The spectrum of twisted Dirac operators on compact flat manifolds
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- by Roberto J. Miatello and Ricardo A. Podestá PDF
- Trans. Amer. Math. Soc. 358 (2006), 4569-4603 Request permission
Abstract:
Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\mathbb {Z}_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the $\eta$-series, in terms of values of Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\mathbb {Z}_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\mathbb {Z}_2^k$-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat $n$-manifolds, pairwise nonhomeomorphic to each other of the order of $a^n$.References
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Additional Information
- Roberto J. Miatello
- Affiliation: FaMAF–CIEM, Universidad Nacional de Córdoba, Argentina
- MR Author ID: 124160
- Email: miatello@mate.uncor.edu
- Ricardo A. Podestá
- Affiliation: FaMAF–CIEM, Universidad Nacional de Córdoba, Argentina
- Email: podesta@mate.uncor.edu
- Received by editor(s): December 8, 2003
- Received by editor(s) in revised form: May 12, 2004, and October 8, 2004
- Published electronically: May 9, 2006
- Additional Notes: This work was supported by Conicet and Secyt-UNC
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4569-4603
- MSC (2000): Primary 58J53; Secondary 58C22, 20H15
- DOI: https://doi.org/10.1090/S0002-9947-06-03873-6
- MathSciNet review: 2231389