A generalization for surfaces using a line of curvature in Lie group

Abstract

In this study, we investigate how to construct surfaces using a line of curvature in a 3dimensional Lie group. Then, by utilizing the Frenet frame, we give the conditions that a curve becomes a line of curvature on a surface when the marching-scale functions are more general expressions. After then, we provide some crucial examples of how efficient our method is on these surfaces.

Surface Family Pair with Bertrand Pair as Mutual Curvature Lines in Three-Dimensional Lie Group

“…The Lie group theory is introduced in this section (see [1][2][3][4][5][6]). Let G be a Lie group with a bi-invariant metric <, >, and ∇ be the Levi-Civita connection of G. If g indicates the Lie algebra, then, for all a, b, c ∈ g, we have…”

“…The shared new approach to geometry heavily depends on research into Lie groups. Consequently, there are several study findings on curves and surfaces in three-dimensional Lie group (3-D Lie group) G [1][2][3][4][5][6].…”