  Consider multidim. universes M= R x M_1 x ... x M_n with D = 1+ d_1 .. + d_n,
where M_i of dimension d_i are of have constant curvature and compact for i>1.
For Lagrangian models L(R,phi) on M which depend only on Ricci curvature R and
a scalar field phi, there exists an explicit description of conformal
equivalence, with the minimal coupling model and the conformal coupling model
as distinguished representatives of a conformal class. For the conformally
coupled model we study classical solutions and their relation to solutions in
the equivalent minimally coupled 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={D-2}/{4(D-1)}. In different
noncritical regions of $xi$ the solutions behave qualitatively different. For
vanishing potential of the minimally coupled scalar field we find a
multidimensional generalization of Kasner's solution. Its scale factor
singularity vanishes in the conformal coupling model. Static internal spaces in
the minimal model become dynamical in the conformal one. The nonsingular
conformal solution has a particular interesting region, where internal spaces
shrink while the external space expands. While the Lorentzian solution relates
to a creation of the universe at finite scale, it Euclidean counterpart is an
(instanton) wormhole. Solving the Wheeler de Witt equation we obtain the
quantum counterparts to the classical solutions. A real Euclidean quantum
wormhole is obtained in a special case.
