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

Gordon, Carolyn S.; Rossetti, Juan Pablo

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

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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