On Backlund transformation and motion of null Cartan curves

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]

Generalized Bertrand Curves in Minkowski 3-Space

We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves.

Smarandache curves for spherical indicatrix of the Bertrand curves pair

In this paper, we investigate special Smarandache curves with regard to Sabban frame for Bertrand partner curve spherical indicatrix. Some results have been obtained. These results were expressed depending on the Bertrand curve. Besides, we are given examples of our results.

Some properties of Bertrand curves in Lorentzian n-space 𝕃n

In this paper, Bertrand curves in [Formula: see text]-dimensional Lorentz space [Formula: see text] are defined and some of their properties are determined. Various relationships and characterizations are found between higher order curvatures and their derivatives for Bertrand curve pair. In addition, some relationships are obtained between these curves and general helix, harmonic curvature.

New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

Generalized Null Bertrand Curves In Minkowski Space-Time

Çöken and ÇIFTCI proved that a null Cartan curve in Minkowski space-time E41 is a null Bertrand curve if and only if k2 is nonzero constant and k3 is zero. That is, the null curve with non-zero curvature k2 is not a Bertrand curve in Minkowski space-time E41. So, in this paper we defined a new type of Bertrand curve in Minkowski space-time E41 for a null curve with non-zero curvature k3 by using the similar idea of generalized Bertrand curve given by Matsuda and Yorozu and we called it a null (1, 3)-Bertrand curve. Also, we proved that if a null curve with non-zero curvatures in Minkowski space-time E41 is a null (1, 3)-Bertrand curve then it is a null helix. We give an example of such curves.

Bertrand Curves of AW(k)-Type in the Equiform Geometry of the Galilean Space

We consider curves of AW(k)-type (1≤k≤3) in the equiform geometry of the Galilean spaceG3. We give curvature conditions of curves of AW(k)-type. Furthermore, we investigate Bertrand curves in the equiform geometry ofG3. We have shown that Bertrand curve in the equiform geometry ofG3is a circular helix. Besides, considering AW(k)-type curves, we show that there are Bertrand curves of weak AW(2)-type and AW(3)-type. But, there are no such Bertrand curves of weak AW(3)-type and AW(2)-type.