Advances in Applied Clifford Algebras

, Volume 25, Issue 4, pp 799–810 | Cite as

Geometry of the Hyperbolic Spinors Corresponding to Alternative Frame



In this paper, we study to express the theory of curves including a wide section of Lorentzian geometry in terms of spinors with two hyperbolic components which has an important place in the Clifford algebra. In other words, we express the rotation, element of SO(1, 3), between the Frenet frame and the other frame defined as alternatively of the (spacelike or timelike) curves in Minkowski space \({\mathbb{R}_1^3}\) in terms of the rotation, element of \({SU(2,\mathbb{H})}\), with the aid of the hyperbolic spinors.

Mathematics Subject Classification

62J05 62J07 


Clifford algebra Hyperbolic spinors Alternative frame Minkowski space 


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  1. 1.
    Antonuccio, F.: Hyperbolic Numbers and the Dirac Spinor (1998). arXiv:hep-th/9812036v1
  2. 2.
    Balci, Y, Erisir, T., Gungor, M.A.: Hyperbolic spinor darboux equations of spacelike curves in Minkowski 3-space, XII. In: National Geometry Symposium, Bilecik, Turkey, June 23–26 (2014)Google Scholar
  3. 3.
    Bilici M., Caliskan M.: On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space. Int. Math. Forum. 4(31), 1497–1509 (2009)MathSciNetMATHGoogle Scholar
  4. 4.
    Birman G.S., Nomizu K.: Trigonometry in Lorentzian Geometry. Ann. Math. Mont. 91(9), 534–549 (1984)MathSciNetGoogle Scholar
  5. 5.
    Bishop L.R.: There is more than one way to frame a curve. Am. Math. Monthly 82(3), 246–251 (1975)CrossRefMATHGoogle Scholar
  6. 6.
    Brauer R., Weyl H.: Spinors in n dimensions. Am. J. Math 57, 425–449 (1935)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carmeli, M.: Group Theory and General Relativity, Representations of the Lorentz Group and their Applications to the Gravitational Field. McGraw-Hill, Imperial College Press, New York (1977)Google Scholar
  8. 8.
    Cartan, É.: The Theory of Spinors. The M.I.T. Press, Cambridge (1966).Google Scholar
  9. 9.
    Del Castillo G.F.T., Barrales G.S.: Spinor formulation of the differential geometry of curves. Revista Colombiana de Matematicas 38, 27–34 (2004)MathSciNetMATHGoogle Scholar
  10. 10.
    Do Carmo, MP.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)Google Scholar
  11. 11.
    Ergin A.A.: Timelike Darboux curves on a timelike surface \(M \subset M_1^3\). Hadron. J. 24(6), 701–712 (2001)MathSciNetMATHGoogle Scholar
  12. 12.
    Friedrich, T.: Dirac Operators in Riemannian Geometry, American Mathematical Society Providence, Rhode Island, (2000).Google Scholar
  13. 13.
    Hanson, A.J., Ma, H.: Parallel Transport Approach to Curve Framing, Tech. Math. Rep., Indiana University, p. 425 (1995)Google Scholar
  14. 14.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, 2nd ed. Kluwer, The Netherlands (1992).Google Scholar
  15. 15.
    Ikawa T.: On curves and submanifolds in an indefinite-Riemannian manifold. Tsukuba J. Math. 9(2), 353–371 (1985)MathSciNetMATHGoogle Scholar
  16. 16.
    Kisi I., Tosun M.: Spinor Darboux Equations of Curves in Euclidean 3-Space. Math. Morav. 19(1), 87–93 (2015)MathSciNetGoogle Scholar
  17. 17.
    Ketenci, Z., Erisir, T., Gungor, M.A.: Spinor equations of curves in minkowski space, V. In: Congress of the Turkic World Mathematicians, Kyrgyzstan, June 05–07 (2014)Google Scholar
  18. 18.
    Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton University Press, New Jersey (1989)Google Scholar
  19. 19.
    O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, New York (1983)Google Scholar
  20. 20.
    O’Donnell, P.: Introduction to 2-Spinors in General Relativity. World Scientific Publishing Co. Pte. Ltd., London (2003)Google Scholar
  21. 21.
    Ozdemir M., Ergin A.A.: Spacelike Darboux curves in Minkowski 3 space. Differ. Geom. Dyn. Syst. 9, 131–137 (2007)MathSciNetGoogle Scholar
  22. 22.
    Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer, New York (1986).Google Scholar
  23. 23.
    Sobczyk G.: The Hyperbolic Number Plane. College Math. J. 26(4), 268–280 (1995)CrossRefGoogle Scholar
  24. 24.
    Unal D., Kisi I., Tosun M.: Spinor bishop equation of curves in Euclidean 3-space. Adv. Appl. Cliff. Algebr. 23(3), 757–765 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vivarelli M.D.: Development of spinors descriptions of rotational mechanics from Euler’s rigid body displacement theorem. Celes. Mech. 32, 193–207 (1984)MathSciNetCrossRefADSMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Tülay Erişir
    • 1
  • Mehmet Ali Güngör
    • 1
    • 2
  • Murat Tosun
    • 1
  1. 1.Department of MathematicsSakarya UniversitySakaryaTurkey
  2. 2.Faculty of Engineering and Natural SciencesInternational University of SarajevoSarajevoBosnia and Herzegovina

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