On Singularities and Instability for Different Couplings between Scalar Field and Multidimensional Geometry

We consider a multidimensional model of the universe given as a $D$-dimensional geometry, represented by a Riemannian manifold $(M,g)$ with arbitrary signature of $g$, $M= \R\times M_1\times \cdots \times M_n$, where the $M_i$ of dimension $d_i$ are Einstein spaces, compact for $i>1$. For Lagrangian models $L(R,\phi)$ on $M$ which depend only on the Ricci curvature $R$ and a scalar field $\phi$, there exists a conformal equivalence with minimal coupling models. For certain nonminimal models we study classical solutions and their relation to solutions in the equivalent minimal coupling model. The domains of equivalence are separated by certain critical values of the scalar field $\phi$. Furthermore, the coupling constant $\xi$ of the coupling between $\phi$ and $R$ is critical at both, the minimal value $\xi=0$ and the conformal value $\xi_c=\frac{D-2}{4(D-1)}$. In different noncritical regions of $\xi$ the solutions behave qualitatively different. Instability can occure only in certain ranges of $\xi$. {This paper is dedicated to Prof. D. D. Ivanenko.}