In this paper, on the first, we prove $\Delta r=2H$ where $\Delta $ 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 $\ker \omega ,$ where $\omega $ is the Darboux form.
Categories: math.DG