Geometry of curve flows in isotropic spaces

Nevin Gürbüz; Dae Won Yoon; Nevin Gürbüz; Dae Won Yoon

doi:10.3934/math.2020222

AIMS Mathematics

2020, Volume 5, Issue 4: 3434-3445. doi: 10.3934/math.2020222

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Research article

Geometry of curve flows in isotropic spaces

Nevin Gürbüz ¹,
Dae Won Yoon ^{2
,
,}

1.
Department of Mathematics and Computer Science, Eskisehir Osmangazi University, Eskisehir, Turkey
2.
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea

Received: 06 January 2020 Accepted: 26 March 2020 Published: 07 April 2020
MSC : 53B30, 53C40, 53Z05

In this paper, we study inextensible flows of a curve in an isotropic 3-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. Using binormal flows of a space curve we give the Bäcklund transformations of the Schrödinger flows and the extended Harry-Dym flows. Finally, we investigate some geometric properties of Hasimoto surfaces which wiped out by the Schrödinger flows.
- Hasimoto surface,
- Schrödinger flow,
- extended Harry-Dym flow,
- isotropic space
Citation: Nevin Gürbüz, Dae Won Yoon. Geometry of curve flows in isotropic spaces[J]. AIMS Mathematics, 2020, 5(4): 3434-3445. doi: 10.3934/math.2020222

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Abstract

In this paper, we study inextensible flows of a curve in an isotropic 3-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. Using binormal flows of a space curve we give the Bäcklund transformations of the Schrödinger flows and the extended Harry-Dym flows. Finally, we investigate some geometric properties of Hasimoto surfaces which wiped out by the Schrödinger flows.

References

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