Abstract
We consider the Dirac operator on fibrations overS 1 which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.
Moreover, we discuss the dependence of the spectrum of theholonomy and obtain bounds for the multiplicity of the eigenvalues.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agricola, I., Ammann, B. and Friedrich, Th.: A comparison of the eigenvalues of the Dirac and Laplace operator on the two-dimensional torus, Manuscripta Math. 100 (1999), 231-258.
Agricola, I. and Friedrich, Th.: Upper bounds for the Dirac operator on surfaces, J. Geom. Phys. 30 (1999), 1-22.
Alekseevsky, A. V. and Alekseevsky, D. V.: Riemannian G-manifold with one-dimensional orbit space, Ann. Global Anal. Geom. 11 (1993), 197-211.
Ammann, B.: Spin-Strukturen und das Spektrum des Dirac-Operators, Dissertation, Shaker-Verlag, Aachen, 1998.
Ammann, B.: The Dirac operator on collapsing S 1-bundles, Preprint 39/1998 der Mathematischen Fakultät der Universität Freiburg.
Ammann, B. and Bär, C.: The Dirac operator on nilmanifolds and collapsing circle bundles, Ann. Global Anal. Geom. 16 (1998), 221-253.
Bär, C.: Das Spektrum von Dirac-Operatoren, Dissertation, Bonn, 1990.
Bär, C.: Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom. 16 (1998), 573-596.
Bär, C.: Metrics with harmonic spinors, GAFA 6(6), (1996), 899-942.
Baum, H., Friedrich, Th., Grunewald, R. and Kath, I.: Twistors and Killing Spinors on Riemannian Manifolds, Teubner, Stuttgart, 1991.
Eastham, M. S. P.: The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.
Friedrich, Th.: Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, Braunschweig, 1997.
Friedrich, Th.: Die Abhängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math. 48(1) (1984), 57-62.
Friedrich, Th.: On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998), 143-157.
Hitchin, N.: Harmonic spinors, Adv. Math. 14 (1974), 1-55.
Kraus, M.: Lower bounds for the eigenvalues of the Dirac operator on surfaces of rotation, J. Geom. Phys. 31 (1999), 209-216.
Kraus, M.: Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry, J. Geom. Phys. 32(4) (2000), 341-348.
Lawson, H.-B. and Michelsohn, M.-L.: Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989.
Lott, J.: Eigenvalue bounds for the Dirac operator, Pacific J. Math. 125 (1986), 117-126.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kraus, M. Eigenvalues of the Dirac Operator on Fibrations over S1. Annals of Global Analysis and Geometry 19, 235–257 (2001). https://doi.org/10.1023/A:1010738706881
Issue Date:
DOI: https://doi.org/10.1023/A:1010738706881