Osaka Journal of Mathematics

Properties of the Dirac spectrum on three dimensional lens spaces

Sebastian Boldt

Full-text: Open access

Abstract

We present a spectral rigidity result for the Dirac operator on lens spaces. More specifically, we show that each homogeneous lens space and each three dimensional lens space $L(q;p)$ with $q$ prime is completely characterized by its Dirac spectrum in the class of all lens spaces.

Article information

Source
Osaka J. Math., Volume 54, Number 4 (2017), 747-765.

Dates
First available in Project Euclid: 20 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1508486578

Mathematical Reviews number (MathSciNet)
MR3715361

Zentralblatt MATH identifier
06821136

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 58J53: Isospectrality 53C27: Spin and Spin$^c$ geometry 11R18: Cyclotomic extensions

Citation

Boldt, Sebastian. Properties of the Dirac spectrum on three dimensional lens spaces. Osaka J. Math. 54 (2017), no. 4, 747--765. https://projecteuclid.org/euclid.ojm/1508486578


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References

  • C. Bär: Das Spektrum von Dirac-Operatoren, PhD thesis, Naturwissenschaftliche Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn, Bonn, 1991.
  • C. Bär: The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan 48 (1996), 69–83.
  • C. Bär: Dependence of the Dirac spectrum on the Spin structure; in Global analysis and harmonic analysis (Marseille-Luminy, 1999), volume 4 of Sémin. Congr., pages 17–33. Soc. Math. France, Paris, 2000.
  • A. Baker, B.J. Birch and E.A. Wirsing: On a problem of Chowla, J. Number Theory 5 (1973), 224–236.
  • N. Berline, E. Getzler and M. Vergne: Heat kernels and Dirac operators, Springer-Verlag, Berlin, 2004.
  • S. Boldt and E.A. Lauret: An explicit formula for the Dirac multiplicities on Lens spaces, J. Geom. Anal. 27 (2017), 689–725.
  • T. Bröcker and T. tom Dieck: Representations of compact Lie groups, Springer-Verlag, New York, 1995.
  • S. Chowla. The nonexistence of nontrivial linear relations between the roots of a certain irreducible equation, J. Number Theory 2 (1970), 120–123.
  • M.M. Cohen: A course in simple-homotopy theory, Springer-Verlag, New York-Heidelberg-Berlin, 1973.
  • A. Franc: Spin structures and Killing spinors on lens spaces, J. Geom. Phys. 4 (1987), 277–287.
  • N. Ginoux: The Dirac spectrum, Lecture Notes in Mathematics 1976, Springer-Verlag, Berlin, 2009.
  • N. Hitchin: Harmonic spinors, Advances in Math. 14 (1974), 1–55.
  • A. Ikeda and Y. Yamamoto: On the spectra of 3-dimensional lens spaces, Osaka J. Math. 16 (1979), 447–469.
  • K. Katase: On the value of Dedekind sums and eta-invariants for 3-dimensional lens spaces, Tokyo J. Math. 10 (1987), 327–347.
  • H.B. Lawson and M.-L. Michelsohn: Spin geometry, Princeton Mathematical Series Princeton University Press, Princeton, NJ, 1989.
  • P. Ribenboim: The theory of classical valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999.
  • P. Ribenboim: Classical theory of algebraic numbers, Universitext, Springer-Verlag, New York, 2001.
  • S. Sulanke: Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre, PhD thesis, Humboldt-Universität zu Berlin, Berlin, 1979.
  • J.A. Wolf: Spaces of constant curvature, 5th ed., Publish or Perish, Inc., Wilmington, Delaware (U.S.A.), 1984.
  • Y. Yamamoto: On the number of lattice points in the square $|x|+|y|\leq u$ with a certain congruence condition, Osaka J. Math. 17 (1980), 9–21.