Spinor Equations of Successor Curves

Year 2022, Volume: 5 Issue: 1, 32 - 41, 15.03.2022

Tülay Erişir , Hilal Köse Öztaş

https://doi.org/10.32323/ujma.1070029

Cited By: 7

Abstract

The aim of this study is to give spinor representation of successive curves in three-dimensional Euclidean space. In three dimensional Euclidean Space, the spinor representations of a curve with unit speed and a successor curve with the same arc length parameter as this curve has been studied. For this, first of all, the curve with unit speed and its successor curve have been corresponded to two different spinors. Then, using the relationships between these curves, the relationships between the spinors corresponding to these curves have been given. Therefore, geometric interpretations of these curves and corresponding spinors have been made. In addition, different spinor equations of the mates and derivatives of spinors have been examined and geometric interpretations of these spinor equations have been given. Then, spinor equations have been obtained in case the successive curves are helices. Consequently, two examples have been given.

Keywords

Spinors, Euclidean space, successor curve

References

• [1] Y. Balcı, T. Erisir, M.A. Gungor, Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-Space, J. Chungcheong Math. Soc., 28(4) (2015), 525-535.
• [2] S. Bilinski, Uber eine Erweiterungsm¨oglichkeit der Kurventheorie, Monatsh. Math., 67 (1963), 289-302.
• [3] E. Cartan, The Theory of Spinors, Dover Publications, New York, 1966.
• [4] A. Cakmak, V. Sahin, Characterizations of adjoint curves according to alternative moving frame, Fun. J. Math. Appl., 5(1) (2022), 42-50.
• [5] R. Delanghe, F. Sommon, V. Soucek, Clifford Algebra and Spinor-Valued Functions: A function Theory for The Dirac Operator, Dover Publications, New York, 1966.
• [6] T. Erisir, M. A. Gungor, M. Tosun, Geometry of the hyperbolic spinors corresponding to alternative frame, Adv. Appl. Clifford Algebr., 25(4) (2015), 799-810.
• [7] T. Erisir, N. C. Kardag, Spinor representations of involute evolute curves in E3, Fun. J. Math. Appl., 2(2) (2019), 148-155.
• [8] T. Erisir, On spinor construction of Bertrand curves, AIMS Mathematics, 6(4) (2021), 3583-3591.
• [9] Z. Ketenci, T. Eris¸ir, M. A. G¨ung¨or, A Construction of hyperbolic spinors according to Frenet frame in Minkowski space, J. Dyn. Syst. Geom. Theor., 13(2) (2015), 179-193.
• [10] I. Kisi, M. Tosun, Spinor Darboux equations of curves in Euclidean 3-space, Math. Morav., 19(1) (2015), 87-93.
• [11] L. D. Landau, E.M. Lifshitz, Quantum Mechanics (Non-Relavistic Theory), Pergamon Press, Oxford, 1977.
• [12] M. Masal, Curves according to the successor frame in Euclidean 3-Space, SAUJS, 22(6) (2018), 1868-1873.
• [13] A. Menninger, Characterization of the slant helix as successor curves of the general helix, Int. Electron. J. Geom., 7(2) (2014), 84-91.
• [14] R. S. Millman, G.D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
• [15] W. Pauli, Zur Quantenmechanik des magnetischen elektrons, ZS. f. Phys., 43 (1927), 601-623.
• [16] W. T. Payne, Elementary spinor theory, Am. J. Phys., 20 (1952), 253.
• [17] S. S¸enyurt, Y. Altun, Smarandache curves of the evolute curve according to Sabban frame, Commun. Adv. Math. Sci., 3(1) (2020), 1-8.
• [18] G. F. Torres del Castillo, G. S. Barrales, Spinor formulation of the differential geometry of curves, Rev. Colombiana Mat., 38 (2004), 27-34.
• [19] G. F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions and Their Applications, Birkh¨auser, Boston, 2003.
• [20] D. Unal, I. Kis¸i, M. Tosun, Spinor Bishop equation of curves in Euclidean 3-space, Adv. Appl. Clifford Algebr., 23(3) (2013), 757-765.
• [21] M. D. Vivarelli, Development of spinor descriptions of rotational mechanics from Euler’s rigid body displacement theorem, Celestial Mech., 32 (1984), 193-207.
• [22] A. Wachter, Relativistic Quantum Mechanics, Springer, Dordrecht, 2011.
• [23] F. Wang, H. Liu, Mannheim partner curves in 3-Euclidean space, Math. Practice Theory, 1 (2007), 141-143.