Some characterizations of Mannheim partner curves in the Minkowski 3-spaceE₁³

Abstract

In this paper, we give some characterizations of Mannheim partner curves in the Minkowski 3-space E¡. Firstly, we classify these curves in E^. Next, we give some relationships characterizing these curves and we show that the Mannheim theorem is not valid for the Mannheim partner curves in E¡. Moreover, by considering the spherical indicatrix of the Frenet vectors of those curves, we obtain some new relationships between the curvatures and torsions of the Mannheim partner curves in E¡.

Construction of Curve Pairs and Their Applications

“…These curves are defined by some common properties of two curves. Involute-evolute curves, Bertrand curves, Mannheim curves are the well-known examples of such curves and studied largely [3,[11][12][13]17,18].…”

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

“…[1,11,13,20,23] and these special curves, especially helices, are used in many applications [2,9,10,19]. Similarly, by considering two curves, some special curve pairs such as involute-evolute curves, Bertrand curves, Mannheim curves have been defined and studied so far [4,12,14,15,18,21,22].…”

Mannheim Curves in n.dimensional Euclidean Space

“…For instance, In [5], Liu and Wang had obtained the necessary and su¢ cient conditions between the curvature and the torsion for a curve to be the Mannheim partner curves. In [3], Önder, Kazaz and U¼ gurlu had studied some characterizations of Mannheim partner curves in Minkowski 3-space. In [4], K¬z¬ltu¼ g and Yayl¬ had given a study on the quaternionic Mannheim curves of AW(k)-type in Euclidean space.…”

“…the notion of Mannheim curve was discovered by A. Mannheim in 1878. These curves have beeen studied by many mathematicians (see [1] and [3][4][5][6][7][8][9][10][11]). For instance, In [5], Liu and Wang had obtained the necessary and su¢ cient conditions between the curvature and the torsion for a curve to be the Mannheim partner curves.…”