Abstract. To every n-dimensional lens space L, we associate a congruence lattice L in Z m , with n = 2m−1 and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on L with the number of lattice elements of a given · 1 -length in L. As a consequence, we show that two lens spaces are isospectral on functions (resp. isospectral on p-forms for every p) if and only if the associated congruence lattices are · 1 -isospectral (resp. · 1 -isospectral plus a geometric condition). Using this fact, we give, for every dimension n ≥ 5, infinitely many examples of Riemannian manifolds that are isospectral on every level p and are not strongly isospectral.
Abstract. We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n = 2p, p ≥ 2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q = p, 0 ≤ q ≤ n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M ′ of dimension n ≥ 6, which are isospectral on functions and such that β p (M ) < β p (M ′ ), for 0 < p < n and pairs isospectral on p-forms for every p odd, and having different holonomy groups, Z 4 and Z 2 2 respectively.
We study the family of closed Riemannian n-manifolds with holonomy group isomorphic to Z n−1 2 , which we call generalized HantzscheWendt manifolds. We prove results on their structure, compute some invariants, and find relations between them, illustrated in a graph connecting the family.
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