On Characterization of Inextensible Flows of Curves According to Type-2 Bishop Frame E3

Kızıltuğ, Sezai

doi:10.3390/mca19010069

Cited by 5 publications

(4 citation statements)

References 11 publications

(12 reference statements)

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“…Hasimoto [6] studied the motion of a vortex filament (smoke ring) and obtained the equations of the time evolution of the curves. In [7], the inextensible flows of curves (IFC) that move according to the type−2 Bishop frame in R 3 were studied. In [8], a general formulation for IFC on an oriented surface in R 3 was derived.…”

Section: Introductionmentioning

confidence: 99%

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

Gaber

Sorour

2023

Symmetry

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The study of the flows of curves is one of the most fascinating research areas in differential geometry. In this paper, we investigate the geometry of the flows of timelike curves according to the quasi-frame in Minkowski space R2,1 (In this paper, we refer to these curves as “quasi-timelike curves”). We investigate the evolution of quasi-timelike curves using the velocity functions and obtain the necessary and sufficient conditions for inextensibility. Additionally, we obtain the explicit forms of the time evolution equations for the quasi-orthonormal frames (tangent, quasi-normal, and quasi-binormal vectors) of the quasi-timelike curve as well as the time evolution equations of their quasi-curvatures. We present a new application for motion with velocities equal to the quasi-curvatures of the quasi-timelike curve. In this application, the time evolution equations of the quasi-curvatures arise as a system of partial differential equations with the form of the heat equation, and by solving this system, we visualize the evolution of quasi-curvatures and the evolution of the quasi-timelike curve. In addition, the acceleration functions are used to investigate the flows of inextensible quasi-timelike curves, and an application for accelerations equal to the quasi-curvatures is given. Through this application, the position vector of the quasi-timelike curve satisfies the one-dimensional wave equation, and the time evolution equations of the quasi-curvatures arise as a system of transport equations. We obtain the solutions and graph them using Wolfram Mathematica 12.

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Section: Introductionmentioning

confidence: 99%

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

Gaber

Sorour

2023

Symmetry

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“…Q. Ding et al [7], studied the soliton solutions to the KdV equation and they described a motion of spacelike curves in R 2,1 . S. Kzltug [8], studied Inextensible flows of curves (IFC) that moves according to type−2 Bishop frame in R 3 . Yıldız et al [9], derived a general formulation for (IFC) on an oriented surface in R 3 .…”

Section: Introductionmentioning

confidence: 99%

The Geometry of Inextensible Flows of Timelike Curves according to Quasi Frame in Minkowski space $\mathbb{R}^{2,1}$

Gaber

Sorour

2022

Preprint

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In this paper, we study the geometry of the motion of timelike curves by quasi-frame according to the velocity and acceleration fields in Minkowski space $\mathbb{R}^{2,1}$. We study the timelike curves and get sufficient conditions to be inextensible. Also, we obtain the explicit form of the evolution equations for quasi orthonormal vectors (tangent, quasi-normal, and quasi binormal) of the quasi-timelike curve and the evolution equations of the quasi curvatures as a system of partial differential equations. We give new applications to the motion of quasi-timelike curves according to quasi-frame by velocity fields and acceleration fields.Mathematics subject classification 2020: 53A04;53A05; 53E10; 53Z05.

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“…In [9], authors studied characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3space. Kızıltug was investigated characterization of inextensible flows of curves according to Type-2 Bishop frame Euclidean 3-space [4]. In this paper we study inextensible non-null curve flows according to Bishop 2-type frame in Minkowski 3-space.…”

Section: Introductionmentioning

confidence: 99%

Inextensible curve flows according to Bishop 2-type frame in Minkowski 3-space

Gürbüz¹

2016

imf

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On Characterization of Inextensible Flows of Curves According to Type-2 Bishop Frame E3

Abstract: In this paper, we study inextensible flows of curves according to type-2 Bishop frame in Euclidean 3-space. Necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature.

Cited by 5 publications

References 11 publications

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

The Geometry of Inextensible Flows of Timelike Curves according to Quasi Frame in Minkowski space $\mathbb{R}^{2,1}$

Inextensible curve flows according to Bishop 2-type frame in Minkowski 3-space

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