Tetra and Didi
= Tetra and Didi
{c}
A pair of closed orientable flat three-manifolds with equal scalar <Laplace-Beltrami operator> spectra but different first <Betti numbers>, respectively one and zero. Consequently they are not <homeomorphic>. They cannot be isospectral for the <Hodge Laplacian> on one-forms, whose zero-eigenvalue multiplicity is the first <Betti number>. This shows why the spectrum on functions must not be silently replaced by the spectra on all differential forms.