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El. knyga: Cartan Geometries and their Symmetries: A Lie Algebroid Approach

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Cartan Geometries and their Symmetries: A Lie Algebroid ApproachDidesnis paveikslėlis
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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.

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.

Recenzijos

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)

1 Lie Groupoids and Lie Algebroids
1(26)
1.1 Groupoids and Lie Groupoids
1(6)
1.2 Lie Algebroids
7(3)
1.3 The Lie Algebroid of a Lie Groupoid
10(4)
1.4 Projections on Lie Groupoids and Lie Algebroids
14(4)
1.5 Pullbacks of Lie Groupoids and Lie Algebroids
18(3)
1.6 Lie Derivatives on Lie Algebroids
21(6)
2 Connections on Lie Groupoids and Lie Algebroids
27(28)
2.1 Path Connections on Lie Groupoids
27(8)
2.2 Infinitesimal Connections on Lie Algebroids
35(6)
2.3 Connections and Projectability
41(4)
2.4 The Kernel Derivative of an Infinitesimal Connection
45(3)
2.5 Covariant Algebroid Derivatives
48(1)
2.6 Representations and Semidirect Sums
49(3)
2.7 Linearizing a Connection
52(3)
3 Groupoids of Fibre Morphisms
55(22)
3.1 Lie Groupoids and Fibre Bundles
55(4)
3.2 Lie Algebroids and Fibre Bundles
59(6)
3.3 Projectability of Bundle Structures
65(6)
3.4 Connections on Fibre Morphism Groupoids and Algebroids
71(3)
3.5 The Comparison with Principal Bundles
74(3)
4 Four Case Studies
77(28)
4.1 Vector Bundles and Linear Maps
77(2)
4.2 Affine Bundles and Affine Maps
79(6)
4.3 Interlude: Densities and Projective Structures
85(6)
4.4 Projective Bundles and Projective Maps
91(6)
4.5 Vector Bundles, Fibre Metrics and Euclidean Maps
97(8)
5 Symmetries
105(22)
5.1 Bisections and Automorphisms
105(5)
5.2 Symmetries of Connections
110(2)
5.3 Infinitesimal Symmetries of Infinitesimal Connections
112(4)
5.4 Properties of Infinitesimal Symmetries
116(3)
5.5 Case Study: Affine Transformations
119(8)
6 Cartan Geometries
127(26)
6.1 Cartan Geometries as Fibre Morphism Groupoids
127(2)
6.2 The Lie Algebroid of a Cartan Geometry
129(3)
6.3 Lie Algebra Actions
132(4)
6.4 Infinitesimal Cartan Geometries
136(3)
6.5 Nondegeneracy and Soldering
139(3)
6.6 Curvature and Torsion of an Infinitesimal Cartan Geometry
142(2)
6.7 Symmetries of Cartan Geometries
144(3)
6.8 Projectability of Cartan Geometries and Their Symmetries
147(6)
7 A Comparison with Alternative Approaches
153(24)
7.1 Cartan Gauge
153(3)
7.2 The Bundle Definition
156(3)
7.3 The Tractor Connection
159(2)
7.4 The Blaom Connection
161(4)
7.5 The Fundamental Self-Representation
165(2)
7.6 Reconstructing a Cartan Geometry from an Adjoint Tractor Bundle
167(4)
7.7 Locally Symmetric Geometries and Torsion
171(4)
7.8 Comparisons Compared
175(2)
8 Infinitesimal Cartan Geometries on TM
177(20)
8.1 Affine Geometry
178(3)
8.2 Projective Geometry
181(2)
8.3 Riemannian Geometry
183(2)
8.4 Conformal Geometry
185(4)
8.5 The General Theory
189(8)
9 Projective Geometry: The Full Version
197(30)
9.1 Cartan Projective Geometry
197(4)
9.2 The Two Versions Compared: Projective Equivalence
201(5)
9.3 Linear Connections and Radius Vector Fields
206(3)
9.4 Thomas-Whitehead Connections
209(6)
9.5 Infinitesimal Symmetries of TW-Connections and Infinitesimal Cartan Projective Geometries
215(3)
9.6 Infinitesimal Affine Transformations of a TW-Connection
218(3)
9.7 The Projective Tractor Connections
221(6)
10 Conformal Geometry: The Full Version
227(12)
10.1 The Mobius Group
227(4)
10.2 The Generalized F-space
231(2)
10.3 The Cartan Connection
233(3)
10.4 The Conformal Tractor Connection
236(3)
11 Developments and Geodesies
239(18)
11.1 Developments
239(2)
11.2 Vertical Infinitesimal Symmetries
241(2)
11.3 Geodesies in Cartan Affine and Projective Geometry
243(5)
11.4 Generalized Geodesies
248(2)
11.5 Geodesic Sprays and Their Generalizations
250(7)
12 Cartan Theory of Second-Order Differential Equations
257(28)
12.1 Affine Geometry over T°M
257(2)
12.2 Cartan Affine Geometry over T°M
259(2)
12.3 The Cartan--Berwald Geometry of a Spray
261(7)
12.4 The Generalized Geodesic Spray of a Spray
268(1)
12.5 Symmetries of a Cartan--Berwald Geometry
269(3)
12.6 Projective Equivalence of Sprays
272(2)
12.7 The Pullback Projective Bundle
274(2)
12.8 Connection and Curvature
276(3)
12.9 BTW-Connections
279(6)
Bibliography 285(2)
Index 287
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