The Journal of Geometric Analysis

, Volume 13, Issue 4, pp 631–657 | Cite as

Length spectra andp-spectra of compact flat manifolds

  • R. J. Miatello
  • J. P. Rossetti
Article

Abstract

We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials.

We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.

Math Subject Classifications

58J53 58C22 20H15 

Key Words and Phrases

Closed geodesic length spectrum isospectral Bieberbach group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bérard-Bergery, L. Laplacien et géodésic fermées sur les formes d’espace hyperbolique compactes, Séminaire Bourbaki, 24éme année (1971/1972), Exp. No. 406,Lectures Notes in Math.,317(406), 107–122, Springer, Berlin, (1973).Google Scholar
  2. [2]
    Bérard P. and Webb, D. On ne peut entendre l’orientabilité d’une surface,C.R. Acad. Sci. Paris,320, 533–536, (1995).MATHGoogle Scholar
  3. [3]
    Berger, M., Gauduchon, P., and Mazet, E.Le Spectre d’une Variété Riemannienne, LNM,194, Springer-Verlag, New York, (1971).MATHGoogle Scholar
  4. [4]
    Blanchard, H. PhD. Thesis, Institut Fourier, France, (1998).Google Scholar
  5. [5]
    Buser, P.Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser, Boston, (1992).MATHGoogle Scholar
  6. [6]
    Charlap, L.Bieberbach Groups and Flat Manifolds, Springer-Verlag, (1988).Google Scholar
  7. [7]
    Colin de Verdiére, Y. Spectre du laplacien et longueurs des géodésiques périodiques, I, II,Compositio Math.,27, 83–106; 159–184, (1973).MathSciNetMATHGoogle Scholar
  8. [8]
    Croke, C. Rigidity for surfaces of nonpositive curvature,Comment. Math. Helv.,65(1), 150–169, (1990).CrossRefMathSciNetMATHGoogle Scholar
  9. [9]
    Duistermaat, J.J. and Guillemin, V. The spectrum of positive elliptic operators and periodic bicharacteristics,Invent. Math.,29, 30–79, (1975).CrossRefMathSciNetGoogle Scholar
  10. [10]
    Eberlein, P. Geometry of 2-step nilpotent groups with a left invariant metric,Ann. Sci. École Norm. Sup. (4),27, 611–660, (1994).MathSciNetMATHGoogle Scholar
  11. [11]
    Gangolli, R. The length spectra of some compact manifolds of negative curvature,J. Differential Geom.,12, 403–424, (1977).MathSciNetMATHGoogle Scholar
  12. [12]
    Gordon, C. The Laplace spectra versus the length spectra of Riemannian manifolds,Contemp. Math.,51, 63–79, (1986).Google Scholar
  13. [13]
    Gordon, C. and Mao, Y. Comparison of Laplace spectra, length spectra and geodesic flows of some Riemannian manifolds,Math. Res. Lett.,1, 677–688, (1994).MathSciNetMATHGoogle Scholar
  14. [14]
    Gordon, C. and Rossetti, J.P. Boundary volume and length spectra of Riemannian manifolds: What the middle degree Hodge spectrum doesn’t reveal,Ann. Inst. Fourier, to appear.Google Scholar
  15. [15]
    Gornet, R. The marked length spectrum vs thep-form spectrum of Riemannian nilmanifolds,Comment. Math. Helv.,71, 297–329, (1996).CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    Gornet, R. Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms,J. Geom. Anal.,10(2), 281–298, (2000).MathSciNetMATHGoogle Scholar
  17. [17]
    Gornet R. and Mast, M.B. The length spectrum of Riemannian two-step nilmanifolds,Ann. Sci. École Norm. Sup.,33(2), 181–209, (2000).MathSciNetMATHGoogle Scholar
  18. [18]
    Grosswald, E.Representations of Integers as Sums of Squares, Springer-Verlag, New York, (1985).MATHGoogle Scholar
