Communications in Mathematical Physics

, Volume 144, Issue 3, pp 581–599 | Cite as

Spineurs, opérateurs de dirac et variations de métriques

  • Jean-Pierre Bourguignon
  • Paul Gauduchon
Article

Abstract

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Références

  1. 1.
    Atiyah, M. F., Bott, R., Shapiro, A. Clifford modules. Topology3, [Suppl. 1] 3–38 (1964)Google Scholar
  2. 2.
    Bär, C.: Das Spektrum von Dirac-Operatoren. Dissertation, Univ. Bonn, 1990Google Scholar
  3. 3.
    Berger, M.: Sur les premières valeurs propres des variétés riemanniennes. Compositio Math.26, 129–149 (1973)Google Scholar
  4. 4.
    Besse, A. L.: Einstein manifolds. Ergeb. Math. vol.10. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  5. 5.
    Binz, E., Pferschy, R.: The Dirac operator and the change of the metric. C. R. Math. Rep. Acad. Sci. CanadaV 269–274 (1983)Google Scholar
  6. 6.
    Cartan, E.: La théorie des spineurs. Paris: Hermann 1937; and 2éme édition, The theory of spinors. Paris: Hermann 1966Google Scholar
  7. 7.
    Cartan, E.: Notice sur les travaux scientifiques. Paris: Gauthier-Villars 1974Google Scholar
  8. 8.
    Chevalley, C.: Algebraic theory of spinors New York: Columbia University Press 1954Google Scholar
  9. 9.
    Dabrowski, L., Percacci, R.: Spinors and diffeomorphisms. Commun. Math. Phys.106, 691–704 (1986)Google Scholar
  10. 10.
    Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen mannigfaltigkeit nicht negativer Skalarkrümmung. Math. Nach.97, 117–146 (1980)Google Scholar
  11. 11.
    Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys.104, 151–162 (1986)Google Scholar
  12. 12.
    Hitchin, N.: Harmonic spinors. Adv. Math.14, 1–55 (1974)Google Scholar
  13. 13.
    Kato, T.: Perturbation theory for linear operators. Grundl. Math. Wiss. vol.136. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  14. 14.
    Kosmann, Y.: Derivées de Lie des spineurs. Ann. Mat. Pura ed Appl.91, 317–395 (1972)Google Scholar
  15. 15.
    Lawson, H. B., Michelsohn, M. L.: Spin geometry. Princeton Math. Series vol.38, Princeton, NJ: Princeton University Press 1989Google Scholar
  16. 16.
    Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. ParisA257, 7–9 (1963)Google Scholar
  17. 17.
    Milnor, J. W.: Remarks concerning Spin-manifolds. In: Differential and Combinatorial Topology. A symposium in honor of Morse, M., Cairns, S. S. (ed.), pp. 55–62. Princeton, NJ: Princeton University Press 1965Google Scholar
  18. 18.
    Ne'eman, Y.: Spinor-type fields with linear, affine and general coordinate transformations. Ann. Inst. Henri PoincaréXXVIII, 369–378 (1978)Google Scholar
  19. 19.
    Penrose, R., Rindler, R.,: Spinors and space-time. Cambridge: Cambridge University Press 1984Google Scholar
  20. 20.
    Pferschy, R.: Die Abhängigkeit des Dirac-Operators von der Riemannschen Metrik. Dissertation Techn. Univ. Graz 1983Google Scholar
  21. 21.
    Weyl, H.: The classical groups, their invariants and representations. Princeton Math. Series1, Princeton, NJ: Princeton University Press, Rev. ed. 1946; 8th edition, 1973Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Jean-Pierre Bourguignon
    • 1
  • Paul Gauduchon
    • 2
  1. 1.Centre de Mathématiques, Ecole PolytechniqueU.R.A. 169 du C.N.R.S.Palaiseau CedexFrance
  2. 2.Département de MécaniqueUniversité Paris VI, U.R.A. 766 du C.N.R.S.Paris Cedex 05France

Personalised recommendations