Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces

2021/02/01 by Renzo Caddeo, Caddeo, R., Irene I. Onnis +3
Mathematics · Physics and Astronomy · #53A10 #53C40 #53C42 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2102.00873

openalex publication_date 2021/02/01 · openalex created_date 2021/02/15 · openalex updated_date 2026/07/28

Abstract

In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R³ to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

Citations

• The Geometries of 3-Manifolds

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