropic inflation from extra dimensions

Vacuum multidimensional cosmological models with internal spaces being compact n-dimensional Lie group manifolds are considered, Products of 3-spheres and SU(3) manifold (a novelty in cosmology) are studied, It turns out that the dynamical evolution of the internal space drives (power-law) inflation in the external world. This inflationary solution brings two extra bonuses: 1) it is an attractor in phase space (no fine-tuning required); 2) it is determined by the Lie group space solely and not by any arbitrary inflaton potentials, Only scalar fields representing the anisotropic scale factors of the internal space appears in the four dimensions. The size of the volume of the internal space at the end of inflation is compatible with observational constraints. This simple and natural model can be completed by some extended-inflation-like mechanism that ends the inflationary evolution.