English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Two closed Riemannian manifolds are said to be isospectral if the associated Laplace Beltrami operators, acting on smooth functions, have the same eigenvalue spectrum. When two metrics are isospectral in this sense, one may ask whether the metrics can nonetheless be distinguished by additional spectral data—e.g., by the spectra of the Laplacians acting on p-forms. In this article we consider isospectral deformations, i.e., continuous families of isospectral metrics on a given manifold. In 1984, E. N. Wilson and the first author [GW] gave a method for constructing isospectral deformations (M, gt) on nilmanifolds M , generalized slightly in ([Go], [DG2]). The construction of these isospectral families is reviewed in section one. These were the only known examples of isospectral deformations of metrics on closed manifolds until recently when R. Gornet [Gt] constructed new examples of isospectral deformations of metrics on nilmanifolds by a different method. However, the metrics in Gornet’s examples can be distinguished by the spectra of the associated Laplacians acting on 1-forms. Thus we focus here on the earlier deformations. The metrics in these deformations are actually strongly isospectral; i.e., all natural self-adjoint elliptic partial differential operators associated with the metrics are isospectral. (Here natural refers to operators that are preserved by isometries; for the definition, see [St].) We show in section two, answering a question raised to us by P. Gilkey, that they are even strongly π1-isospectral; i.e., all natural self-adjoint elliptic operators with coefficients in locally flat bundles (defined by representations of the fundamental group of M) are isospectral. In contrast, Gilkey [Gi] showed that some strongly isospectral spherical space forms constructed by A. Ikeda are not strongly π1-isospectral. For this reason, we next consider non-flat Hermitian line bundles E over the manifold M . Associated to any metric on M and metric connection ∇ on E is

Distinguishing isospectral nilmanifolds by bundle laplacians