ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 83-XX Dryuma, Valery, Institute of mathematics AS Moldova (Kishinev) Integration of Einstein equation for higher-dimensional space The vacuum Einstein equations for ten- and six- dimensional spaces with orthogonal metrics in form ds^{2} = φ(u^{k}) ∑_{α=1}^{4}[(du^{α})^{2}] ∑_{i=1}^{6}[(H^{i})^{2}(u^{k})(du^{i})^{2}] depending from the lesser number of variables are considered. The full system of Einstein equation in this case consist of equations forming a multidimensional generalization of Lame-Darboux system for triply-orthogonal surfaces in three-dimensional flat space and can be integrate with help of the Inverse Scattering Method. Using the representation of the functions φ(u^{k}) and H^{i}(u^{k}) as the following φ = ϕ(u^{k}, λ) = exp[\frac{A_{1}}{λ} + \frac{B_{1}}{λ−1} + \frac{C_{1}}{λ+1} +λD_{1}] and H_{i} = h_{i}(u^{k}) exp[\frac{A_{2}}{λ} + \frac{B_{2}}{λ−1} + \frac{C_{2}}{λ+1} +λD_{2}], where λ is “spectral” parameter and A_{i}, B_{i}, D_{i} are some functions of variables u^{k}, a large number of exact solutions of Einstein equations are constructed. Considering four-dimensional Einstein space as embedded in higherdimensional spaces the applications of these solutions to the problems of astrophysics and modern cosmology are discussed.