Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal

Gordon, Carolyn S.; Rossetti, Juan Pablo

doi:10.5802/aif.2007

Gordon, Carolyn S.; Rossetti, Juan Pablo

Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2297-2314.

Abstract
Résumé

Let $M$ be a $2 m$ -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 $M^{2 m}$ 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.

MR: 2044174 | Zbl: 1049.58033

DOI: 10.5802/aif.2007
Classification: 58J53, 53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants

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     title = {Boundary volume and length spectra of {Riemannian} manifolds: what the middle degree {Hodge} spectrum doesn't reveal},
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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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