New Type Direction Curves in 3-Dimensional Compact Lie Group

Abstract

In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some characterizations for these curves are obtained. Additionally, connections between the new types of associated curves and the curves, such as helices, general helices, Bertrand, and Mannheim, are given. MSC: 53A04; 22E15 IntroductionMany authors have made significant contributions to the theory of curves from past to present. Some of these studies indicated that the relationships between the curvatures of the space curves are quite remarkable, and the new special curves are also defined via these relations [1][2][3][4][5]. Helices, one of these special space curves, have been studied by many researchers [6][7][8][9]. In addition to special space curves, some of the relationships between the curve pairs are also particularly interesting. The curve pairs are obtained by using the Frenet vectors or curvatures. In this respect, involute-evolute, Bertrand, and Mannheim curves are well-known examples of curve pairs, and many studies have been performed on this topic [10][11][12][13][14][15][16].The Riemannian geometry of a Lie group was studied in [17]. Here, the rich collection of examples that are obtained by providing an arbitrary Lie group G with a Riemannian metric invariant under left translations was given. The semi-Riemannian geometry of a Lie group was examined in [18]. 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. Finally, we determine the relationships between the new types of direction curves (rectifying-direction, osculating-direction, and normal-direction curve curves) and the curves (Bertrand curve, involute-evolute, rectifying curve, etc.).

Osculating mate of a Frenet curve in the Euclidean 3-space

“…Kızıltug et al [13] have defined a new kind of special curves called normal direction curves. Later, C ¸akmak [7] has studied the same subject in a 3dimensional compact Lie group and he has also given two similar new curves.…”

Algebraic Inverses on Lie Algebra Comultiplications

“…From the homotopical point of view, co-H-spaces, which are sometimes also called spaces with a topological comultiplication, are important objects of research; see [4][5][6][7][8][9][10][11]. According to Eckmann and Hilton, the co-H-spaces are the dual notions of H-spaces, and have been playing a pivotal role in algebraic topology for many years; see also [12][13][14][15][16].…”