Cartan Geometries And Their Symmetries: A Lie Algebroid Approach (atlantis Studies In Variational Geometry)
by David Saunders /
2016 / English / PDF
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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.
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.
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.