Multidimensional Gravity with Einstein Internal Spaces

Abstract

A multidimensional gravitational model on the manifold $M=M_0\times {\prod }_{i=1}^n{M}_i$, where M_i are Einstein spaces ($i\ge 1$), is studied. For $N_0=dim{M}_0>2$ the $\sigma$ model representation is considered and it is shown that the corresponding Euclidean Toda-like system does not satisfy the Adler-van-Moerbeke criterion. For $M_0=R^{N_0}$, $N_0=3,4,6$ (and the total dimension $D=dimM=11,10,11$, respectively) nonsingular spherically symmetric solutions to vacuum Einstein equations are obtained and their generalizations to arbitrary signatures are considered. It is proved that for a non-Euclidean signature the Riemann tensor squared of the solutions diverges on certain hypersurfaces in $R^{N_0}$.

10 pages, LaTex