Multidimensional integrable vacuum cosmology with two curvatures

Abstract

The vacuum cosmological model on the manifold describing the evolution of n Einstein spaces of non-zero curvatures is considered. For n = 2 the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when . The Kasner-like behaviour of the solutions near the singularity is considered ( is synchronous time). The exceptional (`Milne-type') solutions are obtained for arbitrary n. For n = 2 these solutions are attractors for other ones, when . For and certain two-parametric families of solutions are obtained from n = 2 ones using the `curvature-splitting' trick. In the case n = 2, a family of non-singular solutions with the topology is found.