TARU PUBLICATIONS
Journal of Dynamical Systems and Geometric Theories cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

WoS  JIF 2026 : 0.5 (Q3)

Powered by:Powered by

Half-Yearly Journal: Publishes research on dynamical systems and geometry, including Random Dynamical Systems, hyperbolic behaviour, complex geometry and Ergodic theory.

Issues up to 2022 co-published with and available at:Taylor & Francis
info@tarupublications.com
Open Access Research Article

SPINOR REPRESENTATIONS OF FRAMED CURVES IN THE THREE-DIMENSIONAL LIE GROUPS

* , ,

* Corresponding author · click or hover a name for details

pp. 61–83Vol. 21Issue 1May 2023DOI: 10.47974/JDSGT-21-1-P14XML
Received:
12 Nov 2022
Published Online:
01 May 2023
Article type:
Research Article
Language:
EN
Article no.:
JDSGT-21-1-P14
Pages:
61–83

Abstract

In this study, we determine the spinor representations of special singular curves (framed curves) in the three-dimensional Lie groups with a biinvariant metric. Also, we construct spinor framed equations for some special cases and obtain the relations between the spinor representations of the general frame and adapted frame along the framed curves in the three-dimensional Lie groups. Then, we give some geometric properties and results with respect to them.

Keywords

Subject Classifications

15A6622E1553A0458K05

References

[1] V. I. Arnold Sur la gomtrie di rentielle des groupes de Lie de dimension in nie et ses applications l'hydrodynamique des  uides parfaits, Ann. Inst. Fourier (Grenoble), Vol. 16, Number 1, 319{361 (1966).
[2] Y. Balc, T. Erisir, M. A. Gungor, Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space, J. Chungcheong Math. Soc., Vol. 28, Number 4, 525{535 (2015).
[3] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, Vol. 82, Number 3, 246{251 (1975).
[4] Z. Bozkurt, _I. Gok, O. Z. Okuyucu, N. Ekmekci, Characterizations of rectifying, normal and osculating curves in the three-dimensional compact Lie groups, Life Science Journal, Vol. 10, Number 3, 819{823 (2013).
[5] R. Brauer, H. Weyl, Spinors in n dimensions, American Journal of Mathematics, Vol. 57, Number 2, 425{449 (1935).
[6] A. C akmak, S. Kzltug, Spherical indicatrices of a Bertrand curve in three-dimensional Lie groups, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., Vol. 68, Number 2, 1930{1938 (2019).
[7] E. Cartan, The Theory of Spinors, Hermann, Paris, 1966 [Dover, New York (reprinted 1981)]). 
[8] W. K. Cli ord, Applications of Grassmann's extensive algebra, Am. J. Math., Vol. 1, Number 4, 350{358 (1878).
[9] A. C. C oken, U. C iftci, A note on the geometry of Lie groups, Nonlinear Anal., Vol. 68, Number 7, 2013{2016, (2008).
[10] P. Crouch, L. F. Silva, The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces, J. Dyn. Control Syst., Vol. 1, Number 2, 177{202 (1995).
[11] U. C iftci, A generalization of Lancert's theorem, J. Geom. Phys., Vol. 59, Number 12, 1597{ 1603 (2009).
[12] G. Darboux, Lecons Sur La Thorie Gnrale Des Surfaces I-II-III-IV, Gauthier-Villars, Paris, 1896.
[13] G. F. T. del Castillo, 3-D Spinors, Spin-Weighted Functions and Their Applications, Vol. 32, Springer Science & Business Media, Boston, 2003.
[14] G. F. T del Castillo, G. S. Barrales, Spinor formulation of the di erential geometry of curve, Rev. Colomb. de Mat., Vol. 38, Number 1, 27{34 (2004).
