ON SPHERICALLY SYMMETRIC SOLUTIONS IN D-DIMENSIONAL DILATON GRAVITY

Exact static, spherically symmetric solutions to the Einstein-Abelian gauge-dilaton equations, in $D$-dimensional gravity with a chain of $n$ Ricci-flat internal spaces are considered, with the gauge field potential having three nonzero components: the temporal, Coulomb-like one, the one pointing to one of the extra dimensions, and the one responsible for a radial magnetic field. For dilaton coupling implied by string theory an $(n+5)$-parametric family of exact solutions is obtained, while for other dilaton couplings only $(n+3)$-parametric ones. The geometric properties and special cases of the solutions are discussed, in particular, those when there are horizons in the space-time. Two types of horizons are distinguished: the conventional black-hole (BH) ones and those at which the physical section of the space-time changes its signature ({\it T-horizons}). Two theorems are proved, one fixing the BH and T-horizon existence conditions, the other discarding the possibility of a regular center. Different conformal gauges are used to characterize the system from the $D$-dimensional and 4-dimensional viewpoints.