Null Quaternionic Bertrand Curves in Minkowski Space R14

Abstract

The geometry of null curves in Minkowski spacetime has played an important role in the development of general relativity, as well as in mathematics and physics of gravitation. Many scientists have used Minkowski space to apply general relativity. There has been an increase in research on null curves in geometry and physics [1]. Bonnor [2] has introduced a Cartan frame for null curves in R 1 and proved the fundamental existence and congruence theorems. Bejancu [3] has given a method for the general study of the geometry of null curves in lightlike manifolds and in semi-Riemannian manifolds. A. Ferrandez, A. Gimenez and P. Lucas [4] have shown that a null Frenet curve, parametrized by the pseudo-arc parameter, is a null helix, if its lightlike curvature is constant. Coken and Ciftci [5] have reconstructed the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and have characterized the Bertrand null curves. And then, Duggal and Jin [6] have studied major developments of null curves, hypersurfaces and their physical use, in their recent book with voluminous bibliography. Quaternions were discovered by Hamilton as an extension to the complex numbers in 1843. Quaternions have found a broad application in many scientific areas: in mechanics of a solid body, for the description of rotation in space, in computer animation, etc. One of the most important tools used to analyze a quaternionic curve is the Frenet frame. Therefore, in [7], Bharathi and Nagaraj have defined Serret-Frenet formulas for a quaternionic curve in E and E, and then Coken and Tuna have studied Serret-Frenet formulas for quaternionic curves and quaternionic inclined curves in semi-euclidean spaces [8]. Moreover, we have studied the differential geometry of null quaternionic curves in semi-euclidean 3-spaces R v and gave the Frenet formula for null quaternionic curves by using spatial quaternions. We have constructed recently the Cartan frame for a null quaternionic curve in the 4-dimensional Minkowski space R 1 [9]. Then, we have established a relation of Bertrand pairs with null quaternionic Cartan helices in R v [10].