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

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