scholarly journals A note about the torsion of null curves in the 3-dimensional Minkowski spacetime and the Schwarzian derivative

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 553-561 ◽  
Author(s):  
Zbigniew Olszak
Keyword(s):  
Schwarzian Derivative ◽  
Minkowski Spacetime ◽  
Main Topic ◽  
3 Dimensional ◽  
Null Curve ◽  
Null Curves

The main topic of this paper is to show that in the 3-dimensional Minkowski spacetime, the torsion of a null curve is equal to the Schwarzian derivative of a certain function appearing in a description of the curve. As applications, we obtain descriptions of the slant helices, and null curves for which the torsion is of the form ? = -2?s, s being the pseudo-arc parameter and ? = const ? 0.

scholarly journals Special Curves According to Bishop Frame in Minkowski 3-Space

2020 ◽  
Vol 5 (1) ◽  
pp. 237-248
Author(s):  
Muhammad Abubakar Isah ◽  
Mihriban Alyamaç Külahçı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Slant Helix ◽  
Null Curves ◽  
Necessary And Sufficient

AbstractPseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space [E_1^3 . We define helix and slant helix according to Bishop frame in [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

scholarly journals Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Vector Fields ◽  
Evolution Equations ◽  
Space Form ◽  
Curve Evolution ◽  
Lie Bracket ◽  
Null Curve ◽  
Induction Equation ◽  
Null Curves ◽  
Local Vector ◽  
Local Symmetries

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.

On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650077 ◽  
Author(s):  
Milica Grbović ◽  
Emilija Nešović
Keyword(s):  
Evolution Equation ◽  
Bäcklund Transformation ◽  
Sufficient Conditions ◽  
Vortex Filament ◽  
Backlund Transformation ◽  
Burger's Equation ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Given

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space

2008 ◽  
Vol 63 (5-6) ◽  
pp. 248-252 ◽  
Author(s):  
Mihriban Külahcı ◽  
Mehmet Bektaş ◽  
Mahmut Ergüt
Keyword(s):  
Lorentzian Space ◽  
3 Dimensional ◽  
Null Curves

We investigate null curves of the AW(k)-type (1 ≤ k ≤ 3) in the 3-dimensional Lorentzian space, L3, and give curvature conditions of these curves by using the Cartan frame. Moreover, we study harmonic curvatures of curves of AW(k)-type and show that if the α Frenet curve is of type AW(1), then α is a null helix.

scholarly journals Integrability aspects of the vortex filament equation for pseudo-null curves

2017 ◽  
Vol 14 (06) ◽  
pp. 1750090 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Space Form ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Recursion Operator ◽  
Vortex Filament ◽  
Lorentzian Space ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Relationship

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

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

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nemat Abazari ◽  
Martin Bohner ◽  
Ilgin Sağer ◽  
Alireza Sedaghatdoost ◽  
Yusuf Yayli
Keyword(s):  
Minkowski Space ◽  
Constant Curvature ◽  
Structure Functions ◽  
Curvature Function ◽  
Frenet Frame ◽  
Null Curve ◽  
Null Curves

Abstract In this paper, we investigate the representation of curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in Minkowski space $\mathbb {R}^{4}_{2}$ R 2 4 by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 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 $\kappa _{2}=0$ κ 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.

scholarly journals Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1256
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Reconstruction Process ◽  
Null Curve ◽  
Straight Line ◽  
Null Curves ◽  
Straight Lines

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

COMPLETE CMC SPACELIKE SURFACES WITH BOUNDED HYPERBOLIC ANGLE IN GENERALIZED ROBERTSON–WALKER SPACETIMES

2010 ◽  
Vol 07 (06) ◽  
pp. 961-978 ◽  
Author(s):  
MAGDALENA CABALLERO ◽  
ALFONSO ROMERO ◽  
RAFAEL M. RUBIO
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Theorems ◽  
Natural Extension ◽  
Minkowski Spacetime ◽  
Spacelike Surface ◽  
3 Dimensional ◽  
Spacelike Surfaces ◽  
Hyperbolic Angle ◽  
Wide Family

Complete spacelike surfaces with constant mean curvature (CMC) and bounded hyperbolic angle in Generalized Robertson–Walker (GRW) spacetimes, obeying certain natural curvature assumptions, are studied. This boundedness assumption arises as a natural extension of the notion of bounded hyperbolic image of a spacelike surface in the 3-dimensional Lorentz–Minkowski spacetime. The results obtained apply to complete CMC spacelike surfaces lying between two spacelike slices in an GRW spacetime, in the steady state spacetime and in a static GRW spacetime. As an application, uniqueness and non-existence theorems for certain CMC spacelike surface differential equations in a wide family of open GRW spacetimes are given.

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