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We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose
corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the
heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on p-forms for all values of p are equivalent to each other. The list shows for any p, 1 ≤ p ≤ 3, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine
all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral
pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral.
Mathematics Subject Classifications (2000): 58J53, 58C22, 20H15. 相似文献
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Alexopoulos T Allen C Anderson EW Areti H Banerjee S Beery PD Biswas NN Bujak A Carmony DD Carter T Cole P Choi Y De Bonte RJ Erwin AR Findeisen C Goshaw AT Gutay LJ Hirsch AS Hojvat C Kenney VP Lindsey CS LoSecco JM McMahon T McManus AP Morgan N Nelson KS Oh SH Piekarz J Porile NT Reeves D Scharenberg RP Stampke SR Stringfellow BC Thompson MA Turkot F Walker WD Wang CH Wesson DK 《Physical review letters》1990,64(9):991-994
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Lazarus EA Navratil GA Greenfield CM Strait EJ Austin ME Burrell KH Casper TA Baker DR DeBoo JC Doyle EJ Durst R Ferron JR Forest CB Gohil P Groebner RJ Heidbrink WW Hong R Houlberg WA Howald AW Hsieh C Hyatt AW Jackson GL Kim J Lao LL Lasnier CJ Leonard AW Lohr J La Haye RJ Maingi R Miller RL Murakami M Osborne TH Perkins LJ Petty CC Rettig CL Rhodes TL Rice BW Sabbagh SA Schissel DP Scoville JT Snider RT Staebler GM Stallard BW Stambaugh RD St John HE Stockdale RE Taylor PL Thomas DM 《Physical review letters》1996,77(13):2714-2717
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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. 相似文献
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We determine the poles and residues of the resolvent kernel of the Laplacian on a Damek-Ricci space We show that all poles are simple and the residues define convolution operators of finite rank. This generalizes a result of Guillopé-Zworski for the real hyperbolic -space. If corresponds to a symmetric space of negative curvature , the image of each residue is a -module with a specific highest weight. We compute the dimension by the Weyl dimension formula.
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