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 [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.
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The author is a postdoctoral researcher supported by the Research Foundation – Flanders (FWO).
Received: December 20, 2006. Revised: March 15, 2007.
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Van der Veken, J. Higher Order Parallel Surfaces in Bianchi–Cartan–Vranceanu Spaces. Result. Math. 51, 339–359 (2008). https://doi.org/10.1007/s00025-007-0282-0
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DOI: https://doi.org/10.1007/s00025-007-0282-0