Comparison of Twisted P-Form Spectra for Flat Manifolds with Diagonal Holonomy

Authors: Miatello R.J.¹; Rossetti J.P.²

Source: Annals of Global Analysis and Geometry, Volume 21, Number 4, June 2002 , pp. 341-376(36)

Publisher: Springer

Abstract:

We give an explicit formula for the multiplicities of the eigenvalues of the Laplacian acting on sections of natural vector bundles over a compact flat Riemannian manifold M_Γ = \ \mathbb R^n, a Bieberbach group. In the case of the Laplacian acting on p-forms, twisted by a unitary character of , when has diagonal holonomy group F \mathbb Z_2^k, these multiplicities have a combinatorial expression in terms of integral values of Krawtchouk polynomials and the so called Sunada numbers. If the Krawtchouk polynomial K_p^n(x) does not have an integral root, this expression can be inverted and conversely, the presence of such roots allows to produce many examples of p-isospectral manifolds that are not isospectral on functions. We compare the notions of twisted p-isospectrality, twisted Sunada isospectrality and twisted finite p-isospectrality, a condition having to do with a finite part of the spectrum, proving several implications among them and getting a converse to Sunada's theorem in our context, for n 8. Furthermore, a finite part of the spectrum determines the full spectrum. We give new pairs of nonhomeomorphic flat manifolds satisfying some kind of isospectrality and not another. For instance: (a) manifolds which are isospectral on p-forms for only a few values of p 0, (b) manifolds which are twisted isospectral for every , a nontrivial character of F, and (c) large twisted isospectral sets.

Keywords: Laplacian; flat manifolds; twisted isospectral; Krawtchouk polynomials; vector bundles

Language: English

Document Type: Regular paper

Affiliations: 1: FaMAF-CIEM, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina. e-mail: miatello@mate.uncor.edu 2: FaMAF-CIEM, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina. e-mail: rossetti@mate.uncor.edu

Publication date: 2002-06-01

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• By this author: Miatello R.J. ;  Rossetti J.P.

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