  Using the effective potential, the large-$N$ nonlinear $O(N)$ sigma model
with the curvature coupled term is studied on $S^2\times R^1$. We show that,
for the conformally coupled case, the dynamical mass generation of the model in
the strong-coupled regime on $R^3$ takes place for any finite scalar curvature
(or radius of the $S^2$). If the coupling constant is larger than that of the
conformally coupled case, there exist a critical curvature (radius) above
(below) which the dynamical mass generation does not take place even in the
strong-coupled regime. Below the critical curvature, the mass generation occurs
as in the model on $R^3$.
