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Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 165–183 | Cite as

A Few Comments on Conformal Spin Structures and Conformal U(1)-Spin Structures on a Pseudo-Riemannian 2r-Dimensional Manifold V

  • Pierre Anglès
Article

Abstract

This self-contained paper wants to precise the study of a real conformal spin structure in a strict sense over a pseudo-Riemannian or Riemannian 2r-dimensional manifold V already made in previous publications (Anglès in Studia Scientiarum Mathematicarum Hungarica 23:115–139, 1988; Anglès in Progress in Mathematical Physics, vol 50. Birkhäuser, Boston, 2008). We give a fundamental diagram (A) concerning U(1)-spin geometry, a notion which has been initiated in Atiyah et al. (Topology 3(suppl 1):3–38 (Pergamon Press), 1964), in a special case. The obstruction class for the existence of a conformal spin structure in a strict sense over V is studied. Necessary and sufficient conditions for the existence of such a structure are recalled, using groups called conformal spinoriality groups in a strict sense. The notion of a conformal U(1)-spin structure over a pseudo-Riemannian or Riemannian 2r-dimensional manifold V is defined and studied. Two fundamental diagrams (B) and (C), relative to the conformal U(1)-spin geometry are given. We study the obstruction class for the existence of a conformal U(1)-spin structure over V. New fiber bundles are defined.

Keywords

Spin structure Conformal spin structure U(1)-Spin structure Conformal U(1)-Spin structure Obstruction class 

Mathematics Subject Classification

11E88 15A66 17B37 20C30 16W55 

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References

  1. 1.
    Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(suppl. 1), 3–38. (1964) (Pergamon Press)Google Scholar
  2. 2.
    Anglès, P.: Construction de revêtements du groupe conforme d’un espace vectoriel muni d’une métrique de type (p, q). Annales de l’Institut Henri Poincaré, Section A XXXIII(1), 33–51 (1980)Google Scholar
  3. 3.
    Anglès, P.: Géométrie spinorielle conforme orthogonale triviale et groupes de Spinorialité conformes. Report-HTKK-MAT-A195, pp. 1–36. Helsinki University of Technology (1982)Google Scholar
  4. 4.
    Anglès, P.: Algèbres de Clifford \({C_{r, s}}\) des espaces quadratiques pseudo-Euclidiens standards \({E_{r, s}}\) et structures correspondantes sur les espaces de spineurs associés. Plongements naturels des quadriques projectives réelles \({\widetilde{Q}(E_{r, s})}\) attachées aux espaces \({E_{r, s}}\). Nato ASI Series, vol. 183, pp. 79–91. D. Reidel Publishing Company (1986)Google Scholar
  5. 5.
    Anglès P.: Real conformal spin structures on manifolds. Studia Scientiarum Mathematicarum Hungarica 23, 115–139 (1988)MathSciNetMATHGoogle Scholar
  6. 6.
    Anglès, P.: Conformal groups in geometry and Spin structures. In: Progress in Mathematical Physics, vol. 50. Birkhäuser, Boston (2008)Google Scholar
  7. 7.
    Anglès P.: The structure of the clifford algebra. Adv. Appl. Clifford Algebras 19(3–4), 585–610 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Blaine-Lawson H. Jr., Michelson M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)Google Scholar
  9. 9.
    Chevalley C.: The Algebraic Theory of Spinors and Clifford Algebras. Springer, Berlin (1999)Google Scholar
  10. 10.
    Deligne, P.: Notes on spinors. In: Quantum Fields and Strings: A Course for Mathematicians, vol. 1, pp. 99–134. American Mathematical Society, USA (1999)Google Scholar
  11. 11.
    Greub, W., Petry, R.: On the lifting of Structure groups. In: Lecture Notes in Mathematics, vol. 676, pp. 217–246, Bonn (1977)Google Scholar
  12. 12.
    Gürlebeck, K., Habetha, K., Spröessig, W.: Holomorphic Functions in the Plane and n-Dimensional Space, vol. 20. Birkhäuser, BaselGoogle Scholar
  13. 13.
    Haantjes J.: Conformal representations of an n-dimensional Euclidean space with a non-definite fundamental form on itself. Nedel. Akad. Wetensch. Proc. 40, 700–705 (1937)MATHGoogle Scholar
  14. 14.
    Haefliger, A.: Sur l’extension du groupe structural d’un espace. C. R. A. S. Paris 243, 588–560 (1956)Google Scholar
  15. 15.
    Helmstetter, J.: Conformal groups and Vahlen matrices (2014) (preprint)Google Scholar
  16. 16.
    Hirzebruch F.: Topological Methods in Algebraic Geometry, 3rd edn. Springer, Berlin (1966)CrossRefMATHGoogle Scholar
  17. 17.
    Karoubi, M.: Algèbres de Clifford et K-Théorie. Ann. scient. E. Norm. Sup. 4ème série, t. 1, pp. 161–270 (1968)Google Scholar
  18. 18.
    Lang S.: Algebra, 3rd edn. pp.–169. Springer, New York (2002)CrossRefMATHGoogle Scholar
  19. 19.
    Lichnerowicz A.: Champs spinoriels et propagateurs en relativité générale. Bull. Soc. Math. Fr. 92, 11–100 (1964)CrossRefMATHGoogle Scholar
  20. 20.
    Lounesto, P.: Clifford algebras and spinors, 2nd edn. In: London Mathematical Society, Lecture Notes Series, vol. 286. Cambridge University Press, Cambridge (2001)Google Scholar
  21. 21.
    Milnor, J.: Spin structure on manifolds. In: Enseignement Mathématique, Genève, 2ème série 9, pp. 198–203 (1963)Google Scholar
  22. 22.
    Rodrigues Jr., W.A., Capelas de Oliveira, E.: The Many Faces of Maxwell, Dirac and Einstein Equations. A Clifford Bundle Approach. Springer, Berlin (2007)Google Scholar
  23. 23.
    Weil, A.: Variétés Kählé riennes, Lemma 2, p. 90, Hermann, Paris (1958)Google Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Center CAIROS, Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul SabatierToulouse Cedex 9France

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