Multidimensional cosmological model describing the evolution of (n + 1) Einstein spaces in the theory with several scalar fields and forms is considered. When a (electro-magnetic composite) p-brane Ansatz is adopted the field equations are reduced to the equations for Toda-like system. The Wheeler-De Witt equation is obtained. In the case when n "internal" spaces are Ricci-flat, one space M 0 has a non-zero curvature, and all p-branes do not "live" in M 0 , the classical and quantum solutions are obtained if certain orthogonality relations on parameters are imposed. Spherically-symmetric solutions with intersecting non-extremal p-branes are singled out. A non-orthogonal generalization of intersection rules corresponding to (open, closed) Toda lattices is obtained. A chain of bosonic D ≥ 11 models (that may be related to hypothetical higher dimensional supergravities and F -theories) is suggested.

This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the product form. The sigma-model approach and exact solutions are reviewed. 1

A multidimensional gravitational model containing several dilatonic scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M 0 × M 1 × . . . × M n , where M i are Einstein spaces (i ≥ 1). The block-diagonal metric is chosen and all fields and scale factors of the metric are functions on M 0 . For the forms composite (electro-magnetic) p-brane ansatz is adopted. The model is reduced to gravitating self-interacting sigma-model with certain constraints. In pure electric and magnetic cases the number of these constraints is n 1 (n 1 − 1)/2 where n 1 is number of 1-dimensional manifolds among M i . In the "electro-magnetic" case for dimM 0 = 1, 3 additional n 1 constraints appear. A family of "MajumdarPapapetrou type" solutions governed by a set of harmonic functions is obtained, when all factor-spaces M ν are Ricci-flat. These solutions are generalized to the case of non-Ricci-flat M 0 when also some additional "internal" Einstein spaces of non-zero curvature are added to M . As an example exact solutions for D = 11 supergravity and related 12-dimensional theory are presented.

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N − 1)-dimensional Lobachevsky space H N −1 , N = n + l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N − 2)-dimensional sphere S N −2 by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in "truncated" D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard.

The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perfect fluid. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity are reduced to a billiard on the (n-1)-dimensional Lobachevsky space . The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n-2)-dimensional sphere by point-like sources. Some generalizations of the considered scheme (including scalar field and quantum generalizations) are considered.