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Cartan Geometries and their Symmetries

A Lie Algebroid Approach

  • Book
  • © 2016

Overview

Authors:
  1. Mike Crampin

    1. Burnham Market, United Kingdom

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  2. David Saunders

    1. University of Ostrava, Ostrava, Czech Republic

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    You can also search for this author in PubMed   Google Scholar

  • Expounds a new approach to the theory of Cartan connections as path connections on a certain class of Lie groupoids, or as infinitesimal connections on corresponding Lie algebroids
  • It contains a comprehensive account of the symmetries of Cartan geometries
  • Based on these ideas it extends Cartan's theory of a single ordinary differential equation to cover systems of such equations
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Variational Geometry (ASVG, volume 4)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Lie Groupoids and Lie Algebroids

    • Mike Crampin, David Saunders
    Pages 1-25

  3. Connections on Lie Groupoids and Lie Algebroids

    • Mike Crampin, David Saunders
    Pages 27-54

  4. Groupoids of Fibre Morphisms

    • Mike Crampin, David Saunders
    Pages 55-75

  5. Four Case Studies

    • Mike Crampin, David Saunders
    Pages 77-103

  6. Symmetries

    • Mike Crampin, David Saunders
    Pages 105-126

  7. Cartan Geometries

    • Mike Crampin, David Saunders
    Pages 127-151

  8. A Comparison with Alternative Approaches

    • Mike Crampin, David Saunders
    Pages 153-176

  9. Infinitesimal Cartan Geometries on \(\textit{TM}\)

    • Mike Crampin, David Saunders
    Pages 177-195

  10. Projective Geometry: The Full Version

    • Mike Crampin, David Saunders
    Pages 197-226

  11. Conformal Geometry: The Full Version

    • Mike Crampin, David Saunders
    Pages 227-238

  12. Developments and Geodesics

    • Mike Crampin, David Saunders
    Pages 239-255

  13. Cartan Theory of Second-Order Differential Equations

    • Mike Crampin, David Saunders
    Pages 257-284

  14. Back Matter

    Pages 285-290
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Keywords

About this book

In this book we first review the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. We next consider Lie groupoids of fibre morphisms of a  fibre bundle, and the connections on such groupoids together with their symmetries. We also see how the infinitesimal approach, using Lie algebroids rather than Lie groupoids, and in particular using Lie algebroids of vector fields along the projection of the fibre bundle, may be of benefit.



We then introduce Cartan geometries, together with a number of tools we shall use to study them. We take, as particular examples, the four classical types of geometry: affine, projective, Riemannian and conformal geometry. We also see how our approach can start to fit into a more general theory. Finally, we specialize to the geometries (affine and projective) associated with path spaces and geodesics, and consider their symmetries and other properties.

Reviews

“The authors expose an alternative approach to Cartan geometries. … the book offers a nice exposition of the approach to Cartan geometries via Lie algebroids, one also has to say that, apart from the approach, the book remains in quite well-known territory.” (Andreas Cap, Mathematical Reviews, April 2017)

Authors and Affiliations

  • Burnham Market, United Kingdom

    Mike Crampin

  • University of Ostrava, Ostrava, Czech Republic

    David Saunders

Bibliographic Information

  • Book Title: Cartan Geometries and their Symmetries

  • Book Subtitle: A Lie Algebroid Approach

  • Authors: Mike Crampin, David Saunders

  • Series Title: Atlantis Studies in Variational Geometry

  • DOI: https://doi.org/10.2991/978-94-6239-192-5

  • Publisher: Atlantis Press Paris

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Atlantis Press and the author(s) 2016

  • Hardcover ISBN: 978-94-6239-191-8Published: 30 May 2016

  • eBook ISBN: 978-94-6239-192-5Published: 20 May 2016

  • Series ISSN: 2214-0700

  • Series E-ISSN: 2214-0719

  • Edition Number: 1

  • Number of Pages: XIV, 290

  • Topics: Differential Geometry

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