annals of physics 248, 246285 (1996) Pure Geometrical Evolution of the Multidimensional Universe Zdzis|aw A. Golda, * Marco Litterio, - Leszek M. Soko|owski,* Luca Amendola,  and Andrzej Dyrek 9 * Astronomical Observatory, Jagellonian University, Orla 171, 30-244 Krako w, Poland ; - Instituto Astronomico, Universita di Roma ``La Sapienza,'' Via Lancisi 29, I-00161 Roma, Italy ;  Osservatorio Astronomico di Roma, Viale del Parco Mellini 84, I-00136 Roma, Italy ; and 9 Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Krako w, Poland Received July 11, 1995 An exhaustive qualitative analysis of cosmological evolution for some multidimensional universes is given. The internal space is taken to be a compact Lie group Riemannian manifold. The space is generically anisotropic; i.e., its cosmological evolution is described by its (time-dependent) volume, the dilaton, and by relative anisotropic deformation factors representing the shear of the internal dimensions during the evolution. Neither the internal space nor its subspaces need to be Einstein spaces. The total spacetime is empty, and the cosmic evolution of the external, four-dimensional world is driven by the geometric ``matter'' consisting of the dilaton and of the deformation factors. Since little is known about any form of matter in the extra dimensions, we do not introduce any ad hoc matter content of the Universe. We derive the four-dimensional Einstein field equations (with a cosmological term) for these geometric sources in full generality, i.e., for any compact Lie group. A detailed analysis is done for some specific internal geometries: products of 3-spheres, and SU(3) space. Asymptotic solutions exhibit power law inflation along with a process of full or partial isotropization. For the SU(3) space all the deformation factors tend to a common value, whereas in the case of S 3 's each sphere isotropizes separately.  1996 Academic Press, Inc. 1. Introduction After 14 years since its first appearance [1], inflation is now a necessary ingredient of the standard cosmological model [2]. Since it will be frequently men- tioned in this paper in order to assess if any given model gives inflation or not, let us identify its most relevant features. For the purposes of the present paper it will be enough recalling that inflation is a period of accelerated expansion such that in a short time interval ( t t10 &32 s for exponential inflation) the scale factor of the universe a( t ) grows from a 0 (that we put in the following equal to unity), a 0 =1, to ln a & 60. This evolution is driven by some scalar field , whose dynamics is gov- erned by the potential V( ,). A successful inflation is achieved if V( ,) has: (i) a flat region, in which , moves slowly, large enough to allow for the 60 e-folding growth of a ; (ii) a minimum around which , oscillates, dissipating its kinetic energy and thus reheating the universe to a temperature just below the critical temperature of grand-unification. The second condition allows the Universe to enter the usual article no. 0059 246 0003-491696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.