New Representation of Hasimoto Surfaces with the Modified Orthogonal Frame

Year 2022, Volume: 10 Issue: 1, 69 - 72, 15.04.2022

Kemal Eren

Abstract

In this study, we investigate Hasimoto surfaces considering the modified orthogonal frame. Firstly, we recall the relations between the Frenet frame and the modified orthogonal frame, and then we give the evolution equations of the modified orthogonal frame. After that, the first and second fundamental forms, mean curvatures, and Gaussian curvatures of the Hasimoto surfaces are determined with respect to the modified orthogonal frame. We give the definitions and some new theorems about Hasimoto surfaces. Finally, we express the properties of parameter curves of Hasimoto surfaces with a modified orthogonal frame in Euclidean 3-space.

Keywords

Binormal motion, evolution of curves, Hasimoto surfaces, modified orthogonal frame, smoke ring equations

Thanks

I would like to thank the editors and referees for evaluating the work.

References

• [1] H. Hasimoto, Motion of a vortex filament and its relation to elastica, J. Phys. Soc. Jpn., 31 (1971), 293-294.
• [2] H. Hasimoto, A soliton on a vortex filament, J. Fluid. Mech. 51 (1972), 477-485.
• [3] N. H. Abdel-All, R. A. Hussien and T. Youssef, Hasimoto surfaces, Life Science Journal, 9(2) (2012), 556–560.
• [4] A. Kelleci, M. Bektas¸ and M. Erg¨ut, The Hasimoto surface according to Bishop frame, Adıyaman University Journal of Science, 9(1) (2019), 13-22.
• [5] K. Eren and A. K. Akbay, On the harmonic evolute surfaces of Hasimoto surfaces,Adyu. J. Sci.,11(1) (2021), 87-100.
• [6] M. Erdogdu and M. O¨ zdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom., 17 (2014), 169-181.
• [7] M. Elzawy, Hasimoto surfaces in Galilean space, J Egypt Math Soc., 29 (2021), 5-14.
• [8] A. C¸ akmak, Parallel surfaces of Hasimoto surfaces in Euclidean 3-Space, BEU Journal of Science, 7(1) (2018), 125-132.
• [9] N. G¨urb¨uz, Hasimoto surfaces according to three classes of curve evolution with Darboux frame in Euclidean space, Gece Publishing, 2018.
• [10] N. Gurbuz and D. W. Yoon, Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space, Demonstratio Mathematica, 53(1) (2020), 277-284.
• [11] T.A. Ivey, Helices, Hasimoto surfaces and B¨acklund transformations, Canad. Math. Bull., 43(4) (2000), 427-439.
• [12] M. Grbovi´c and E. Neˇsovi´c, On B¨acklund transformation and vortex filament equation for null Cartan curve in Minkowski 3-space, Math. Phys. Anal. Geom., 23 (2016), 1-15.
• [13] M. Grbovi´c and E. Neˇsovi´c, On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space, J. Math. Anal. Appl., 461(1) (2018), 219–233.
• [14] T. Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J., 36 (1984), 17–24.
• [15] B. B¨ukc¨u and M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3- Space, Math. Sci. Appl. E-Notes, 4 (2016), 184–188.
• [16] K. Eren and H. H. K¨osal, Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Mathematics, 5(3) (2020), 2027-2039.
• [17] K. E. O¨ zen, M. Gu¨ner and M. Tosun, A note on the acceleration and Jerk in motion along a space curve, An. S¸ tiin¸t. Univ. “Ovidius” Constan¸ta Ser. Mat., 28(1) (2020), 151-164.