Let be a -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on -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 -spectrum also does not distinguish orbifolds from manifolds.
Soit une variété riemannienne compacte. On montre que le spectre du laplacien de Hodge opérant sur les -formes ne détermine pas si 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 -spectre de Hodge ne distingue pas entre orbifolds et variétés.
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}, 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/} }
TY - JOUR TI - Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal JO - Annales de l'Institut Fourier PY - 2003 DA - 2003/// SP - 2297 EP - 2314 VL - 53 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2007/ UR - https://www.ams.org/mathscinet-getitem?mr=2044174 UR - https://zbmath.org/?q=an%3A1049.58033 UR - https://doi.org/10.5802/aif.2007 DO - 10.5802/aif.2007 LA - en ID - AIF_2003__53_7_2297_0 ER -
%0 Journal Article %T Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal %J Annales de l'Institut Fourier %D 2003 %P 2297-2314 %V 53 %N 7 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2007 %R 10.5802/aif.2007 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2007/
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