On the Harmonic Evolute Surfaces of Hasimoto Surfaces

In this study, firstly by considering the evolution of a moving space curve, we give some related definitions and some new results about Hasimoto surfaces in Euclidean 3-spaces. Secondly, we examine harmonic evolute surfaces of Hasimoto surfaces in Euclidean 3-spaces and also, we give some geometric properties of these type surfaces. Moreover, we express the properties of parameter curves of harmonic evolute surfaces in Euclidean space. Finally, we give an explicit example of Hasimoto surface and its harmonic evolute surface and also we plot these surfaces.

Keywords:

Hasimoto surfaces;, harmonic evolute surface;, binormal motion;, evolution of curves and surfaces; Gaussian curvature;, mean curvature.,

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[1] Sipus, Z.M., Vladimir, V., The harmonic evolute of a surface in Minkowski 3-space, Mathematical Communications, 19, 43-55, 2014.
[2] Lopez, R., Sipus, Z.M., Gajcic, L.P., Protrka, I., Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz-Minkowski space, International Journal of Geometric Methods in Modern Physics, 16 (5), 1950076, 2019.
[3] Körpinar, T., Kaymanli, G.U., On the harmonic evolute of quasi normal surfaces, Journal of Science and Arts, 1 (50), 55-64, 2020.
[4] Eren, K., Kösal, H.H., Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Mathematics, 5 (3), 2027-2039, 2020.
[5] Kelleci, A., Eren, K., On evolution of some associated type ruled surfaces, Mathematical Sciences and Applications E-Notes, 8 (2), 178-186, 2020.
[6] Hasimoto, H., Motion of a vortex filament and its relation to elastica, Journal of the Physical Society of Japan, 31, 293-294, 1971.
[7] Hasimoto, H., A soliton on a vortex filament, Journal of Fluid Mechanics, 51 (3), 477-485, 1972.
[8] Rogers, C., Schief, W.K., Bäcklund and Darboux transformations, Cambridge University Press, 432, 2002.
[9] Abdel-All, N.H., Hussien, R.A., Youssef, T., Hasimoto surfaces, Life Science Journal, 9 (3), 556-560, 2012.
[10] Kelleci, A., Bektaş, M., Ergüt, M., The Hasimoto surface according to Bishop frame, Adıyaman University Journal of Science, 9 (1), 13-22, 2019.
[11] Erdoğdu, M., Özdemir, M., Geometry of Hasimoto surfaces in Minkowski 3-space, Mathematical Physics, Analysis and Geometry, 17, 169-181, 2014.
[12] Çakmak, A., Öklid 3-uzayında Hasimoto yüzeylerinin paralel yüzeyleri, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, 7 (1), 125-132, 2018.