A generalization of Lancret’s theorem

On Mannheim partner curves in three dimensional Lie groups

“…Moreover, they obtanied a naturally reductive homogeneous semi-Riemannian space using the Lie group. Then Ç iftçi [3] defined general helices in three dimensional Lie groups with a bi-invariant metric and obtained a generalization of Lancret's theorem and gave a relation between the geodesics of the so-called cylinders and general helices.…”

New Type Direction Curves in 3-Dimensional Compact Lie Group

“…They also obtained the sectional curvature in terms of Lie invariants based on the semisimple case. Furthermore, the curves mentioned above have been handled in Lie group theory by many authors [14,[19][20][21][22].…”

“…They also obtained the sectional curvature in terms of Lie invariants based on the semisimple case. Furthermore, the curves mentioned above have been handled in Lie group theory by many authors [14,[19][20][21][22].In [23], the authors explained the notions of both the principal (binormal)-direction curve and principal (binormal)-donor curve of a Frenet curve in E 3 . They characterized some special curves in E 3 by using the relationships between the curves.In this study, within the framework of the definition of associated curves, we introduce new types of direction curves in a three-dimensional Lie group G, and we characterize these curves.…”

“…where X, Y, Z ∈ g and D is the Levi-Civita connection of G [20]. Let us assume that α : I ⊂ R → G is a curve parameterized by arc-length and {X 1 , X 2 , ..., X n } is an orthonormal basis of g. Now, we consider vector fields V = n ∑ i=1 w i X i and Z = n ∑ i=1 z i X i throughout the curve α such that w i , z i : I → R are smooth functions.…”

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

“…They have obtained a naturally reductive homogeneous semi-Riemannian space using the Lie group. Later, some of subjects given above have been considered in three dimensional Lie groups and some characterizations for these curves have been obtained in a three dimensional Lie group [3,5,[16][17][18].…”