Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces

Abstract

We give a complete classification of surfaces with parallel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces.

A loop group method for minimal surfaces in the three-dimensional Heisenberg group

“…It is clear from the definition that surfaces tangent to E 3 are Hopf cylinders, [5]. Thus a surface which has zero support, that is h ≡ 0, is a Hopf cylinder.…”

“…293-306]. Thus the family E(κ, τ ) with κ ∈ R and τ ≥ 0 is referred to as the Bianchi-Cartan-Vranceanu family, [5]. By what was said above the Bianchi-Cartan-Vranceanu family includes all local three-dimensional homogeneous Riemannian metrics whose isometry groups have dimension greater than 3 except constant negative curvature metrics.…”

Slant Curves in Contact Pseudo-Hermitian 3-Manifolds

“…Then we see that the structure (ϕ, ξ, η, g c ) is Sasakian and further that (D, g c ) is of constant holomorphic sectional curvature H = −3 + 2c (see [1,13] …”

A Note on Almost Contact Riemannian 3-Manifolds Ii

“…Sasakian manifolds Nil 3 = R 3 (−3) and SL 2 R, cosymplectic 3-manifolds S 2 (κ) × E 1 and H 2 (κ) × E 1 are model spaces of Thurston geometries. Moreover these space are included in the 2-parameter family of homogeneous Riemannian spaces referred as to the Bianchi-Cartan-Vranceanu family, see [1].…”