We characterize the Riemannian manifolds whose the twistor space satisfies the geometric properties necessary to the existence of some sigma model with a Wess-Zumino term on this twistor space. We prove that these manifolds are space forms. Then we study the Riemannian manifolds for which there exists a subbundle of the twistor space which satisfies these geometric properties and prove that in most cases these manifolds are locally homogeneous. In our study, we are led to prove some theorems about metric connections with parallel curvature: we prove for example that a metric connection with parallel curvature and with restricted holonomy group $SO(n)$ must be the Levi-Civita connection and therefore the Riemannian manifold is a space form. We also propose a general method to study metric connections with parallel curvature.
