2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222384We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. In the route, we also discover new characterizations of space-forms. We next generalize results concerning metric rigidity via equations involving vector fields.Differential Geometry53C20text