Inextensible Flows of Curves on Lightlike Surfaces

Zühal Küçükarslan Yüzbaşı,Dae Yoon

doi:10.3390/math6110224

Zühal Küçükarslan Yüzbaşı, Dae Yoon

Open Access

https://doi.org/10.3390/math6110224

Copy DOI

Journal: Mathematics	Publication Date: Oct 29, 2018
Citations: 7	License type: CC BY 4.0

Affiliation: Fırat University, Gyeongsang National University

Abstract

In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface.

Highlights

It is well known that many nonlinear phenomena in physics, chemistry and biology are described by dynamics of shapes, such as curves and surfaces, and the time evolution of a curve and a surface has significance in computer vision and image processing
The time evolution of a curve and a surface is described by flows, in particular inextensible flows of a curve and a surface
The outline of the paper is organized as follows: In Section 2, we give some geometric concepts in Minkowski space and present the pseudo-Darboux frames of a spacelike curve and a lightlike curve on a lightlike surface

Summary

Introduction

It is well known that many nonlinear phenomena in physics, chemistry and biology are described by dynamics of shapes, such as curves and surfaces, and the time evolution of a curve and a surface has significance in computer vision and image processing. The swinging motion of a cord of fixed length or of a piece of paper carried by the wind can be described by inextensible flows of a curve and a surface. In [5], the authors derived the time evolution equations for an inextensible flow of a space curve and studied inextensible flows of a developable ruled surface. The outline of the paper is organized as follows: In Section 2, we give some geometric concepts in Minkowski space and present the pseudo-Darboux frames of a spacelike curve and a lightlike curve on a lightlike surface. We classify lightlike ruled surfaces and study inextensible flows of lightlike tangent developable surfaces

Preliminaries

Inextensible Flows of a Spacelike Curve

Inextensible Flows of a Lightlike Curve

Lightlike Ruled Surfaces

Conclusions