  We study a three dimensional conformal field theory in terms of its partition
function on arbitrary curved spaces. The large $N$ limit of the nonlinear sigma
model at the non-trivial fixed point is shown to be an example of a conformal
field theory, using zeta--function regularization. We compute the critical
properties of this model in various spaces of constant curvature ($R^2 \times
S^1$, $S^1\times S^1 \times R$, $S^2\times R$, $H^2\times R$, $S^1 \times S^1
\times S^1$ and $S^2 \times S^1$) and we argue that what distinguishes the
different cases is not the Riemann curvature but the conformal class of the
metric. In the case $H^2\times R$ (constant negative curvature), the $O(N)$
symmetry is spontaneously broken at the critical point. In the case $S^2\times
R$ (constant positive curvature) we find that the free energy vanishes,
consistent with conformal equivalence of this manifold to $R^3$, although the
correlation length is finite. In the zero curvature cases, the correlation
length is finite due to finite size effects. These results describe two
dimensional quantum phase transitions or three dimensional classical ones.
