The Journal of Geometric Analysis

, Volume 13, Issue 4, pp 631–657

Length spectra andp-spectra of compact flat manifolds

  • R. J. Miatello
  • J. P. Rossetti

DOI: 10.1007/BF02921882

Cite this article as:
Miatello, R.J. & Rossetti, J.P. J Geom Anal (2003) 13: 631. doi:10.1007/BF02921882


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 

Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

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

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