Cartan geometries and their symmetries. A Lie algebroid approach

✍ Scribed by Cartan, Élie; Crampin, M.; Saunders, David

Publisher: Atlantis Press
Year: 2016
Tongue: English
Leaves: 298
Series: Atlantis Studies in Variational Geometry 4; Atlantis Studies in Variational Geometry
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xiv
Lie Groupoids and Lie Algebroids....Pages 1-25
Connections on Lie Groupoids and Lie Algebroids....Pages 27-54
Groupoids of Fibre Morphisms....Pages 55-75
Four Case Studies....Pages 77-103
Symmetries....Pages 105-126
Cartan Geometries....Pages 127-151
A Comparison with Alternative Approaches....Pages 153-176
Infinitesimal Cartan Geometries on (\textit{TM}) ....Pages 177-195
Projective Geometry: The Full Version....Pages 197-226
Conformal Geometry: The Full Version....Pages 227-238
Developments and Geodesics....Pages 239-255
Cartan Theory of Second-Order Differential Equations....Pages 257-284
Back Matter....Pages 285-290

✦ Subjects

Cartan, Élie, -- (1869-1951);Symétrie;Géométrie

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