Totally umbilic surfaces in homogeneous 3-manifolds

Abstract

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in ℍ2 × ℝ, \mathbb{S}2 × ℝ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.