  A dynamical mechanism for symmetry breaking is investigated under the
circumstances with the finite curvature, finite size and non-trivial topology.
A four- and eight-fermion interaction model is considered as a prototype model
which induces symmetry breaking at GUT era. Evaluating the effective potential
in the leading order of the 1/N-expansion by using the dimensional
regularization, we explicitly calculate the phase boundary which divides the
symmetric and the broken phase in a weakly curved space-time and a flat
space-time with non-trivial topology, $R^{D-1} \otimes S^1$.
