  We canonically quantize $O(D+2)$ nonlinear sigma models (NLSMs) with theta
term on arbitrary smooth, closed, connected, oriented $D$-dimensional spatial
manifolds $\mathcal{M}$, with the goal of proving the suitability of these
models for describing symmetry-protected topological (SPT) phases of bosons in
$D$ spatial dimensions. We show that in the disordered phase of the NLSM, and
when the coefficient $\theta$ of the theta term is an integer multiple of
$2\pi$, the theory on $\mathcal{M}$ has a unique ground state and a finite
energy gap to all excitations. We also construct the ground state wave
functional of the NLSM in this parameter regime, and we show that it is
independent of the metric on $\mathcal{M}$ and given by the exponential of a
Wess-Zumino term for the NLSM field, in agreement with previous results on flat
space. Our results show that the NLSM in the disordered phase and at
$\theta=2\pi k$, $k\in\mathbb{Z}$, has a symmetry-preserving ground state but
no topological order (i.e., no topology-dependent ground state degeneracy),
making it an ideal model for describing SPT phases of bosons. Thus, our work
places previous results on SPT phases derived using NLSMs on solid theoretical
ground. To canonically quantize the NLSM on $\mathcal{M}$ we use Dirac's method
for the quantization of systems with second class constraints, suitably
modified to account for the curvature of space. In a series of four appendices
we provide the technical background needed to follow the discussion in the main
sections of the paper.