[15] N. do Espirito-Santo, S. Fornari, K. Frensel, J. Ripoll, Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscr. Math., Vol. 111, 459{470 (2003).
[16] B. Dogan Yazc, S.  O. Karakus, M. Tosun, Framed normal curves in Euclidean space, Tbilisi Math. J., Sciendo, 27{37 (2020).
[17] B. Dogan Yazc, S.  O. Karakus, M. Tosun, On the classi cation of framed rectifying curves in Euclidean space, Math. Meth. Appl. Sci., Vol. 45, Number 18, 12089{12098 (2022).
[18] B. Dogan Yazc, O. Z. Okuyucu, M. Tosun, Framed curves in three-dimensional Lie groups and a Berry phase model, J. Geom. Phys., Vol. 182, 104682 (2022).
[19] B. Dogan Yazc, Z. _ Isbilir, M. Tosun, Spinor representation of framed Mannheim curves, Turk. J. Math., Vol. 46, Number 7, 2690{2700 (2022).
[20] T. Erisir, On spinor construction of Bertrand curves, AIMS Mathematics, Vol. 6, Number 4, 3583{3591 (2021).
[21] T. Erisir, M. A. Gungor, M. Tosun, Geometry of the hyperbolic spinors corresponding to alternative frame, Adv. Appl. Cli ord Algebras, Vol. 25, 799{810 (2015).
[22] T. Erisir, N. C. Kardag, Spinor representations of involute evolute curves in E3, Fundam. J. Math. Appl., Vol. 2, Number 2, 148{155 (2019).
[23] T. Erisir, H. K.  Oztas, Spinor equations of successor curves, Univers. J. Math. Appl., Vol. 5, Number 1, 32{41 (2022).
[24] T. Erisir, M. A. Gungor, On Fibonacci spinors, Int. J. Geom. Methods Mod. Phys., Vol. 17, Number 04, 2050065 (2020).
[25] F. Frenet, Sur les courbes a double courbure, Journal de mathmatiques pures et appliques, (1852), 437{447.
[26] T. Fukunaga, M. Takahashi, Existence conditions of framed curves for smooth curves, J. Geom., Vol. 108, 763{774 (2017).
[27] _I. Gok, O. Z. Okuyucu, N. Ekmekci, Y. Yayl, On Mannheim partner curves in the threedimensional Lie groups, Miskolc Math. Notes, Vol. 15, Number 2, 467{479 (2014).
[28] J. Hladik, Spinors in Physics, Springer Science & Business Media, New York, 1999.
[29] S. Honda, Rectifying developable surfaces of framed base curves and framed helices, Singularities in Generic Geometry, Mathematical Society of Japan, (2018), 273{292.
[30] S. Honda, M. Takahashi, Framed curves in the Euclidean space, Adv. Geom., Vol. 16, 265{276 (2016).
[31] S. Honda, M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3- dimensional Euclidean space, Turk. J. Math., Vol. 44, Number 3, 883{899 (2020).
[32] S. Honda, M. Takahashi, Evolutes and focal surfaces of framed immersions in the Euclidean space, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 150, Number 1, 497{516 (2020).
[33] J. Huang, D. Pei, Singularities of non-developable surfaces in three-dimensional Euclidean space, Mathematics, Vol. 7, Number 11, 1106 (2019).
[34] Z. _ Isbilir, B. Dogan Yazc, M. Tosun, The spinor representations of framed Bertrand curves, Filomat, Vol. 37, Number 9, 2831{2842 (2023).
[35] Z. Ketenci, T. Erisir, M. A. Gungor, A construction of hyperbolic spinors according to the Frenet frame in Minkowski space, Journal of Dynamical Systems and Geometric Theories, Vol. 13, Number 2, 179{193 (2015).
[36] Z. Ketenci, T. Erisir, M. A. Gungor, Spinor equations of curves in Minkowski space, V. In: Congress of the Turkic World Mathematicians, Kyrgyzstan, June 05-07 (2014).
[37] _I. Kisi, M. Tosun, Spinor Darboux equations of curves in Euclidean 3-space, Mathematica Moravica, Vol. 19, Number 1, 87{93 (2015).
