  We study the $O(N)$ nonlinear $\sigma$ model on a three-dimensional compact
space $S^1 \times S^2$ (of radii $L$ and $R$ respectively) by means of large
$N$ expansion, focusing on the finite size effects and conformal symmetries of
this model at the critical point. We evaluate the correlation length and the
Casimir energy of this model and study their dependence on $L$ and $R$. We
examine the modular transformation properties of the partition function, and
study the dependence of the specific heat on the mass gap in view of possible
extension of the $C-$theorem to three dimensions.
