Cartan Geometries and their Symmetries: A Lie Algebroid Approach (Atlantis Studies in Variational Geometry Book 4)

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 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. Topics Differential Geometry 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