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Length spectra andp-spectra of compact flat manifolds

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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.

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References

  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. Bérard P. and Webb, D. On ne peut entendre l’orientabilité d’une surface,C.R. Acad. Sci. Paris,320, 533–536, (1995).

    MATH  Google Scholar 

  3. Berger, M., Gauduchon, P., and Mazet, E.Le Spectre d’une Variété Riemannienne, LNM,194, Springer-Verlag, New York, (1971).

    MATH  Google Scholar 

  4. Blanchard, H. PhD. Thesis, Institut Fourier, France, (1998).

    Google Scholar 

  5. Buser, P.Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser, Boston, (1992).

    MATH  Google Scholar 

  6. Charlap, L.Bieberbach Groups and Flat Manifolds, Springer-Verlag, (1988).

  7. Colin de Verdiére, Y. Spectre du laplacien et longueurs des géodésiques périodiques, I, II,Compositio Math.,27, 83–106; 159–184, (1973).

    MathSciNet  MATH  Google Scholar 

  8. Croke, C. Rigidity for surfaces of nonpositive curvature,Comment. Math. Helv.,65(1), 150–169, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  9. Duistermaat, J.J. and Guillemin, V. The spectrum of positive elliptic operators and periodic bicharacteristics,Invent. Math.,29, 30–79, (1975).

    Article  MathSciNet  Google Scholar 

  10. Eberlein, P. Geometry of 2-step nilpotent groups with a left invariant metric,Ann. Sci. École Norm. Sup. (4),27, 611–660, (1994).

    MathSciNet  MATH  Google Scholar 

  11. Gangolli, R. The length spectra of some compact manifolds of negative curvature,J. Differential Geom.,12, 403–424, (1977).

    MathSciNet  MATH  Google Scholar 

  12. Gordon, C. The Laplace spectra versus the length spectra of Riemannian manifolds,Contemp. Math.,51, 63–79, (1986).

    Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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.

  15. Gornet, R. The marked length spectrum vs thep-form spectrum of Riemannian nilmanifolds,Comment. Math. Helv.,71, 297–329, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  16. Gornet, R. Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms,J. Geom. Anal.,10(2), 281–298, (2000).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  18. Grosswald, E.Representations of Integers as Sums of Squares, Springer-Verlag, New York, (1985).

    MATH  Google Scholar 

  19. Günther, P. The Poisson formula for Euclidean space groups and some of its applications, I,Z. Anal. Anwendungen,1, 13–23, (1982).

    MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  21. Hamenstädt, U. Cocycles, symplectic structures and intersection,Geom. Funct. Anal.,9(1), 90–140, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  22. Huber, H. Zur analytischen théorie hyperbolischer raumformen und bewegungsgruppen,Math. Ann.,138, 1–26, (1959).

    Article  MathSciNet  MATH  Google Scholar 

  23. Huber, H. Zur analytischen theorie hyperbolischer raumformen und bewegungsgruppen II,Math. Ann.,143, 463–464, (1961).

    Article  MathSciNet  MATH  Google Scholar 

  24. Jost, J.Riemannian Geometry and Geometric Analysis, Springer, Berlin-Heidelberg, (1998).

    MATH  Google Scholar 

  25. Krasikov, I. and Litsyn, S. On integral zeros of Krawtchouk polynomials,J. Combin. Theory A,74, 71–99, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  26. Meyerhoff, G.R. The ortho-length spectrum for hyperbolic 3-manifolds,Quart. J. Math. Oxford (2),47, 349–359, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  27. Miatello, R. and Rossetti, J.P. Isospectral Hantzsche-Wendt manifolds,J. Reine Angew. Math.,515, 1–23, (1999).

    MathSciNet  MATH  Google Scholar 

  28. Miatello, R. and Rossetti, J.P. Flat manifolds isospectral onp-forms,J. Geom. Anal,11(4), 649–667, (2001).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  30. Otal, J.P. Le spectre marqué des longueurs des surfaces à courbure négative,Ann. of Math.,131(1), 151–162, (1990).

    Article  MathSciNet  Google Scholar 

  31. Pesce, H. Une formule de Poisson pour les variétés de Heisenberg,Duke Math. J.,73, 515–538, (1994).

    Article  MathSciNet  Google Scholar 

  32. Reid, A. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds,Duke Math. J.,65, 215–228, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  33. Salvai, M. On the Laplace and complex length spectra of locally symmetric spaces of negative curvature,Math. Nachr.,239/240, 198–203, (2002).

    Article  MathSciNet  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  35. Serre, J.P.A Course in Arithmetic, Springer-Verlag, New York, Heidelberg, (1973).

    MATH  Google Scholar 

  36. Sunada, T. Spectrum of a compact flat manifold,Comment. Math. Helv.,53, 613–621, (1978).

    Article  MathSciNet  MATH  Google Scholar 

  37. Sunada, T. Riemannian coverings and isospectral manifolds,Annals of Math.,121, 169–186, (1985).

    Article  MathSciNet  Google Scholar 

  38. Wolf, J.Spaces of Constant Curvature, Mc Graw-Hill, New York, (1967).

    MATH  Google Scholar 

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Authors and Affiliations

  1. FaMAF-CIEM, Universidad Nacional de Córdoba, 5000, Córdoba, Argentina

    R. J. Miatello & J. P. Rossetti

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  1. R. J. Miatello
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  2. J. P. Rossetti
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Miatello, R.J., Rossetti, J.P. Length spectra andp-spectra of compact flat manifolds. J Geom Anal 13, 631–657 (2003). https://doi.org/10.1007/BF02921882

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