AIMS Mathematics, 2020, 5(4): 3434-3445. doi: 10.3934/math.2020222

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Geometry of curve flows in isotropic spaces

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

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.
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1. J. Arroyo, O. J. Garay, A. Pámpano, Binormal motion of curves with constant torsion in 3-spaces, Adv. Math. Phys., 2017 (2017), 1-8.

2. M. Desbrun, M. P. Cani-Gascuel, Active implicit surface for animation, In: Proc. Graphics Interface-Canadian Inf. Process. Soc., 1998, 143-150.

3. R. E. Goldstein, D. M. Petrich, The Kortewege-de Vries hierachy as dynamics of closed curves in the plane, Phys. Rev. Lett., 67 (1991), 3203-3206.    

4. N. Gurbuz, Inextensible flows of spacelike, timelike and null curves, Int. J. Contemp. Math. Sci., 4 (2009), 1599-1604.

5. H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.    

6. R. A. Hussien, S. G. Mohamed, Generated surfaces via inextensible flows of curves in $\mathbb{R}^3$, J. Appl. Math., 2016 (2016), 1-8.

7. M. Kass, A. Witkin, D. Terzopoulos, Snakes: Active contour models, In: Proc. 1st Int. Conference on Computer Vision, 1987, 259-268.

8. G. L. Lamb Jr, Elements of Soliton Theory, JohnWiley and Sons, New York, 1980.

9. S. G. Mohamed, Binormal motions of inextensible curves in de-sitter space $\mathbb{S}^{2,1}$, J. Egyptian Math. Soc., 25 (2017), 313-318.

10. C. Qu, J. Han, J. Kang, Bäcklund transformations for integrable geometric curve flows, Symmetry, 7 (2015), 1376-1394.    

11. W. K. Schief, C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, Proc. R. Soc. London Ser. A, 455 (1999), 3163-3188.

12. Ž. M. ŠipuŠ, Translation surfaces of constant curvatures in a simpley isotropic space, Period. Math. Hungar., 68 (2014), 160-175.    

13. M. Yeneroglu, On new characterization of inextensible flows of space-like curves in de Sitter space, Open Math., 14 (2016), 946-954.    

14. D. W. Yoon, J. W. Lee, Linear Weingarten helicoidal surfaces in isotropic space, Symmerty, 8 (2016), 1-7.

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