Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A generalisation of g-rectifying and g-normal curves in Lorentzian n-space

Fatma Almaz, Hazel Diken

Abstract

In this paper, we introduce and analyze $g-$rectifying curves (spacelike and null curves) and $\ g-$normal curves in Lorentzian $n$-space, building upon the established notion of rectifying curves and normal curve, respectively. Our generalization extends this definition by considering an $% g-$position vector field, $ξ_{g}(s)=\int g(s)dξ$, where $g$ is an integrable function in the arc-length parameter $s$. An $g$-rectifying curves(or $g-$normal curves) are then defined as an arc-length parametrized curve $ξ$ in Lorentzian $n-$space such that its $g$-position vector consistently lies within its rectifying space(or normal space). The primary objective of this work is to provide a comprehensive characterization and classification of these $g$-rectifying curves and $g-$normal curves, thereby expanding the geometric understanding of curves in Lorentzian $n$-spaces.

A generalisation of g-rectifying and g-normal curves in Lorentzian n-space

Abstract

In this paper, we introduce and analyze rectifying curves (spacelike and null curves) and normal curves in Lorentzian -space, building upon the established notion of rectifying curves and normal curve, respectively. Our generalization extends this definition by considering an position vector field, , where is an integrable function in the arc-length parameter . An -rectifying curves(or normal curves) are then defined as an arc-length parametrized curve in Lorentzian space such that its -position vector consistently lies within its rectifying space(or normal space). The primary objective of this work is to provide a comprehensive characterization and classification of these -rectifying curves and normal curves, thereby expanding the geometric understanding of curves in Lorentzian -spaces.

Paper Structure

This paper contains 6 sections, 4 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation of $g-$rectifying curves in Lorentzian $n-$space $L^{n}$
  4. Representation of null $g-$rectifying curves in Lorentzian $n-$space $L^{n}$
  5. Representation of $g-$normal curves in Lorentzian $n-$space $L^{n}$
  6. Conclusion

Key Result

Theorem 1

Let $\xi :I\subset \mathbb{R} \longrightarrow L^{n}$ be a unit-speed spacelike curve having no where vanishing $n-1$ curvatures $\kappa _{1},\kappa _{2},...,\kappa _{n-1},$ and let $g:I\rightarrow R$ be a no where vanishing integrable function with at least $(n-2)$-times differentiable primitive fun 5) The spacelike $g-$rectifying curve of $\xi$ is given as where $G(s)$ represents the primitive f

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Definition 4
  • ...and 2 more