  A number of 2d and 3d four-fermion models which are renormalizable ---in the
$1/N$ expansion--- in a maximally symmetric constant curvature space, are
investigated. To this purpose, a powerful method for the exact study of spinor
heat kernels and propagators on maximally symmetric spaces is reviewed. The
renormalized effective potential is found for any value of the curvature and
its asymptotic expansion is given explicitly, both for small and for strong
curvature. The influence of gravity on the dynamical symmetry breaking pattern
of some U(2) flavor-like and discrete symmetries is described in detail. %It is
seen explicitly that the effect of a %negative curvature is similar to that of
a magnetic field. The phase diagram in $S^2$ is constructed and it is shown
that, for any value of the coupling constant, a curvature exists above which
chiral symmetry is restored. For the case of $H^2$, chiral symmetry is always
broken. In three dimensions, in the case of positive curvature, $S^3$, it is
seen that some values of the coupling constants lead to a gap equation which
has no solutions. Both for $H^3$ and $S^3$ the configuration given by the
auxiliary fields equated to zero is not a solution of the gap equation.
