  We derive explicit forms of the two--point correlation functions of the
$O(N)$ non-linear sigma model at the critical point, in the large $N$ limit, on
various three dimensional manifolds of constant curvature. The two--point
correlation function, $G(x, y)$, is the only $n$-point correlation function
which survives in this limit. We analyze the short distance and long distance
behaviour of $G(x, y)$. It is shown that $G(x, y)$ decays exponentially with
the Riemannian distance on the spaces $R^2 \times S^1,~S^1 \times S^1 \times R,
~S^2 \times R,~H^2 \times R$. The decay on $R^3$ is of course a power law. We
show that the scale for the correlation length is given by the geometry of the
space and therefore the long distance behaviour of the critical correlation
function is not necessarily a power law even though the manifold is of infinite
extent in all directions; this is the case of the hyperbolic space where the
radius of curvature plays the role of a scale parameter. We also verify that
the scalar field in this theory is a primary field with weight $\delta=-{1
\over 2}$; we illustrate this using the example of the manifold $S^2 \times R$
whose metric is conformally equivalent to that of $R^3-\{0\}$ up to a
reparametrization.