[38] S. Kzltug, S. C akal, Bertrand curves of AW(k)-type in three dimensional Lie groups, J. Math. Comput. Sci., Vol. 7, Number 4, 806{816 (2017).
[39] B. Kolev, Lie groups and mechanics: An introduction, J. Nonlinear Math. Phys., Vol. 11, Number 4, 480{498 (2004).
[40] Y. Li, S. Liu, Z. Wang, Tangent developables and Darboux developables of framed curves, Topol. Appl., Vol. 301, 107526 (2021).
[41] P. Lounesto, Cli ord Algebras and Spinors. In: Cli ord Algebras and Their Applications in Mathematical Physics, Vol. 183, Springer, Dordrecht, 1986.
[42] P. Lounesto, Cli ord Algebras and Spinors, Cambridge University Press, 2001.
[43] O. Z. Okuyucu, _I. Gok, Y. Yayl, N. Ekmekci, Bertrand curves in the three-dimensional Lie groups, Miskolc Math. Notes, Vol. 17, Number 2, 999{1010 (2017).
[44] O. Z. Okuyucu, _I. Gok, Y. Yayl, N. Ekmekci, Slant helices in the three-dimensional Lie groups, Appl. Math. Comput., Vol. 221, 672{683 (2013).
[45]  O. G. Yldz, O. Z. Okuyucu, Inextensible  ows of curves in Lie groups, Casp. J. Math. Sci., Vol. 2, Number 1, 23{32 (2013).
[46] O. Z. Okuyucu, C., Degirmen,  O. G. Yldz, Smarandache curves in the three-dimensional Lie groups, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., Vol. 68, Number 1, 1175{1185 (2019).
[47] O. Z. Okuyucu,  O. G. Yldz, M. Tosun, Spinor Frenet equations in the three-dimensional Lie groups, Adv. Appl. Cli ord Algebras, Vol. 26, 1341{1348 (2016).
[48] Okuyucu O. Z., Dogan Yazc B. Generalized Bertrand and Mannheim curves in 3D Lie groups, Fundam. J. Math. Appl., Vol. 5, Number 3, 201{209 (2022).
[49] W. Pauli, Zur Quantenmechanik des magnetischen elektrons, Zeitschrift fr Physik, Vol. 43, 601{632 (1927).
[50] J. A. Serret, Sur quelques formules relatives la thorie des courbes double courbure, Journal de Mathmatiques Pures et Appliques, (1851), 193{207.
[51] S. Senyurt, A. C alskan, Spinor formulation of Sabban frame of curve on S2, Pure Math. Sci., Vol. 4, Number 1, 37{42 (2015).
[52] M. A. Soliman, N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of space curves using type-3 Bishop frame, Casp. J. Math. Sci., Vol. 8, Number 1, 58{73 (2019).
[53] M. Takahashi, Legendre curves in the unit spherical bundle over the unit sphere and evolutes, Contemp. Math., Vol. 675, 337{355 (2016).
[54] S. I. Tomonaga, The Story of Spin, University of Chicago Press, 1997.
[55] D. Unal, _I. Kisi, M. Tosun, Spinor Bishop equations of curves in Euclidean 3-space, Adv. Appl. Cli ord Algebras, Vol. 23, 757{765 (2013).
[56] D. Unal, Spinor Q-equations in Lorentzian 3-space E3 1 , Bitlis Eren Universitesi Fen Bilimleri Dergisi, Vol. 11, 294{300 (2022).
[57] J. Vaz, R. da Rocha, An Introduction to Cli ord Algebras and Spinors, Oxford University Press, 2016.
[58] M. D. Vivarelli, Development of spinors descriptions of rotational mechanics from Eulers rigid body displacement theorem, Celestial Mechanics, Vol. 32, 193{207 (1984).
[59] Y. Wang, D. Pei, R. Gao, Generic properties of framed rectifying curves, Mathematics, Vol. 7, Number 1, 37 (2019).
[60]  O. G. Yldz, M. Akyigit, M. Tosun, On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Methods Appl. Sci., Vol. 44, 7463{7470 (2021).
[61] S. Ylmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., Vol. 371, Number 2, 764{776 (2010).
[62] D. W. Yoon, General helices of AW(k)-type in the Lie group, J. Appl. Math., Vol. 10, 535123 (2012).

Views: 140Downloads: 66Citations: 0