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

Carolyn S. Gordon[1]; Juan Pablo Rossetti[2]

  • [1] Dartmouth College, Hanover, N.H. 03755 (USA)
  • [2] Universidad Nacional de Córdoba, FAMAF, Ciudad Universitaria, 5000 Córdoba (Argentine)

Annales de l'Institut Fourier (2003)

  • Volume: 53, Issue: 7, page 2297-2314
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gordon, Carolyn S., and 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 53.7 (2003): 2297-2314. <http://eudml.org/doc/116100>.

@article{Gordon2003,
abstract = {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.},
affiliation = {Dartmouth College, Hanover, N.H. 03755 (USA); Universidad Nacional de Córdoba, FAMAF, Ciudad Universitaria, 5000 Córdoba (Argentine)},
author = {Gordon, Carolyn S., Rossetti, Juan Pablo},
journal = {Annales de l'Institut Fourier},
keywords = {spectral geometry; Hodge Laplacian; isospectral manifolds; heat invariants; absolute boundary conditions; middle degree spectrum},
language = {eng},
number = {7},
pages = {2297-2314},
publisher = {Association des Annales de l'Institut Fourier},
title = {Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal},
url = {http://eudml.org/doc/116100},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Gordon, Carolyn S.
AU - Rossetti, Juan Pablo
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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 7
SP - 2297
EP - 2314
AB - 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.
LA - eng
KW - spectral geometry; Hodge Laplacian; isospectral manifolds; heat invariants; absolute boundary conditions; middle degree spectrum
UR - http://eudml.org/doc/116100
ER -

References

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