A natural Frenet frame for null curves on the lightlike cone in Minkowski space mathbb{R} ^{4}_{2}

Nemat Abazari,Ilgin Sağer,Martin Bohner,Alireza Sedaghatdoost,Yusuf Yayli

doi:10.1186/s13660-020-02500-y

Nemat Abazari, Ilgin Sağer + Show 3 more

Open Access

https://doi.org/10.1186/s13660-020-02500-y

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Abstract

In this paper, we investigate the representation of curves on the lightlike cone mathbb {Q}^{3}_{2} in Minkowski space mathbb {R}^{4}_{2} by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone mathbb {Q}^{3}_{2} in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function kappa _{2}=0, and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

Highlights

The study of semi-Riemannian manifolds plays an important role in differential geometry and physics, especially in the theory of relativity
Bohner, Sağer, and Yayli [2] studied the relationship between Frenet elements of the stationary acceleration curve in four-dimensional Euclidean space
We show that the structure functions and the curvature functions of a null curve on the lightlike cone Q32 satisfy a special secondorder differential equation, and by this natural Frenet frame, we conclude that any null curve on the lightlike cone is a helix

Summary

Introduction

The study of semi-Riemannian manifolds plays an important role in differential geometry and physics, especially in the theory of relativity. For a null curve on the lightlike cone, we construct a natural Frenet frame and calculate its curvature functions.

Results

Conclusion