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 $H^2\times R$, ${\mathbb{S}}^2\times R$ and the $Sol$ group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.