  We discuss the phase structure (in the $1/N$-expansion) of the
Nambu-Jona-Lasinio model in curved spacetime with non-trivial topology ${\cal
M}^3 \times {\rm S}^1$. The evaluation of the effective potential of the
composite field $\bar{\psi} \psi$ is presented in the linear curvature
approximation (topology is treated exactly) and in the leading order of the
$1/N$-expansion. The combined influence of topology and curvature to the phase
transitions is investigated. It is shown, in particular, that at zero curvature
and for small radius of the torus there is a second order phase transition from
the chiral symmetric to the chiral non-symmetric phase. When the curvature
grows and (or) the radius of ${\rm S}^1$ decreases, then the phase transition
is in general of first order. The dynamical fermionic mass is also calculated
in a number of different situations.
