Abstract
We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that βp(M)<βp(M’), for 0<p<n and pairs isospectral on p-forms for every p odd, and having different holonomy groups, ℤ4 and ℤ 22 , respectively.
Similar content being viewed by others
References
Berger, M., Gauduchon, P., and Mazet, E.Le Spectre d'une Variété Riemannienne, LNM 194, Springer-Verlag, New York, 1971.
Brown, H., Bülow, R., Neubüser, J., Wondratschok, H., and Zassenhaus, H.Crystallographic Groups of Four-Dimensional Space, John Wiley & Sons, New York, 1978.
Charlap, L.Bieberbach Groups and Flat Manifolds, Springer-Verlag, 1988.
Chihara, L. and Stanton, D. Zeros of generalized Krawtchouk polynomials,J. Approx. Theory,60, 53–57, (1990).
Conway, J.H. and Sloane, N.J.A. Four-dimensional lattices with the same theta series,Intl. Math. Res. Notes,24, 93–96, (1992).
Deturck, D. and Gordon, C. Isospectral deformations I: Riemannian structures on two-step nilspaces,Comm. Pure App. Math.,40, 367–387, (1987).
Dotti, I. and Miatello, R. Isospectral compact flat manifolds,Duke Math. J.,68, 489–498, (1992).
Gilkey, P. On spherical space forms with metacyclic fundamental group which are isospectral but not equivarianly cobordant,Compositio Mathematica,56, 171–200, (1985).
Gordon, C. Riemannian manifolds isospectral on functions but not on 1-forms,J. Diff. Geom.,24, 79–96, (1986).
Gornet, R. Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms,J. Geom. Anal.,10(2), 281–298, (2000).
Hiller, H. Cohomology of Bieberbach groups,Mathematika,32, 55–59, (1985).
Ikeda, A. Riemannian manifoldsp-isospectral but not (p+1)-isospectral,Perspectives in Math.,8, 159–184, (1988).
Krasikov, I. and Litsyn, S. On integral zeros of Krawtchouk polynomials,J. Combin. Theory A,74, 71–99, (1996).
Miatello, R. and Rossetti, J.P.. Isospectral Hantzsche-Wendt manifolds,J. für die Reine Angewandte Mathematik, J. Reine Angew. Math.,515, 1–23, (1999).
Mitello, R. and Rossetti, J.P. Hantzsche-Wendt manifolds of dimension 7,Diff. Geom. Appl., Proceedings of the 7th International Conference, Masaryk Univ., Brno, 379–391, (1999).
Miatello, R. and Rossetti, J.P. Comparison of twistedp-form spectra for flat manifolds with diagonal holomony,Ann. Global Anal. Geom., to appear.
Pesce, H. Une reciproque generique du theorème of Sunada,Compos. Math.,109, 357–365, (1997).
Rossetti, J.P. and Tirao, P. Compact flat manifolds with holonomy group ℤ2⊕ℤ2.Rendiconti del Sem. Matem. de Padova,101, 99–136, (1999).
Schueth, D. Continuous families of isospectral metrics on simply connected manifolds,Ann. of Math.,149, 287–308, (1999).
Sunada, T. Riemannian coverings and isospectral manifolds,Ann. of Math.,121, 169–186, (1985).
Vigneras, M.F. Varietés Riemanniennes isospectrales et non isométriques,Ann. of Math.,112, 21–32, (1980).
Wallach, N.R.Harmonic Analysis on Homogeneous Spaces. Marcel-Dekker, New York, 1973.
Wells, R.Differential Analysis on Complex Manifolds, Springer-Verlag, GTM 65, Berlin-New York, 1967.
Wolf, J.Spaces of Constant Curvature, McGraw-Hill, New York, 1967.
Author information
Authors and Affiliations
Additional information
Communicated by Carolyn Gordon
Rights and permissions
About this article
Cite this article
Miatello, R.J., Rossetti, J.P. Flat manifolds isospectral on p-forms. J Geom Anal 11, 649–667 (2001). https://doi.org/10.1007/BF02930761
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02930761