Classiffications of special curves in the three-dimensional Lie group

Abstract

In the present paper, we define a Bertrand curve in the three-dimensional Lie group G with a bi-invariant metric, and we show a Frenet curve α with Frenet curvatures k 1 and k 2 in G is a Bertrand curve if and only if it satisfies Ak 1 + B(k 2 +k 2) = 1, where A and B are some constants andk 2 = 1/2 [V 1 , V 2 ], V 3. Also, we investigate a Bertrand curve using the Frenet curvature conditions of AW(k)-type (k = 1, 2, 3) curves in G.

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…”

“…where t = γ = dγ ds and a = n Σ i=1 a i s i , where a i = da i ds . Here "dash" indicates the derivative with respect to the parameter s. It is imperative to note that if a is the left-invariant vector field to the curve then a = 0 (see for details [5][6][7][8]).…”

“…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].…”