2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/54124We study the $O(N)$ non-linear $σ$ model on three-dimensional manifolds of constant curvature by means of the large $N$ expansion at the critical point. We examine saddle point equations imposing anti-periodic boundary condition in time direction. In the case $S^1 \times S^2$ we find that a solution is inevitably unstable. We briefly refer to the case $S^1 \times S^1 \times S^1$.10pages,LatexHigh Energy Physics - Theorytext