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

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

2001-11-02

arxiv.org/abs/math/0111016v2

Mathematics

Differential Geometry

18 pages, 4 figures. This new version expands and generalizes results in the first version. Annales de l'Institut Fourier (to

Scientific paper

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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