Annals of Global Analysis and Geometry

, Volume 16, Issue 3, pp 221–253 | Cite as

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

  • Bernd Ammann
  • Christian Bär


We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± ∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.

Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

circle bundles collapse Dirac operator Heisenberg manifolds isospectral deformation nilmanifolds 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B är, C.: Das Spektrum von Dirac-Operatoren, Bonner Math. Schr. 217 (1991).Google Scholar
  2. 2.
    B är, C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces, Arch. Math. 59 (1992), 65-79.Google Scholar
  3. 3.
    B är, C.: The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan 48 (1994), 69-83.Google Scholar
  4. 4.
    B är, C.: Metrics with harmonic spinors, GAFA 6 (1996), 899-942.Google Scholar
  5. 5.
    B érard Bergery, L. and Bourguignon, J.-P.: Riemannian submersions with totally geodesic fibers, Illinois J. Math. 26 (1982), 181-200.Google Scholar
  6. 6.
    Berline, N., Getzler, E., and Vergne, M.: Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1991.Google Scholar
  7. 7.
    Bourguignon, J.-P. and Gauduchon, P.: Spineurs, op érateurs de Dirac et variations de m étriques, Commun. Math. Phys. 144 (1992), 581-599.Google Scholar
  8. 8.
    Bunke, U.: Upper bounds of small eigenvalues of the Dirac operator and isometric immersions, Ann. Global Anal. Geom. 9 (1991), 109-116.Google Scholar
  9. 9.
    Cahen, M., Franc, A., and Gutt, S.: Spectrum of the Dirac operator on complex projective space P 2q-1(ℂ), Lett. Math. Phys. 18 (1989), 165-176.Google Scholar
  10. 10.
    Cahen, M., Franc, A., and Gutt, S.: Erratum to "spectrum of the Dirac operator on complex projective space P 2q-1(ℂ)', Lett. Math. Phys. 32 (1994), 365-368.Google Scholar
  11. 11.
    Camporesi, R. and Higuchi, A.: On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996), 1-18.Google Scholar
  12. 12.
    Corwin, L., and Greenleaf, F. P.: Representations of Nilpotent Lie Groups and Their Applications, Part 1: Basic Theory and Examples, Cambridge University Press, Cambridge, 1990.Google Scholar
  13. 13.
    Eberlein, P.: Geometry of two-step nilpotent groups with a left invariant metric, Ann. Scien. de l'Ecole Norm. Sup. 27 (1994), 611-660.Google Scholar
  14. 14.
    Fegan, H.: The spectrum of the Dirac operator on a simply connected compact Lie group, Simon Stevin 61 (1987), 97-108.Google Scholar
  15. 15.
    Friedrich, T.: Zur Abh ängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math. 48 (1984), 57-62.Google Scholar
  16. 16.
    Gornet, R.: A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996), 159-188.Google Scholar
  17. 17.
    Gornet, R.: Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal. (to appear).Google Scholar
  18. 18.
    Gordon, C. and Wilson, E.: Isospectral deformations of compact solvmanifolds, J. Diff. Geom. 19 (1984), 241-256.Google Scholar
  19. 19.
    Gordon, C. and Wilson, E.: The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), 253-271.Google Scholar
  20. 20.
    Hitchin, N.: Harmonic spinors, Adv. Math. 14 (1974), 1–55.Google Scholar
  21. 21.
    Lawson, H.-B. and Michelsohn, M.-L.: Spin Geometry, Princeton University Press, Princeton, NJ, 1989.Google Scholar
  22. 22.
    Milhorat, J.-L.: Spectre de l'op érateur de Dirac sur les espaces projectifs quaternioniens, C. R. Acad. Sci., Paris 1 (1992), 69-72.Google Scholar
  23. 23.
    Milhorat, J.-L.: Spectrum of the Dirac operator on quaternion-K ähler spin manifolds: The case of the symmetric space Gr 2(ℂm+2), J. Math. Phys. 39 (1998), 594-609.Google Scholar
  24. 24.
    Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. 51 (1964), 542.Google Scholar
  25. 25.
    Moroianu, A.: Op érateur de Dirac et submersions riemanniennes, Thesis, École Polytechnique, Palaiseau, 1996.Google Scholar
  26. 26.
    Moore, C. G. and Wolf, J. A.: Square integrable representations of nilpotent groups, Trans. AMS 185 (1973), 445-462.Google Scholar
  27. 27.
    Raghunathan, M. S.: Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer-Verlag, Berlin, 1972.Google Scholar
  28. 28.
    Seeger, L.: Der Dirac-Operator auf kompakten symmetrischen R äumen, Diplomarbeit, Universit ät Bonn, 1997.Google Scholar
  29. 29.
    Seeger, L.: The Dirac operator on oriented Grassmann manifolds and G 2 /SO(4), Preprint, 1997.Google Scholar
  30. 30.
    Seifarth, S. and Semmelmann, U.: The spectrum of the Dirac operator on the odd dimensional complex projective space P 2m-1 . (ℂ), SFB 288 Preprint No. 95, 1993.Google Scholar
  31. 31.
    Strese, H.: Über den Dirac-Operator auf Graßmann-Mannigfaltigkeiten, Math. Nachr. 98 (1980), 53-59.Google Scholar
  32. 32.
    Strese, H.: Spektren symmetrischer R äume, Math. Nachr. 98 (1980), 75-82.Google Scholar
  33. 33.
    Sulanke, S.: Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sph äre, Doktorarbeit, Humboldt-Universit ät Berlin, 1979.Google Scholar
  34. 34.
    Trautman, A.: Spin structures on hypersurfaces and the spectrum of the Dirac operator on spheres, in Oziewicz, B. (ed.), Spinors, Twistors, Clifford Algebras and Quantum Deformations, Kluwer Academic Publishers, Dordrecht, 1993, pp. 25-29.Google Scholar
  35. 35.
    Trautman, A.: The Dirac operator on hypersurfaces, Acta Phys. Polon. B 26 (1995), 1283-1310.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Bernd Ammann
    • 1
  • Christian Bär
    • 1
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburgGermany

Personalised recommendations