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

Caddeo, Renzo

Member of the Collaboration Group

;Onnis, Irene I.;Piu, Paola

2022-01-01

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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.