Cartan Geometries and their Symmetries

A Lie Algebroid Approach

  • Mike Crampin
  • David Saunders
Book

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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Mike Crampin, David Saunders
    Pages 1-25
  3. Mike Crampin, David Saunders
    Pages 27-54
  4. Mike Crampin, David Saunders
    Pages 55-75
  5. Mike Crampin, David Saunders
    Pages 77-103
  6. Mike Crampin, David Saunders
    Pages 105-126
  7. Mike Crampin, David Saunders
    Pages 127-151
  8. Mike Crampin, David Saunders
    Pages 153-176
  9. Mike Crampin, David Saunders
    Pages 177-195
  10. Mike Crampin, David Saunders
    Pages 197-226
  11. Mike Crampin, David Saunders
    Pages 227-238
  12. Mike Crampin, David Saunders
    Pages 239-255
  13. Mike Crampin, David Saunders
    Pages 257-284
  14. Back Matter
    Pages 285-290

About this book

Introduction

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.

Keywords

Cartan Geometry Lie Groupoid Lie Algebroid Infinitesimal Symmetry Differential Geometry

Authors and affiliations

  • Mike Crampin
    • 1
  • David Saunders
    • 2
  1. 1.Burnham MarketUnited Kingdom
  2. 2.University of OstravaOstravaCzech Republic

Bibliographic information

  • DOI https://doi.org/10.2991/978-94-6239-192-5
  • Copyright Information Atlantis Press and the author(s) 2016
  • Publisher Name Atlantis Press, Paris
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-94-6239-191-8
  • Online ISBN 978-94-6239-192-5
  • Series Print ISSN 2214-0700
  • Series Online ISSN 2214-0719
  • About this book