Ruled surfaces of finite type in 3-dimensional Heisenberg group
In this paper, on the first, we prove $Δr=2H$ where $Δ$ is the Laplacian operator, $r=\left( r_{1},r_{2},r_{3}\right) $ the position vector field and $H$ is the mean curvature vector field of a surface $\mathcal{S}$ in the 3-dimensional Heisenberg group $H_{3}.$ In the second, we classify the ruled surfaces by straight geodesic lines, which are of finite type in $H_{3}.$ The straight geodesic lines belong to $\ker ω,$ where $ω$ is the Darboux form.
math.DG↗