Investigation of a curve using Frenet frame in the lightlike cone

Abstract

Abstract:Often times the language of mathematics is used to formulate physical theories. For example, as in this paper, while Minkowski space or the theory of special relativity were studying, their formulation was given by means of mathematical methods. In this manuscript, we study spacelike normal curves lying entirely in the 2-dimensional and 3-dimensional lightlike cone. In particular, some related theorems and definitions are also given. The study of representations of spacelike normal curves in the lightlike cone has led to the existence of different areas of mathematics and physics.

ℚ𝟐 𝟑 ⊂ 𝐄𝟐 𝟒 Lightlike Koni 3-Uzayında Null Eğrilerin Elastik Olmayan Akış(İnextensible Flow) Eğrileri

“…Liu tarafından lightlike konide eğriler üzerine bazı temel ifadeler verilmiştir [16]. Ayrıca, lightlike koni uzayında farklı tipden eğriler [2,3,12,13] çalışmalarında ele alınmıştır. Uygulama alanlarında elastik olmayan eğri(inextensible flows) ve yüzey akışları, bilgisayar destekli ve bilgisayar animasyonu gibi ve hatta mekanik ile ilgili bir çok alanda problem olarak ortaya çıkmakta ve araştırmacılar tarafından ele alınmıştır [7,8,20,21].…”

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

“…To study the behavior of a curve in two-and threedimensional lightlike cone, Liu and Meng [13] defined structure functions for a spacelike curve, and by these structure functions, they obtained representation formulas of spacelike curves in the lightlike cones Q 2 1 and Q 3 1 of Lorentzian space R 3 1 and R 4 1 , respectively. Also, Külahci, Bektaş and Ergüt [11] considered AW(k)-type curves in the 3-dimensional lightlike cone, and recently Külahci [10] considered spacelike normal curves on the lightlike cones Q 2 1 and Q 3 1 . Since for a null curve that lies on the lightlike cone, any order derivative vectors are null vectors [15], for a null curve x on the lightlike cone Q 3 2 , there exists a natural Frenet frame {x, x , N, W } [6,15].…”