On the construction of generalized rectifying ruled surfaces in 3-dimensional Lie groups

In this study, we investigate the geometry of generalized rectifying ruled surfaces in the 3-dimensional Lie group $G$. We construct geometric structures such as singular point sets, cylindrical surfaces, striction curves, developable surfaces, geodesic and asymptotic curves, as well as the Gauss and mean curvatures of generalized rectifying ruled surfaces in $G$. Then, we present the shape operator matrix and some related characterizations of developable generalized rectifying ruled surfaces in the 3-dimensional Lie group $G$. We also discuss how generalized rectifying ruled surfaces in 3-dimensional Lie groups correspond, in special cases, to tangent developable ruled surfaces, binormal ruled surfaces, and rectifying ruled surfaces both in 3-dimensional Lie groups and in 3-dimensional Euclidean space.