A note on Einstein metrics and Riemannian twistor spaces

Inspired by the problem of classifying Einstein manifolds with positive scalar curvature, we prove that an Einstein four-manifold whose associated twistor space has scalar curvature constant on the fibers of the twistor bundle is half conformally flat: in particular, the only compact Einstein four-manifolds with positive scalar curvature satisfying this twistorial condition are S⁴ and CP². We also generalize a well-known result due to Friedrich and Grunewald, providing a classification of complete four-manifolds whose twistor space is Ricci parallel.