The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space

Jianguo Sun,Yanping Zhao

doi:10.3390/math9243294

Abstract

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively.

Highlights

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The first author and Pei, D.H., obtained the singularities and other characterizations of null curves on the 3-null cone in [4,6]. The authors focused their attention on the characterizations of the pseudo-spherical null curves and Bertrand null curves in [8]
Synthesizing the above views, in this paper, we study the existence conditions of the Bertrand curves of the null curves in semi-Euclidean 4-space

Summary

Introduction

Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space. In case of semi-Euclidean space, there are three categories of curves: spacelike curves, timelike curves, and null (lightlike) curves. The first author and Pei, D.H., obtained the singularities and other characterizations of null curves on the 3-null cone in [4,6] The authors focused their attention on the characterizations of the pseudo-spherical null curves and Bertrand null curves in [8]. Synthesizing the above views, in this paper, we study the existence conditions of the Bertrand curves of the null curves in semi-Euclidean 4-space. We present some geometrical properties of the Bertrand curves of the null curves, and we provide the existence conditions of the Bertrand curves for two different cases. Two examples are given to demonstrate the correctness of the conclusions in view of the geometric intuition

Preliminaries

The Bertrand Curves of Null Curves

Some Examples