Twisted Isospectrality, Homological Wideness, and Isometry

A Sample of Algebraic Methods in Isospectrality

Gunther Cornelissen, Norbert Peyerimhoff


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The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings).

The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology.

The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds.

The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality. 

This is an open access book.

Détails
Auteur
Gunther  Cornelissen , Norbert  Peyerimhoff
Marque
Springer
Collection
SpringerBriefs in Mathematics
Parution
2023-05-10
Pages
111 pages
EAN papier
9783031277030
Langue
Anglais
Informations ebook
EAN PDF
9783031277047
Prix
0,00 €
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Nb pages copiables 1
Nb pages imprimables 11
Taille du fichier 3523 Ko
Ce livre est accessible aux handicaps   Informations sur l'accessibilité
EAN EPUB
9783031277047
Prix
0,00 €
En savoir plus
Nb pages copiables 1
Nb pages imprimables 11
Taille du fichier 4352 Ko
Ce livre est accessible aux handicaps   Informations sur l'accessibilité
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