2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/171533Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of S^2xR and H^2xR is proved.30 pages. Corrreted typos, added a theorem and a sectionDifferential Geometry53B25, 53C40text