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Old and New Aspects in Spectral Geometry

  • Mircea Craioveanu
  • Mircea Puta
  • Themistocles M. Rassias

Part of the Mathematics and Its Applications book series (MAIA, volume 534)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 1-73
  3. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 75-117
  4. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 119-211
  5. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 213-271
  6. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 273-326
  7. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 327-353
  8. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 355-391
  9. Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias
    Pages 393-407
  10. Back Matter
    Pages 409-445

About this book

Introduction

It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera­ tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col­ loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva­ lent.

Keywords

Eigenvalue Matrix Matrix Theory Multilinear Algebra Riemannian geometry Riemannian manifold differential geometry linear algebra manifold

Authors and affiliations

  • Mircea Craioveanu
    • 1
  • Mircea Puta
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Facultatea de MatematicăUniversitatea de Vest din TimişoaraTimişoaraRomania
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-2475-3
  • Copyright Information Springer Science+Business Media B.V. 2001
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5837-9
  • Online ISBN 978-94-017-2475-3
  • Buy this book on publisher's site