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

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.

arXiv comment18 pages, 4 figures. Annales de l'Institut Fourier, Tome 53 (2003) no. 7 p. 2297-2314. This new version includes a couple of corrections to the original version, which will appear soon in the same journal as "Correction to Boundary...''