Ruled surfaces of finite type in 3-dimensional Heisenberg group

TL;DR

This paper investigates ruled surfaces in the 3D Heisenberg group, proving a key Laplacian relation and classifying finite type ruled surfaces formed by straight geodesic lines.

Contribution

It establishes a Laplacian relation for surfaces in the Heisenberg group and classifies finite type ruled surfaces generated by straight geodesic lines.

Findings

Proves $ riangle r=2H$ for surfaces in $H_3$

Classifies finite type ruled surfaces by straight geodesic lines

Identifies straight geodesic lines as belonging to $ ext{ker}\, ext{omega} $

Abstract

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.