Gordon, Carolyn S.; Rossetti, Juan Pablo
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, Tome 53 (2003) no. 7 , p. 2297-2314
Zbl 1049.58033 | MR 2044174
doi : 10.5802/aif.2007
URL stable : http://www.numdam.org/item?id=AIF_2003__53_7_2297_0

Classification:  58J53,  53C20
Mots clés: géométrie spectrale, laplacien de Hodge, variétés isospectrales, invariants de la chaleur
Soit M 2m une variété riemannienne compacte. On montre que le spectre du laplacien de Hodge opérant sur les m-formes ne détermine pas si M est à bord, ni les longueurs des géodésiques périodiques. Parmi les nombreux exemples il y a un espace projectif et un hémisphère qui ont le même spectre de Hodge sur les 1-formes, et des espaces hyperboliques, mutuellement isospectraux sur les 1-formes, qui ont des rayons d’injectivité différents. On montre aussi que le m-spectre de Hodge ne distingue pas entre orbifolds et variétés.
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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