Ruled surfaces of finite type in 3-dimensional Heisenberg group

In this paper, on the first, we prove $\mathrm{\Delta }r=2H$ where $\mathrm{\Delta }$ is the Laplacian operator, $r=(r_1,r_2,r_3)$ the position vector field and $H$ is the mean curvature vector field of a surface $\mathcal{S}$ in the 3-dimensional Heisenberg group $H_3\mathrm{.}$ In the second, we classify the ruled surfaces by straight geodesic lines, which are of finite type in $H_3\mathrm{.}$ The straight geodesic lines belong to $\ker \omega ,$ where $\omega$ is the Darboux form.