Ruled Surfaces in Three Dimensional Lie Groups

TL;DR

This paper explores the properties and classifications of ruled surfaces within three-dimensional Lie groups equipped with a bi-variant metric, analyzing their geometric characteristics and curvature properties.

Contribution

It introduces a comprehensive framework for understanding ruled surfaces in Lie groups, including characterizations and specific types like normal and Darboux developable surfaces.

Findings

Characterization of ruled surfaces in Lie groups

Formulas for Gaussian and mean curvatures

Analysis of geodesic and torsion properties

Abstract

Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, binormal surface, tangent developable surface, rectifying developable surface and Darboux developable surface are worked. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsion of the base curves with respect to these surfaces depend on three dimensional Lie groups.