The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

Annals of Global Analysis and Geometry

Volumn 16, Issue 3, 1998, Pages 221-253

  Ammann, Bernd ^a   Bär, Christian ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Circle bundles; Collapse; Dirac operator; Heisenberg manifolds; Isospectral deformation; Nilmanifolds

Indexed keywords

EID: 0009139895     PISSN: 0232704X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1006553302362     Document Type: Article

References (35)

1. Bär, C.: Das Spektrum von Dirac-Operatoren, Bonner Math. Schr. 217 (1991).
2. Bär, C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces, Arch. Math. 59 (1992), 65-79.
3. Bär, C.: The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan 48 (1994), 69-83.
4. Bär, C.: Metrics with harmonic spinors, GAFA 6 (1996), 899-942.
5. Bérard Bergery, L. and Bourguignon, J.-P.: Riemannian submersions with totally geodesic fibers, Illinois J. Math. 26 (1982), 181-200.
6. Berline, N., Getzler, E., and Vergne, M.: Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1991.
7. Bourguignon, J.-P. and Gauduchon, P.: Spineurs, opérateurs de Dirac et variations de métriques, Commun. Math. Phys. 144 (1992), 581-599.
8. Bunke, U.: Upper bounds of small eigenvalues of the Dirac operator and isometric immersions, Ann. Global Anal. Geom. 9 (1991), 109-116.
9. q-1(ℂ), Lett. Math. Phys. 18 (1989), 165-176.
10. 2q-1(ℂ)', Lett. Math. Phys. 32 (1994), 365-368.
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.
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.
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.
14. Fegan, H.: The spectrum of the Dirac operator on a simply connected compact Lie group, Simon Stevin 61 (1987), 97-108.
15. Friedrich, T.: Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math. 48 (1984), 57-62.
16. Gornet, R.: A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996), 159-188.
17. Gornet, R.: Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal, (to appear).
18. Gordon, C. and Wilson, E.: Isospectral deformations of compact solvmanifolds, J. Diff. Geom. 19 (1984), 241-256.
19. Gordon, C. and Wilson, E.: The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), 253-271.
20. Hitchin, N.: Harmonic spinors, Adv. Math. 14 (1974), 1-55.
21. Lawson, H.-B. and Michelsohn, M.-L.: Spin Geometry, Princeton University Press, Princeton, NJ, 1989.
22. Milhorat, J.-L.: Spectre de l'opérateur de Dirac sur les espaces projectifs quaternioniens, C. R. Acad. Sci., Paris 1 (1992), 69-72.
23. m+2), J. Math. Phys. 39 (1998), 594-609.
24. Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. 51 (1964), 542.
25. Moroianu, A.: Opérateur de Dirac et submersions riemanniennes, Thesis, École Polytechnique, Palaiseau, 1996.
26. Moore, C. G. and Wolf, J. A.: Square integrable representations of nilpotent groups, Trans. AMS 185 (1973), 445-462.
27. Raghunathan, M. S.: Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer-Verlag, Berlin, 1972.
28. Seeger, L.: Der Dirac-Operator auf kompakten symmetrischen Räumen, Diplomarbeit, Universität Bonn, 1997.
29. 2/SO(4), Preprint, 1997.
30. 2m-1 (ℂ), SFB 288 Preprint No. 95, 1993.
31. Strese, H.: Über den Dirac-Operator auf Graßmann-Mannigfaltigkeiten, Math. Nachr. 98 (1980), 53-59.
32. Strese, H.: Spektren symmetrischer Räume, Math. Nachr. 98 (1980), 75-82.
33. Sulanke, S.: Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre, Doktorarbeit, Humboldt-Universität Berlin, 1979.
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.
35. Trautman, A.: The Dirac operator on hypersurfaces, Acta Phys. Polon. B 26 (1995), 1283-1310.