Length spectra and<Emphasis Type="Italic">p</Emphasis>-spectra of compact flat manifolds |
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Authors: | R J Miatello J P Rossetti |
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Institution: | (1) FaMAF-CIEM, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina |
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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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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 58J53 58C22 20H15 |
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