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.

This book is open access, which means that you have free and unlimited access Offers a solid background on the theory of twisting Laplace operators on Riemannian manifolds Includes many examples and several open problems

Autorentext

Gunther Cornelissen holds the chair of geometry and number theory at Utrecht University. He earned his PhD from Ghent University in 1997 and has held visiting positions at institutions such as the Max Planck Institute for Mathematics in Bonn, Leuven University, California Institute of Technology and the University of Warwick. His research focuses on Arithmetic Geometry, particularly in positive characteristic, and also branches off into areas such as Spectral Geometry, Undecidability, Algebraic Dynamics and Graph Algorithms.

Norbert Peyerimhoff received his PhD in Mathematics in 1993 from the University of Augsburg. He held postdoctoral positions at the City University of New York, was an Assistant at the University of Basel and at the Ruhr University Bochum, before moving to Durham University (United Kingdom) in 2004. He has been a Professor of Geometry at Durham University since 2013, and his research interests include Differential Geometry, Discrete Geometry, Lie groups, Dynamical Systems, Spectral Theory and X-Ray Crystallography.

Klappentext

Chapter. 1. Introduction.- Part I: Leitfaden.- Chapter. 2. Manifold and orbifold constructions.- Chapter. 3. Spectra, group representations and twisted Laplacians.- Chapter. 4. Detecting representation isomorphism through twisted spectra.- Chapter. 5. Representations with a unique monomial structure.- Chapter. 6. Construction of suitable covers and proof of the main theorem.- Chapter. 7. Geometric construction of the covering manifold.- Chapter. 8. Homological wideness.- Chapter. 9. Examples of homologically wide actions.- Chapter. 10. Homological wideness, "class field theory" for covers, and a number theoretical analogue.- Chapter. 11. Examples concerning the main result.- Chapter. 12. Length spectrum.- References.- Index.

Inhalt
Chapter. 1. Introduction.- Part I: Leitfaden.- Chapter. 2. Manifold and orbifold constructions.- Chapter. 3. Spectra, group representations and twisted Laplacians.- Chapter. 4. Detecting representation isomorphism through twisted spectra.- Chapter. 5. Representations with a unique monomial structure.- Chapter. 6. Construction of suitable covers and proof of the main theorem.- Chapter. 7. Geometric construction of the covering manifold.- Chapter. 8. Homological wideness.- Chapter. 9. Examples of homologically wide actions.- Chapter. 10. Homological wideness, class field theory for covers, and a number theoretical analogue.- Chapter. 11. Examples concerning the main result.- Chapter. 12. Length spectrum.- References.- Index.