Cartan geometries and their symmetries. A Lie algebroid approach (Q2803013)

Roughly speaking, the monograph is devoted to an alternative approach to Cartan geometries and their symmetries. The book has 290 pages, is divided into 12 chapters and falls essentially into four parts.NEWLINENEWLINEThe first part (Chapter 1: Lie groupoids and Lie algebroids and Chapter 2: Connections on Lie groupoids and Lie algebroids) reviews the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. In particular, Chapter 2 is devoted to several different notions of connection. As well as introducing connections on Lie groupoids (path connection) and on Lie algebroids (infinitesimal connections), it is clarified how these ideas are related to the classical concepts of covariant derivatives and, more generally, connections on vector bundles.NEWLINENEWLINEThe second part (Chapter 3: Groupoids on fibre morphisms, Chapter 4: Four case studies and Chapter 5: Symmetries) is devoted to the relation to fiber bundles and to symmetries. The authors consider here what might be called ``pre-Cartan geometries'', where they consider in particular the Lie groupoids on fibre morphisms of a given fibre bundle, and the connections on such groupoids together with their symmetries. In this second part, it is also shown how the infinitesimal approach, using Lie algebroids rather than Lie groupoids, and, in particular, using Lie algebroids of projectable vector fields along the fibers of the bundle, may be of benefit.NEWLINENEWLINEIn the third part (Chapter 6: Cartan geometries, Chapter 7: A comparison with alternative approaches, Chapter 8: Infinitesimal Cartan geometries on \(TM\), Chapter 9: Projective geometry: the full version and Chapter 10: Conformal geometry: the full version) Cartan geometries are introduced properly together with the number of tools. The author's version of Cartan geometry is explained in Chapter 6. In particular, Cartan geometry is defined as a special kind of fibre-morphism groupoid with a path connection. As particular examples there are taken the four classical types of geometry: affine, projective, Riemannian and conformal geometry.NEWLINENEWLINEFinally, the last part (Chapter 11: Developments of geodesics and Chapter 12: Cartan theory of second-order differential equations) specializes to the geometries (affine and projective) associated with path spaces and geodesics, their symmetries and other properties.NEWLINENEWLINEThe book can be recommended to specialists in mathematics or physics and also to Ph.D. students in differential geometry.