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A number of 2d and 3d four-fermion models which are renormalizable ---in the$1/N$ expansion--- in a maximally symmetric constant curvature space, areinvestigated. To this purpose, a powerful method for the exact study of spinorheat kernels and propagators on maximally symmetric spaces is reviewed. Therenormalized effective potential is found for any value of the curvature andits asymptotic expansion is given explicitly, both for small and for strongcurvature. The influence of gravity on the dynamical symmetry breaking patternof some U(2) flavor-like and discrete symmetries is described in detail. %It isseen explicitly that the effect of a %negative curvature is similar to that ofa magnetic field. The phase diagram in $S^2$ is constructed and it is shownthat, for any value of the coupling constant, a curvature exists above whichchiral symmetry is restored. For the case of $H^2$, chiral symmetry is alwaysbroken. In three dimensions, in the case of positive curvature, $S^3$, it isseen that some values of the coupling constants lead to a gap equation whichhas no solutions. Both for $H^3$ and $S^3$ the configuration given by theauxiliary fields equated to zero is not a solution of the gap equation.Comment: Improvements in the text, one of the figures replace

Phase structure of renormalizable four-fermion models in spacetimes of constant curvature