Higher Order Parallel Surfaces in Bianchi–Cartan–Vranceanu Spaces

We 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 2]. As a partial result, we prove that totally umbilical surfaces only exist if the ambient Bianchi–Cartan–Vranceanu 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. Received: December 20, 2006. Revised: March 15, 2007.