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The Spinor Expressions of Mannheim Curves in Euclidean 3-Space

Yıl 2023, Cilt: 16 Sayı: 1, 111 - 121, 30.04.2023

Tülay Erişir Zeynep İsabeyoğlu

https://doi.org/10.36890/iejg.1210442

Öz

In this paper, the spinor formulations of Mannheim curve pair are investigated. First of all, two spinors matching to Mannheim curve pair are given and by considering the relationships between the Frenet frames of Mannheim curve pair, the relationship between two spinors matching to this curve pair are gotten. Therefore, a geometric interpretations of spinors are obtained using the Mannheim curve pair and considering Mannheim curve as helix the spinor formulations of Mannheim curve pair are given. Moreover, the spinor formulations are also obtained for the curvatures of the Mannheim curve pair. Consequently, an example of these spinors is obtained. Therefore, it is thought that this study will make an important contribution to the mathematical analysis and geometric interpretation of spinors, which have many uses in physics.

Anahtar Kelimeler

Spinors Clifford Algebra Euclidean space Mannheim curve pair

Kaynakça

  • [1] Akyigit, M., Ersoy, S., Özgür, I., Tosun, M.: Generalized Timelike Mannheim Curves in Minkowski Space-Time $E^4_1$. Mathematical Problems in Engineering. 1, 1-19 (2011).
  • [2] Akyigit, Azak, A. Z., Tosun, M.: Admissible Mannheim Curves in Pseudo-Galilean Space G31 . Afr. Diaspora J. of Math. 10, 58 (2010).
  • [3] Azak, A. Z.: On Timelike Mannheim Curve Pair in Three-Dimensional Lorentz Space. Sakarya University Journal of Science and Arts. 11 (2), 35-45 (2009).
  • [4] Balcı, Y., Eri¸sir, T., Güngör, M. A.: Hyperbolic Spinor Darboux Equations of Spacelike Curves in Minkowski 3-Space. J. Chungcheong Math. Soc. 28 (4), 525-535 (2015).
  • [5] Blum R.: A Remarkable Class of Mannheim-Curves. Canadian Mathematical Bulletin. 9, 223-228 (1966).
  • [6] Cartan, E.: The Theory of Spinors. Dover Publications. New York (1966).
  • [7] Torres del Castillo, G. F. T., Barrales, G. S.: Spinor Formulation of the Differential Geometry of Curves. Revista Colombiana de Matematicas. 38, 27-34 (2004).
  • [8] Delanghe, R., Sommon, F., Soucek, V.: Clifford Algebra and Spinor-Valued Functions: A function Theory for the Dirac Operator. Dover Publications. New York (1966).
  • [9] Dirac, P. A. M.: The Quantum Theory of the Electron. Proceeding of the Royal Society A. 117 (778), 610-624 (1928).
  • [10] Erişir, T., Güngör, M. A., Tosun, M.: Geometry of the Hyperbolic Spinors Corresponding to Alternative Frame. Adv. Appl. Clifford Algebr. 25 (4), 799-810 (2015).
  • [11] Erişir, T., Kardağ, N. C.: Spinor Representations of Involute Evolute Curves in E3. Funndam J. Math Appl. 2 (2), 148-155 (2019).
  • [12] Erişir, T.: On Spinor Construction of Bertrand Curves. AIMS Mathematics. 6 (4), 3583-3591 (2021).
  • [13] Ersoy, S., Tosun, M., Matsuda, H.: Generalized Mannheim Curves in Minkowski Space-Time E41 . Hokkaido Math. J. 41 (3), 441-461 (2012).
  • [14] Ersoy, S., Masal, M., Tosun, M.: On Mannheim Partner Curves of AW (k)-type. Uzbek. Mat. Zh. 1, 97-107 (2014).
  • [15] Ersoy, S., Akyigit, M., Tosun, M.: A Note an Admissible Mannheim Curves in Galilean Space. Int. J. Math. Comb. 1, 88-93 (2011).
  • [16] Güngör, M. A., Tosun, M.: A Study on Dual Mannheim Partner Curves. International Mathematical Forum. 5 (47), 2319-2330 (2010).
  • [17] Ketenci, Z., Erişir, T., Güngör M. A.: A Construction of Hyperbolic Spinors According to Frenet Frame in Minkowski Space. J. Dyn. Syst. Geom. Theor. 13 (2), 179-193 (2015).
  • [18] Kisi, I., Tosun, M.: Spinor Darboux Equations of Curves in Euclidean 3-Space. Math. Morav. 19 (1), 87-93 (2015).
  • [19] Landau, L.D., Lifshitz, E. M.: Quantum Mechanics (Non-relavistic Theory). Pergamon Press. Oxford (1977).
  • [20] Liu, H., Wang, F.: Mannheim Partner Curves in 3-Space. J. Geom. 88, 120-126 (2008).
  • [21] Mannheim, A.: Paris C. R., 86, 1254-1256 (1878).
  • [22] Masal, M., Azak A. Z.: Mannheim B-Curves in the Euclidean 3-Space E3. Kuwait J. Sci. 44 (1), 36-41 (2017).
  • [23] Matsuda, H., Yorozu, S.: On Genaralized Mannheim Curves in Euclidean 4-Space. Nihonkai Math. J. 20, 33-56 (2009).
  • [24] Orbay, K., Kasap, E.: On Mannheim Partner Curves in E3. International Journal of the Physical Sciences. 4 (5), 261-264 (2009).
  • [25] Pauli, W.: Zur Quantenmechanik des Magnetischen Elektrons. Zeitschrift für Physik. 43, 601-623 (1927).
  • [26] Payne, W. T.: Elementary Spinor Theory. American Journal of Physics. 20, 253 (1952).
  • [27] Senyurt, S., Bektas, O.: Timelike-Spacelike Mannheim Partner Curves in E31 . Int. J. Phys. Sci. 7 (1), 100-106 (2012).
  • [28] Ünal, D., Ki¸si, I., Tosun, M.: Spinor Bishop Equation of Curves in Euclidean 3-Space. Adv. Appl. Clifford Algebr. 23 (3), 757-765 (2013).
  • [29] Vivarelli, M. D.: Development of Spinor Descriptions of Rotational Mechanics from Euler’s Rigid Body Displacement Theorem. Celestial Mechanics.32, 193-207 (1984).
  • [30] Wachter, A.: Relativistic Quantum Mechanics. Springer. Dordrecht (2011).
  • [31] Wang, F., Liu, H.: Mannheim Partner Curves in 3-Euclidean Space. Mathematics in Practice and Theory. 1, 141-143 (2007).

