2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163270We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.28 pages, v2 : corrected typos, new section added. To appear in Commentarii Math. HelvetDifferential Geometry53C30, 53B25text