  19. [19]
    Günther, P. The Poisson formula for Euclidean space groups and some of its applications, I,Z. Anal. Anwendungen,1, 13–23, (1982).MATHGoogle Scholar
  20. [20]
    Günther, P. The Poisson formula for Euclidean space groups and some of its applications, II, The Jacobi transformation for flat manifolds,Z. Anal. Anwendungen,4, 341–352, (1985).MathSciNetMATHGoogle Scholar
  21. [21]
    Hamenstädt, U. Cocycles, symplectic structures and intersection,Geom. Funct. Anal.,9(1), 90–140, (1999).CrossRefMathSciNetMATHGoogle Scholar
  22. [22]
    Huber, H. Zur analytischen théorie hyperbolischer raumformen und bewegungsgruppen,Math. Ann.,138, 1–26, (1959).CrossRefMathSciNetMATHGoogle Scholar
  23. [23]
    Huber, H. Zur analytischen theorie hyperbolischer raumformen und bewegungsgruppen II,Math. Ann.,143, 463–464, (1961).CrossRefMathSciNetMATHGoogle Scholar
  24. [24]
    Jost, J.Riemannian Geometry and Geometric Analysis, Springer, Berlin-Heidelberg, (1998).MATHGoogle Scholar
  25. [25]
    Krasikov, I. and Litsyn, S. On integral zeros of Krawtchouk polynomials,J. Combin. Theory A,74, 71–99, (1996).CrossRefMathSciNetMATHGoogle Scholar
  26. [26]
    Meyerhoff, G.R. The ortho-length spectrum for hyperbolic 3-manifolds,Quart. J. Math. Oxford (2),47, 349–359, (1996).CrossRefMathSciNetMATHGoogle Scholar
  27. [27]
    Miatello, R. and Rossetti, J.P. Isospectral Hantzsche-Wendt manifolds,J. Reine Angew. Math.,515, 1–23, (1999).MathSciNetMATHGoogle Scholar
  28. [28]
    Miatello, R. and Rossetti, J.P. Flat manifolds isospectral onp-forms,J. Geom. Anal,11(4), 649–667, (2001).MathSciNetMATHGoogle Scholar
  29. [29]
    Miatello, R. and Rossetti, J.P. Comparison of twistedP-form spectra for flat manifolds with diagonal holonomy,Ann. Global Anal. Geom.,21, 341–376, (2002).CrossRefMathSciNetMATHGoogle Scholar
  30. [30]
    Otal, J.P. Le spectre marqué des longueurs des surfaces à courbure négative,Ann. of Math.,131(1), 151–162, (1990).CrossRefMathSciNetGoogle Scholar
  31. [31]
    Pesce, H. Une formule de Poisson pour les variétés de Heisenberg,Duke Math. J.,73, 515–538, (1994).CrossRefMathSciNetGoogle Scholar
  32. [32]
    Reid, A. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds,Duke Math. J.,65, 215–228, (1992).CrossRefMathSciNetMATHGoogle Scholar
  33. [33]
    Salvai, M. On the Laplace and complex length spectra of locally symmetric spaces of negative curvature,Math. Nachr.,239/240, 198–203, (2002).CrossRefMathSciNetGoogle Scholar
  34. [34]
    Selberg, A. Harmonic Analysys and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,J. Indian Math. Soc. (N.S.),20, 47–87, (1956).MathSciNetMATHGoogle Scholar
  35. [35]
    Serre, J.P.A Course in Arithmetic, Springer-Verlag, New York, Heidelberg, (1973).MATHGoogle Scholar
  36. [36]
    Sunada, T. Spectrum of a compact flat manifold,Comment. Math. Helv.,53, 613–621, (1978).CrossRefMathSciNetMATHGoogle Scholar
  37. [37]
    Sunada, T. Riemannian coverings and isospectral manifolds,Annals of Math.,121, 169–186, (1985).CrossRefMathSciNetGoogle Scholar
  38. [38]
    Wolf, J.Spaces of Constant Curvature, Mc Graw-Hill, New York, (1967).MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  • R. J. Miatello
    • 1
  • J. P. Rossetti
    • 1
  1. 1.FaMAF-CIEMUniversidad Nacional de CórdobaCórdobaArgentina

Personalised recommendations