Cartan Geometries and their Symmetries: A Lie Algebroid Approach

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Springer, May 20, 2016 - Mathematics - 290 pages
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.
 

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Contents

1 Lie Groupoids and Lie Algebroids
1
2 Connections on Lie Groupoids and Lie Algebroids
26
3 Groupoids of Fibre Morphisms
55
4 Four Case Studies
77
5 Symmetries
104
6 Cartan Geometries
127
7 A Comparison with Alternative Approaches
153
8 Infinitesimal Cartan Geometries on TM
177
The Full Version
196
The Full Version
227
11 Developments and Geodesics
239
12 Cartan Theory of SecondOrder Differential Equations
257
Bibliography
285
Index
287
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