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Carolyn S. Gordon; Juan Pablo Rossetti
Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
(Volume du bord et spectre de longueurs des variétés riemanniennes : les invariants que le spectre de Hodge de degré moyen ne révèle pas)
Annales de l'institut Fourier, 53 no. 7 (2003), p. 2297-2314, doi: 10.5802/aif.2007
Article PDF | Reviews MR 2044174 | Zbl 1049.58033
Class. Math.: 58J53, 53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants

Résumé - Abstract

Let $M$ be a $2m$-dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on $m$-forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the same Hodge spectrum on 1- forms, and hyperbolic surfaces, mutually isospectral on 1-forms, with different injectivity radii. The Hodge $m$-spectrum also does not distinguish orbifolds from manifolds.

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