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Preface

Published online by Cambridge University Press:  05 January 2012

Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
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Summary

Hyperbolic geometry is a classical subject of pure mathematics. It was invented as the first example of a non-euclidean geometry, and hence played a significant role in the development of modern geometry. Soon, however, its manifold connections with other branches of mathematics and science in general became apparent. Subsequently, it played a major role in such diverse fields as number theory, representation theory, complex functions, ergodic theory, dynamical systems, string theory, quantum chaos, and cosmology.

One of the earliest appearances of hyperbolic geometry in applications arose in connection with ergodic theory and dynamical systems, when in 1898 Hadamard realised that the negative curvature of hyperbolic surfaces produced an unstable and erratic behaviour of the geodesic motion. As a specific example, in 1924 Artin considered the geodesic billiard motion in an infinite triangle on the hyperbolic plane. He constructed a symbolic coding of the geodesics in terms of continued fractions and for the first time proved a quasi ergodic behaviour of the geodesics. Subsequently, many developments in ergodic theory and dynamical systems began with studies of geodesic flows on hyperbolic manifolds.

Beginning in the 1940s, a further far reaching development was initiated by the work of Maass, Selberg and others. Maass considered the spectral theory of the Laplace Beltrami operator on a hyperbolic surface, and Selberg discovered a close connection between the eigenvalues of the Laplacian and the closed geodesics on the surface, as expressed by the celebrated Selberg trace formula.

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Publisher: Cambridge University Press
Print publication year: 2011

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  • Preface
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.001
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  • Preface
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Preface
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.001
Available formats
×