Advertisement

Geometriae Dedicata

, Volume 197, Issue 1, pp 107–122 | Cite as

The spectrum on p-forms of a lens space

  • Emilio A. LauretEmail author
Original Paper

Abstract

We give an explicit description of the spectrum of the Hodge–Laplace operator on p-forms of an arbitrary lens space for any p. We write the two generating functions encoding the p-spectrum as rational functions. As a consequence, we prove a geometric characterization of lens spaces that are p-isospectral for every p in an interval of the form \([0,p_0]\).

Keywords

Spectrum Lens space Isospectrality One-norm 

Mathematics Subject Classification (2010)

58J50 58J53 

References

  1. 1.
    Boldt, S., Lauret, E.A.: An explicit formula for the Dirac multiplicities on lens spaces. J. Geom. Anal. 27, 689–725 (2017).  https://doi.org/10.1007/s12220-016-9695-x MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cohen, M.: A course in simple-homotopy theory. GraduateTexts in Mathematics, vol. 10. Springer, New York (1970)Google Scholar
  3. 3.
    DeFord, D., Doyle, P.: Cyclic groups with the same Hodge series. Rev. Un. Mat. Argentina 59(2), 241–254 (2018)Google Scholar
  4. 4.
    DeTurck, D., Gordon, C.: Isospectral deformations II: trace formulas, metrics, and potentials. Commun. Pure Appl. Math. 42(8), 1067–1095 (1989).  https://doi.org/10.1002/cpa.3160420803 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gordon, C.: Riemannian manifolds isospectral on functions but not on 1-forms. J. Differ. Geom. 24(1), 79–96 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gornet, R.: Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms. J. Geom. Anal. 10(2), 281–298 (2000).  https://doi.org/10.1007/BF02921826 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gornet, R., McGowan, J.: Lens spaces, isospectral on forms but not on functions. LMS J. Comput. Math. 9, 270–286 (2006).  https://doi.org/10.1112/S1461157000001273 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ikeda, A.: On lens spaces which are isospectral but not isometric. Ann. Sci. École Norm. Sup. (4) 13(3), 303–315 (1980)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ikeda, A.: Riemannian manifolds \(p\)-isospectral but not \(p+1\)-isospectral. In: Geometry of Manifolds (Matsumoto, 1988). Perspect. Math. 8, 383–417 (1989)Google Scholar
  10. 10.
    Ikeda, A., Taniguchi, Y.: Spectra and eigenforms of the Laplacian on \(S^n\) and \(P^n(\mathbb{C})\). Osaka J. Math. 15(3), 515–546 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ikeda, A., Yamamoto, Y.: On the spectra of 3-dimensional lens spaces. Osaka J. Math. 16(2), 447–469 (1979)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lauret, E.A.: Spectra of orbifolds with cyclic fundamental groups. Ann. Glob. Anal. Geom. 50(1), 1–28 (2016).  https://doi.org/10.1007/s10455-016-9498-0 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lauret, E.A.: A computational study on lens spaces isospectral on forms (2017). arXiv:1703.03077
  14. 14.
    Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Representation equivalence and p-spectrum of constant curvature space forms. J. Geom. Anal. 25(1), 564–591 (2015).  https://doi.org/10.1007/s12220-013-9439-0 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Spectra of lens spaces from 1-norm spectra of congruence lattices. Int. Math. Res. Not. IMRN 2016(4), 1054–1089 (2016).  https://doi.org/10.1093/imrn/rnv159 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Non-strongly isospectral spherical space forms. In: Mathematical Congress of the Americas, Contemporary Mathematics, vol. 656. American Mathematical Society, Providence, RI (2016).  https://doi.org/10.1090/conm/656/13104
  17. 17.
    Lauret, E.A., Rossi Bertone, F.: Multiplicity formulas for fundamental strings of representations of classical Lie algebras. J. Math. Phys. 58, 111703 (2017).  https://doi.org/10.1063/1.4993851 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miatello, R.J., Rossetti, J.P.: Flat manifolds isospectral on \(p\)-forms. J. Geom. Anal. 11(4), 649–667 (2001).  https://doi.org/10.1007/BF02930761 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51(4), 542 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mohades, H., Honari, B.: Harmonic-counting measures and spectral theory of lens spaces. C. R. Math. Acad. Sci. Paris 354(12), 1145–1150 (2016).  https://doi.org/10.1016/j.crma.2016.10.016 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mohades, H., Honari, B.: On a relation between spectral theory of lens spaces and Ehrhart theory. Indag. Math. 28(2), 556–565 (2017).  https://doi.org/10.1016/j.indag.2017.01.003 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shams Ul Bari: N.: Orbifold lens spaces that are isospectral but not isometric. Osaka J. Math. 48(1), 1–40 (2011)Google Scholar
  23. 23.
    Solé, P.: Counting lattice points in pyramids. Discrete Math. 139, 381–392 (1995).  https://doi.org/10.1016/0012-365X(94)00142-6 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math. (2) 121(1), 169–186 (1985).  https://doi.org/10.2307/1971195 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vignéras, M.: Variétés riemanniennes isospectrales et non isométriques. Ann. Math. (2) 112(1), 21–32 (1980).  https://doi.org/10.2307/1971319 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany
  2. 2.CIEM–FaMAF (CONICET)Universidad Nacional de CórdobaCórdobaArgentina

Personalised recommendations