Cartan Geometries and their Symmetries

A Lie Algebroid Approach
2016
ISBN: 978-94-6239-191-8
Verlag: Atlantis Press

Buch, Englisch, Band 4, 290 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 5797 g

Reihe: Atlantis Studies in Variational Geometry

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.

Zielgruppe

Research

Autoren/Hrsg.

Saunders, David

Crampin, Mike

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

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