2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154359We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a conjecture formulated in 2002. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Gaussian curvature and the real line, and we give a local parametrization of all totally umbilical surfaces.To appear under the title "Higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces" in Result. Math. (in press)Differential Geometry53B25 (Primary), 53C40 (Secondary)text