Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Real spin-Clifford bundle and the spinor structure of space-time

  • 68 Accesses

  • 12 Citations


By analyzing the conditions for the existence on a space-time ℒ of a global algebraic spinor field, we prove the following result, known as Geroch's theorem: A necessary and sufficient condition for ℒ to admit a spinor structure is that the orthonormal frame bundleF 0(ℒ) have a global section. Our proof, which does not use in any stage the complexification of ℝ1,3 (the space-time Clifford algebra), is simple, requiring only the explicit construction of the algebraic spinor and the spinorial metric within ℝ1,3 and elementary facts about associated bundles and the bundle reduction process. This is to be compared with the original proof, which uses the full algebraic topology machinery. We also clarify the relation of the covariant spinor structure and Graf'se-spinor structure.

This is a preview of subscription content, log in to check access.


  1. Atiyah, M. F., Bott, R., and Shapiro, A. (1964).Topology 3(Suppl. 1), 3.

  2. Been, I. M., and Tucker, R. W. (1985).Communications in Mathematical Physics,98, 53.

  3. Bichteler, K. (1968).Journal of Mathematical Physics,9, 813.

  4. Blaine Lawson, Jr., H., and Michelsohn, M. L. (1983).Spin Geometry, Universidad Federal do Ceará, Brazil.

  5. Blau, M. (1987).Letters in Mathematical Physics,13, 83.

  6. Bugajska, K. (1968). InClifford Algebras and their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds., D. Reidel, Dordrecht.

  7. Bugajska, K. (1979).International Journal of Theoretical Physics,18, 77.

  8. Choquet-Bruhat, Y., Dewitt-Morette, C., Dillard-Bleick, M. (1982).Analysis, Manifolds and Physics, rev. ed. North-Holland, Amsterdam.

  9. Crumeyrole, A. (1969).Annales de l'Institut Henri Poincaré,XI, 19.

  10. Figueiredo, V. L., Oliveira, E. C., and Rodrigues, Jr., W. A. (1990).International Journal of Theoretical Physics, this issue.

  11. Geroch, R. (1968).Mathematical Physics,9, 1739.

  12. Graf, W. (1978).Annales de l'Institut Henri Poincaré,XXIV, 85.

  13. Lee, K. K. (1973).General Relativity and Gravitation,6, 421.

  14. Milnor, J. (1963).L'Enseignement Mathematique,9, 198.

  15. Popovici, I. (1976).Annales de l'Institut Henri Poincaré,XXV, 35.

  16. Porteous, I. R. (1981).Topological Geometry, 2nd ed., Cambridge University Press, Cambridge.

  17. Rodrigues, Jr., W. A., Faria-Rosa, M. A., Maia, Jr., A., and Recami, E. (1989). R. T., IMECC-UNICAMP, to appear inHadronic Journal.

  18. Sachs, R. K., and Wu, H. (1977).General Relativity for Mathematicians, Springer-Verlag, New York.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rodrigues, W.A., Figueiredo, V.L. Real spin-Clifford bundle and the spinor structure of space-time. Int J Theor Phys 29, 413–424 (1990). https://doi.org/10.1007/BF00674440

Download citation


  • Field Theory
  • Elementary Particle
  • Quantum Field Theory
  • Reduction Process
  • Spinor Structure