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A review on the Vn-slant helices in lightlike cone Qn+1 En+2

Fatma Almaz, Handan Oztekin

TL;DR

Addresses the problem of characterizing $V_n$-slant helices on the lightlike cone $Q^{n+1}$ in a Lorentzian setting. It defines harmonic curvature functions $H_i$ and uses an asymptotic orthonormal frame to derive a differential system governing these helices and to express the axis $W$ in frame coordinates. It provides necessary and sufficient conditions for a curve to be a $V_n$-slant helix and reveals a key relation among the $H_i$ and axis components. The results extend slant-helix theory to degenerate metric spaces and offer geometric insights for light propagation and spacetime geometry.

Abstract

In this paper, Vn-slant helices and the harmonic curvature functions of Vn-slant helices are de ned in lightlike cone Qn+1, and the differential equations of the harmonic curvature functions Hi, 1<i<n-2 of Vn-slant helices are expressed by using constant vector field W that is the axis of Vn slant helices. Also, the necessary and sufficient conditions are given according to the condition of being Vn-slant helices by using the asymptotic orthonormal frame in Qn+1.

A review on the Vn-slant helices in lightlike cone Qn+1 En+2

TL;DR

Addresses the problem of characterizing -slant helices on the lightlike cone in a Lorentzian setting. It defines harmonic curvature functions and uses an asymptotic orthonormal frame to derive a differential system governing these helices and to express the axis in frame coordinates. It provides necessary and sufficient conditions for a curve to be a -slant helix and reveals a key relation among the and axis components. The results extend slant-helix theory to degenerate metric spaces and offer geometric insights for light propagation and spacetime geometry.

Abstract

In this paper, Vn-slant helices and the harmonic curvature functions of Vn-slant helices are de ned in lightlike cone Qn+1, and the differential equations of the harmonic curvature functions Hi, 1<i<n-2 of Vn-slant helices are expressed by using constant vector field W that is the axis of Vn slant helices. Also, the necessary and sufficient conditions are given according to the condition of being Vn-slant helices by using the asymptotic orthonormal frame in Qn+1.
Paper Structure (4 sections, 3 theorems, 70 equations)

This paper contains 4 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation of $V_{n}-$slant helices in the lightlike cone $\mathbb{Q} ^{n+1}$
  4. Conclusion

Key Result

Theorem 1

Let $\varkappa :I\longrightarrow \mathbb{Q} ^{n+1}\subset E_{1}^{n+2}$ be a unit speed non-null curve with non zero curvatures $\kappa _{i}(s)$, $\tau _{j}(s)$, $(i=1,2,...,n;$$j=1,2,...,n-1)$ given as the asymptotic orthonormal frame $\{\varkappa (s),V_{1}(s),V_{2}(s),V_{3}(s),...,V_{n}(s),y(s)\}$ where $H_{1}=\frac{\tau _{1}}{\kappa _{2}}.$

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof