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

Curvature and topological effects on dynamical symmetry breaking in a four- and eight-fermion interaction model