Spinors permeate all of modern physics and have also an important place in mathematics. Spinors are used intensively in modern theoretical physics and differential geometry. In this study, spinors are used for a different representation of differential geometry in $ \mathbb{E}^3 $. The goal of this study is also the spinor structure lying at the basis of differential geometry. In this paper, firstly, spinors are introduced algebraically. Then, the spinor construction of Bertrand curves is defined. Moreover, the angle notion for these spinors is given. In this way, a different geometric construction of spinors is established in this paper.
Citation: Tülay Erişir. On spinor construction of Bertrand curves[J]. AIMS Mathematics, 2021, 6(4): 3583-3591. doi: 10.3934/math.2021213
Spinors permeate all of modern physics and have also an important place in mathematics. Spinors are used intensively in modern theoretical physics and differential geometry. In this study, spinors are used for a different representation of differential geometry in $ \mathbb{E}^3 $. The goal of this study is also the spinor structure lying at the basis of differential geometry. In this paper, firstly, spinors are introduced algebraically. Then, the spinor construction of Bertrand curves is defined. Moreover, the angle notion for these spinors is given. In this way, a different geometric construction of spinors is established in this paper.
| [1] |
Y. Balci, T. Erisir, M. A. Gungor, Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space, J. Chungcheong Math. Soc., 28 (2015), 525–535. doi: 10.14403/jcms.2015.28.4.525
|
| [2] | J. Bertrand, Memoire sur la théorie des courbes a double courbure, J. Math. Pure. Appl., 15 (1850), 332–350. |
| [3] |
J. F. Burke, Bertrand curves associated with a pair of curves, Mathematics Magazine, 34 (1960), 60–62. doi: 10.1080/0025570X.1960.11975181
|
| [4] | É. Cartan, The theory of spinors, New York: Dover Publications, 1966. |
| [5] |
Y. M. Cheng, C. C. Lin, On the generalized Bertrand curves in Euclidean n-spaces, Note Mat., 29 (2009), 33–39. doi: 10.7202/039440ar
|
| [6] | G. F. T. del Castillo, G. S. Barrales, Spinor formulation of the differential geometry of curves, Rev. Colombiana Mat., 38 (2004), 27–34. |
| [7] | R. Delanghe, F. Sommen, V. Soucek, Clifford algebra and spinor-valued functions: A function theory for the Dirac operator, New York: Dover Publications, 1966. |
| [8] | P. A. M. Dirac, The quantum theory of the electron, P. Roy. Soc. A, 117 (1928), 610–624. |
| [9] | F. N. Ekmekci, K. Ilarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 3 (2001), 17–24. |
| [10] |
T. Erisir, M. A. Gungor, M. Tosun, Geometry of the hyperbolic spinors corresponding to alternative frame, Adv. Appl. Clifford Algebr., 25 (2015), 799–810. doi: 10.1007/s00006-015-0552-y
|
| [11] | T. Erisir, N. C. Kardag, Spinor representations of Involute Evolute curves in $\mathbb{E}^3$, Fundam. J. Math. Appl., 2 (2019), 148–155. |
| [12] | H. H. Hacisalihoglu, Differential geometry, Ankara University, Faculty of Science, Ankara, 1996. |
| [13] | G. Juvet, Opérateurs de Dirac et équations de Maxwell, Oxford Scholarship Online, 2016. Available from: 10.1093/acprof: oso/9780198782926.001.0001. |
| [14] | Z. Ketenci, T. Erisir, M. A. Gungor, A construction of hyperbolic spinors according to Frenet frame in Minkowski space, J. Dyn. Syst. Geom. Theor., 13 (2015), 179–193. |
| [15] |
I. Kişi, M. Tosun, Spinor Darboux equations of curves in Euclidean 3-space, Math. Morav., 19 (2015), 87–93. doi: 10.5937/MatMor1501087K
|
| [16] | L. D. Landau, E. M. Lifshitz, Quantum mechanics: Non-relativistic theory, Oxford: Pergamon Press, 1977. |
| [17] | P. Lounesto, Clifford algebras and spinors, Cambridge University Press, 2001. |
| [18] | P. Lounesto, Crumeyrolle's bivectors and spinors, In: Clifford algebras and spinor structures, Dordrecht: Springer, 1995,137–166. |
| [19] |
P. Lucas, J. A. Ortega-Yagües, Bertrand curves in the three-dimensional sphere, J. Geom. Phys., 62 (2012), 1903–1914. doi: 10.1016/j.geomphys.2012.04.007
|
| [20] |
W. Pauli, Zur Quantenmechanik des Magnetischen Elektrons, Zeitschrift für Physik, 43 (1927), 601–623. doi: 10.1007/BF01397326
|
| [21] |
F. Sauter, Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren, Zeitschrift für Physik, 63 (1930), 803–814. doi: 10.1007/BF01339277
|
| [22] |
D. Unal, I. Kisi, M. Tosun, Spinor Bishop equation of curves in Euclidean 3-space, Adv. Appl. Clifford Algebr., 23 (2013), 757–765. doi: 10.1007/s00006-013-0390-8
|
| [23] | J. Vaz, R. da Rocha, An introduction to Clifford algebras and spinors, Oxford Scholarship Online, 2016. Available from: 10.1093/acprof: oso/9780198782926.001.0001. |
| [24] | A. Wachter, Relativistic quantum mechanics, Dordrecht: Springer, 2011. |
Tülay Erişir. On spinor construction of Bertrand curves[J]. AIMS Mathematics, 2021, 6(4): 3583-3591. doi: 10.3934/math.2021213
DownLoad: