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

Gordon, Carolyn S.; Rossetti, Juan Pablo

Gordon, Carolyn S. ; Rossetti, Juan Pablo

Annales de l'Institut Fourier, Volume 53 (2003) no. 7, p. 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 : https://doi.org/10.5802/aif.2007
Classification: 58J53, 53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants

@article{AIF_2003__53_7_2297_0,
     author = {Gordon, Carolyn S. and Rossetti, Juan Pablo},
     title = {Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     pages = {2297-2314},
     doi = {10.5802/aif.2007},
     zbl = {1049.58033},
     mrnumber = {2044174},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_7_2297_0}
}

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. http://www.numdam.org/item/AIF_2003__53_7_2297_0/

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