Isospectral hyperbolic surfaces have matching geodesics.
Tetra and Didi, the cosmic spectral twins.
Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
Let be a -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on -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 -spectrum also does not distinguish orbifolds from manifolds....
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