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Curvature induced phase transition is thoroughly investigated in a four-fermion theory with $N$ components of fermions for arbitrary space-timedimensions $(2 \leq D < 4)$. We adopt the $1/N$ expansion method and calculatethe effective potential for a composite operator $\bar{\psi}\psi$. Theresulting effective potential is expanded asymptotically in terms of thespace-time curvature $R$ by using the Riemann normal coordinate. We assume thatthe space-time curves slowly and keep only terms independent of $R$ and termslinear in $R$. Evaluating the effective potential it is found that thefirst-order phase transition is caused and the broken chiral symmetry isrestored for a large positive curvature. In the space-time with a negativecurvature the chiral symmetry is broken down even if the coupling constant ofthe four-fermion interaction is sufficiently small. We present the behavior ofthe dynamically generated fermion mass. The critical curvature, $R_{cr}$, whichdivides the symmetric and asymmetric phases is obtained analytically as afunction of the space-time dimension $D$. At the four-dimensional limit ourresult $R_{cr}$ agrees with the exact results known in de Sitter space andEinstein universe.Comment: 19 pages, uses LaTeX, eepic.st

CURVATURE-INDUCED PHASE TRANSITION IN A FOUR-FERMION THEORY USING THE WEAK CURVATURE EXPANSION