no. 7

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, 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.

DOI : https://doi.org/10.5802/aif.2007
Classification : 58J53,  53C20
Mots clés : géométrie spectrale, laplacien de Hodge, variétés isospectrales, invariants de la chaleur 
@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},
     pages = {2297--2314},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     doi = {10.5802/aif.2007},
     mrnumber = {2044174},
     zbl = {1049.58033},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2007/}
}
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, Tome 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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