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

Carolyn S. Gordon; J. P. Rossetti

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

Carolyn S. Gordon·J. P. Rossetti

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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.

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BibTeX RIS

Carolyn S. Gordon, J. P. Rossetti. 2021-06-14. Boundary volume and length spectra of Riemannian manifolds: What the middle degree Hodge spectrum doesn't reveal. https://arxiv.org/abs/math/0111016

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