Cartan Geometries and their Symmetries: A Lie Algebroid Approach
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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2 Connections on Lie Groupoids and Lie Algebroids
3 Groupoids of Fibre Morphisms
4 Four Case Studies
6 Cartan Geometries
7 A Comparison with Alternative Approaches
8 Infinitesimal Cartan Geometries on TM
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action of G algebra bracket attachment section automorphism bisection canonical Cartan projective geometry coefficients condition consider construction Corollary corresponding covariant derivative curvature defined denote diffeomorphism dimension Ehresmann connection element equations fibre bundle fibre coordinates fibre F fibre morphisms follows geodesic given group G groupoid G homogeneous horizontal lift identity infinitesimal Cartan geometry infinitesimal Cartan projective infinitesimal connection infinitesimal symmetry integral curve isomorphism Lemma Lie algebra bracket Lie algebroid Lie derivative Lie group Lie groupoid linear connection linear map locally trivial manifold neighbourhood path connection principal bundle projectable vector field projective geometry Proof Proposition PVVM restriction satisfies smooth soldering spray standard fibre surjective surjective submersion tangent vector tensor field Theorem TēM transitive Lie algebroid trivial Lie groupoid TW-connection vector bundle morphism vertical lift