Yıl 2023, Cilt: 16 Sayı: 1, 111 - 121, 30.04.2023

Tülay Erişir Zeynep İsabeyoğlu

https://doi.org/10.36890/iejg.1210442

Öz

Kaynakça

  • [1] Akyigit, M., Ersoy, S., Özgür, I., Tosun, M.: Generalized Timelike Mannheim Curves in Minkowski Space-Time $E^4_1$. Mathematical Problems in Engineering. 1, 1-19 (2011).
  • [2] Akyigit, Azak, A. Z., Tosun, M.: Admissible Mannheim Curves in Pseudo-Galilean Space G31 . Afr. Diaspora J. of Math. 10, 58 (2010).
  • [3] Azak, A. Z.: On Timelike Mannheim Curve Pair in Three-Dimensional Lorentz Space. Sakarya University Journal of Science and Arts. 11 (2), 35-45 (2009).
  • [4] Balcı, Y., Eri¸sir, T., Güngör, M. A.: Hyperbolic Spinor Darboux Equations of Spacelike Curves in Minkowski 3-Space. J. Chungcheong Math. Soc. 28 (4), 525-535 (2015).
  • [5] Blum R.: A Remarkable Class of Mannheim-Curves. Canadian Mathematical Bulletin. 9, 223-228 (1966).
  • [6] Cartan, E.: The Theory of Spinors. Dover Publications. New York (1966).
  • [7] Torres del Castillo, G. F. T., Barrales, G. S.: Spinor Formulation of the Differential Geometry of Curves. Revista Colombiana de Matematicas. 38, 27-34 (2004).
  • [8] Delanghe, R., Sommon, F., Soucek, V.: Clifford Algebra and Spinor-Valued Functions: A function Theory for the Dirac Operator. Dover Publications. New York (1966).
  • [9] Dirac, P. A. M.: The Quantum Theory of the Electron. Proceeding of the Royal Society A. 117 (778), 610-624 (1928).
  • [10] Erişir, T., Güngör, M. A., Tosun, M.: Geometry of the Hyperbolic Spinors Corresponding to Alternative Frame. Adv. Appl. Clifford Algebr. 25 (4), 799-810 (2015).
  • [11] Erişir, T., Kardağ, N. C.: Spinor Representations of Involute Evolute Curves in E3. Funndam J. Math Appl. 2 (2), 148-155 (2019).
  • [12] Erişir, T.: On Spinor Construction of Bertrand Curves. AIMS Mathematics. 6 (4), 3583-3591 (2021).
  • [13] Ersoy, S., Tosun, M., Matsuda, H.: Generalized Mannheim Curves in Minkowski Space-Time E41 . Hokkaido Math. J. 41 (3), 441-461 (2012).
  • [14] Ersoy, S., Masal, M., Tosun, M.: On Mannheim Partner Curves of AW (k)-type. Uzbek. Mat. Zh. 1, 97-107 (2014).
  • [15] Ersoy, S., Akyigit, M., Tosun, M.: A Note an Admissible Mannheim Curves in Galilean Space. Int. J. Math. Comb. 1, 88-93 (2011).
  • [16] Güngör, M. A., Tosun, M.: A Study on Dual Mannheim Partner Curves. International Mathematical Forum. 5 (47), 2319-2330 (2010).
  • [17] Ketenci, Z., Erişir, T., Güngör M. A.: A Construction of Hyperbolic Spinors According to Frenet Frame in Minkowski Space. J. Dyn. Syst. Geom. Theor. 13 (2), 179-193 (2015).
  • [18] Kisi, I., Tosun, M.: Spinor Darboux Equations of Curves in Euclidean 3-Space. Math. Morav. 19 (1), 87-93 (2015).
  • [19] Landau, L.D., Lifshitz, E. M.: Quantum Mechanics (Non-relavistic Theory). Pergamon Press. Oxford (1977).
  • [20] Liu, H., Wang, F.: Mannheim Partner Curves in 3-Space. J. Geom. 88, 120-126 (2008).
  • [21] Mannheim, A.: Paris C. R., 86, 1254-1256 (1878).
  • [22] Masal, M., Azak A. Z.: Mannheim B-Curves in the Euclidean 3-Space E3. Kuwait J. Sci. 44 (1), 36-41 (2017).
