Totally umbilic surfaces in homogeneous 3-manifolds

Abstract

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 R, 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. Mathematics Subject Classification (2000). 53C30, 53B25.

ON EXTRINSICALLY SYMMETRIC HYPERSURFACES OF ℍn×ℝ

“…We remark that the third author has classified totally umbilical surfaces in S 2 × R and H 2 × R, by means of an explicit parametrization, in [8]. Independently, another description of the same family of surfaces was obtained in [7]. Moreover, together with Vrancken, the third author classified totally umbilical hypersurfaces in S n × R in [9].…”

Umbilical surfaces of products of space forms

“…The case of S n ×R (respectively, H n × R) was carried out in [4] (respectively, [5]), extending previous results in [7] and [8] (respectively, [1]) for hypersurfaces.…”

On stable constant mean curvature surfaces in $\mathbb S^2\times \mathbb R$ and $\mathbb H^2\times \mathbb R $

“…Proposition 3.1 applies in particular to the product spaces S 2 × R and H 2 × R as they are conformally flat. More precisely S 2 × R is conformal to R 3 \{(0, 0, 0)} and H 2 ×(0, π) is conformal to H 3 (see [21]). The totally umbilic surfaces in these spaces and more generally in homogeneous 3-manifolds were classified recently by Toubiana and the author in [22].…”