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

2001/11/02 by Carolyn S. Gordon, Gordon, Carolyn S., Juan Pablo Rossetti +1
Mathematics · #53C20 #58J53 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.math/0111016

openalex publication_date 2001/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Geometries of 3-Manifolds

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