Contents
Preface vii
Part 1. Background 1
Chapter 1. Cartan geometries 3
1.1. Prologue a few examples of homogeneous spaces 4
1.2. Some background from differential geometry 15
1.3. A survey on connections 35
1.4. Geometry of homogeneous spaces 49
1.5. Cartan connections 70
1.6. Conformal Riemannian structures 112
Chapter 2. Semisimple Lie algebras and Lie groups 141
2.1. Basic structure theory of Lie algebras 141
2.2. Complex semisimple Lie algebras and their representations 160
2.3. Real semisimple Lie algebras and their representations 199
Part 2. General theory 231
Chapter 3. Parabolic geometries 233
3.1. Underlying structures and normalization 234
3.2. Structure theory and classification 290
3.3. Kostant’s version of the Bott–Borel–Weil theorem 339
Historical remarks and references for Chapter 3 360
Chapter 4. A panorama of examples 363
4.1. Structures corresponding to |1|–gradings 363
4.2. Parabolic contact structures 402
4.3. Examples of general parabolic geometries 426
4.4. Correspondence spaces and twistor spaces 455
4.5. Analogs of the Fefferman construction 478
Chapter 5. Distinguished connections and curves 497
5.1. Weyl structures and scales 498
5.2. Characterization of Weyl structures 517
5.3. Canonical curves 558
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