On Spontaneous Compactification in Multidimensional Cosmology

Mutidimensional cosmological models with $n(n\ge 2)$ Einstein spaces $M_i(i=1,\dots ,n)$ are investigated. The cosmological constant and homogeneous minimally coupled scalar field as a matter sources are considered. The scalar field has a potential of general form depending on the scalar field as well as on the scale factors of $M_i$ . The general condition of existence of the solutions with spontaneous compactification is obtained and applied to physically important cases. Two particular kinds of solutions are found. One of them has a wormhole-type continuation into the Euclidean region. Another one represents the generalization of the de Sitter universe to the multidimensional model under consideration.