Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2297-2314.

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.

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.

DOI: 10.5802/aif.2007
Classification: 58J53,  53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants
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Gordon, Carolyn S.; Rossetti, Juan Pablo. Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2297-2314. doi : 10.5802/aif.2007. https://aif.centre-mersenne.org/articles/10.5802/aif.2007/

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