  The O(N) non-linear sigma model in a $D$-dimensional space of the form ${\bf
R}^{D-M} \times {\bf T}^M$, ${\bf R}^{D-M} \times {\bf S}^M$, or ${\bf T}^M
\times {\bf S}^P$ is studied, where ${\bf R}^M$, ${\bf T}^M$ and ${\bf S}^M$
correspond to flat space, a torus and a sphere, respectively. Using zeta
regularization and the $1/N$ expansion, the corresponding partition functions
and the gap equations are obtained. Numerical solutions of the gap equations at
the critical coupling constants are given, for several values of $D$. The
properties of the partition function and its asymptotic behaviour for large $D$
are discussed. In a similar way, a higher-derivative non-linear sigma model is
investigated too. The physical relevance of our results is discussed.
