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The Spinor Expressions of Mannheim Curves in Euclidean 3-Space

Year 2023, Volume: 16 Issue: 1, 111 - 121, 30.04.2023
https://doi.org/10.36890/iejg.1210442

Abstract

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.

References

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  • [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).
Year 2023, Volume: 16 Issue: 1, 111 - 121, 30.04.2023
https://doi.org/10.36890/iejg.1210442

Abstract

References

  • [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).
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Tülay Erişir 0000-0001-6444-1460

Zeynep İsabeyoğlu This is me 0000-0003-4020-3315

Publication Date April 30, 2023
Acceptance Date April 10, 2023
Published in Issue Year 2023 Volume: 16 Issue: 1

Cite

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. April 2023;16(1):111-121. doi:10.36890/iejg.1210442
Chicago Erişir, Tülay, and Zeynep İsabeyoğlu. “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”. International Electronic Journal of Geometry 16, no. 1 (April 2023): 111-21. https://doi.org/10.36890/iejg.1210442.
EndNote Erişir T, İsabeyoğlu Z (April 1, 2023) The Spinor Expressions of Mannheim Curves in Euclidean 3-Space. International Electronic Journal of Geometry 16 1 111–121.
IEEE T. Erişir and Z. İsabeyoğlu, “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”, Int. Electron. J. Geom., vol. 16, no. 1, pp. 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 (April 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 and Zeynep İsabeyoğlu. “The Spinor Expressions of Mannheim Curves in Euclidean 3-Space”. International Electronic Journal of Geometry, vol. 16, no. 1, 2023, pp. 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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