Higher order parallel surfaces in three-dimensional homogeneous spaces

TL;DR

This paper classifies higher order parallel surfaces in a specific family of three-dimensional homogeneous spaces, confirming a conjecture from 2002 and characterizing totally umbilical surfaces within these spaces.

Contribution

It provides a complete classification of higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces and characterizes totally umbilical surfaces in these geometries.

Findings

Complete classification of higher order parallel surfaces.

Proof that totally umbilical surfaces only exist in Riemannian product spaces.

Local parametrization of all totally umbilical surfaces.

Abstract

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 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.