Curvature induced phase transition is thoroughly investigated in a four- fermion theory with N components of fermions for arbitrary space-time dimensions (2 ≤ D < 4). We adopt the 1/N expansion method and calculate the effective potential for a composite operator ψ̅ψ. The resulting effective potential is expanded asymptotically in terms of the space-time curvature R by using the Riemann normal coordinate. We assume that the space-time curves slowly and keep only terms independent of R and terms

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... linear in R. Evaluating the effective potential it is found that the first-order phase transition is caused and the broken chiral symmetry is restored for a large positive curvature. In the space-time with a negative curvature the chiral symmetry is broken down even if the coupling constant of the four-fermion interaction is sufficiently small. We present the behavior of the dynamically generated fermion mass. The critical curvature, R_cr, which divides the symmetric and asymmetric phases is obtained analytically as a function of the space-time dimension D. At the four-dimensional limit our result R_cr agrees with the exact results known in de Sitter space and Einstein universe.