  We study the nonlinear $O(N)$ sigma model on $S^2$ with the gravitational
coupling term, by evaluating the effective potential in the large-$N$ limit. It
is shown that there is a critical curvature $R_c$ of $S^2$ for any positive
gravitational coupling constant $\xi$, and the dynamical mass generation takes
place only when $R<R_c$. The critical curvature is analytically found as a
function of $\xi$ $(>0)$, which leads us to define a function looking like a
natural generalization of Euler-Mascheroni constant.
