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Cartan Geometries and their Symmetries: A Lie Algebroid Approach: Atlantis Studies in Variational Geometry, cartea 4

Autor Mike Crampin, David Saunders
en Limba Engleză Hardback – 30 mai 2016
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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Specificații

ISBN-13: 9789462391918
ISBN-10: 9462391912
Pagini: 233
Ilustrații: XIV, 290 p.
Dimensiuni: 155 x 235 x 19 mm
Greutate: 0.6 kg
Ediția:1st ed. 2016
Editura: ATLANTIS PRESS
Colecția Atlantis Press
Seria Atlantis Studies in Variational Geometry

Locul publicării:Paris, Netherlands

Cuprins

Lie groupoids and Lie algebroids.- Connections on Lie groupoids and Lie algebroids.-  Groupoids of fibre morphisms.- Four case studies.- Symmetries.- Cartan geometries.- A comparison with alternative approaches.- Infinitesimal Cartan geometries on TM.- Projective geometry: the full version.- Conformal geometry: the full version.- Developments and geodesics.- Cartan theory of second-order differential equations.    

Recenzii

“The authors expose an alternative approach to Cartan geometries. … the book offers a nice exposition of the approach to Cartan geometries via Lie algebroids, one also has to say that, apart from the approach, the book remains in quite well-known territory.” (Andreas Cap, Mathematical Reviews, April 2017)

Textul de pe ultima copertă

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.

Caracteristici

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 Includes supplementary material: sn.pub/extras