arXiv subjects

J. P. Rossetti

Publications and source records attributed to J. P. Rossetti.

5 recordsLinked to original sources

arXiv2021

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

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.

math.DG↗

arXiv2006

Hearing the platycosms

A `platycosm' is a flat Riemannian 3-manifold without boundary. In this paper we prove that there is (up to scale) a unique isospectral pair of compact platycosms.

math.DG↗

arXiv2004

(Z_2^k)-manifolds are isospectral on forms

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, $Δ_f$, acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z_2^k, with 0<k<n. This formula implies that any two compact flat manifolds with holonomy group Z_2^k having isospectral lattices of translations are isospectral on forms, that is, with respect to $Δ_f$. As a consequence, we construct a large family of pairwise $Δ_f$-isospectral and nonhomeomorphic n-manifolds of cardinality greater than $2^{(n-1)(n-2)/2}$.

math.DG↗

arXiv2003

Flat Manifolds Isospectral on p-Forms

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, for instance, pairs of flat manifolds of dimension n=2p, p>1, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q different from p. 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 Kahler, and the other not admitting any Kahler structure. We also construct pairs, M, M' of dimension n>5, which are isospectral on functions and such that the Betti numbers of M are less than those of M' for every 0<p<n; and pairs isospectral on p-forms for every p odd, and having different holonomy groups, Z_4 and Z_2+Z_2 respectively.

math.DG↗

arXiv2001

Length spectra and p-spectra of compact flat manifolds

We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having different lengths of closed geodesics and in some cases, different injectivity radius and different first eigenvalue. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the spectrum determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for flat manifolds. We give a variety of examples, for instance, a pair of isospectral (resp. Sunada isospectral) manifolds with different length spectra and a pair with the same complex length spectra and not p-isospectral for any p, or else p-isospectral for only one value of p different from 0.

math.DG↗