# 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

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topGordon, 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 -

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