  • [23] Matsuda, H., Yorozu, S.: On Genaralized Mannheim Curves in Euclidean 4-Space. Nihonkai Math. J. 20, 33-56 (2009).
  • [24] Orbay, K., Kasap, E.: On Mannheim Partner Curves in E3. International Journal of the Physical Sciences. 4 (5), 261-264 (2009).
  • [25] Pauli, W.: Zur Quantenmechanik des Magnetischen Elektrons. Zeitschrift für Physik. 43, 601-623 (1927).
  • [26] Payne, W. T.: Elementary Spinor Theory. American Journal of Physics. 20, 253 (1952).
  • [27] Senyurt, S., Bektas, O.: Timelike-Spacelike Mannheim Partner Curves in E31 . Int. J. Phys. Sci. 7 (1), 100-106 (2012).
  • [28] Ünal, D., Ki¸si, I., Tosun, M.: Spinor Bishop Equation of Curves in Euclidean 3-Space. Adv. Appl. Clifford Algebr. 23 (3), 757-765 (2013).
  • [29] Vivarelli, M. D.: Development of Spinor Descriptions of Rotational Mechanics from Euler’s Rigid Body Displacement Theorem. Celestial Mechanics.32, 193-207 (1984).
  • [30] Wachter, A.: Relativistic Quantum Mechanics. Springer. Dordrecht (2011).
  • [31] Wang, F., Liu, H.: Mannheim Partner Curves in 3-Euclidean Space. Mathematics in Practice and Theory. 1, 141-143 (2007).
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Tülay Erişir 0000-0001-6444-1460

Zeynep İsabeyoğlu Bu kişi benim 0000-0003-4020-3315

Yayımlanma Tarihi 30 Nisan 2023
Kabul Tarihi 10 Nisan 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 16 Sayı: 1

Kaynak Göster

APA Erişir, T., & İsabeyoğlu, Z. (2023). The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. International Electronic Journal of Geometry, 16(1), 111-121. https://doi.org/10.36890/iejg.1210442
AMA Erişir T, İsabeyoğlu Z. The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. Int. Electron. J. Geom. Nisan 2023;16(1):111-121. doi:10.36890/iejg.1210442
Chicago Erişir, Tülay, ve Zeynep İsabeyoğlu. “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”. International Electronic Journal of Geometry 16, sy. 1 (Nisan 2023): 111-21. https://doi.org/10.36890/iejg.1210442.
EndNote Erişir T, İsabeyoğlu Z (01 Nisan 2023) The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. International Electronic Journal of Geometry 16 1 111–121.
IEEE T. Erişir ve Z. İsabeyoğlu, “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”, Int. Electron. J. Geom., c. 16, sy. 1, ss. 111–121, 2023, doi: 10.36890/iejg.1210442.
ISNAD Erişir, Tülay - İsabeyoğlu, Zeynep. “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”. International Electronic Journal of Geometry 16/1 (Nisan 2023), 111-121. https://doi.org/10.36890/iejg.1210442.
JAMA Erişir T, İsabeyoğlu Z. The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. Int. Electron. J. Geom. 2023;16:111–121.
MLA Erişir, Tülay ve Zeynep İsabeyoğlu. “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”. International Electronic Journal of Geometry, c. 16, sy. 1, 2023, ss. 111-2, doi:10.36890/iejg.1210442.
Vancouver Erişir T, İsabeyoğlu Z. The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. Int. Electron. J. Geom. 2023;16(1):111-2.
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