Contents 6
Preface 10
Introduction 12
Chapter 1. Lie Groups 18
1. An example of a Lie group 18
2. Smooth manifolds: A review 19
3. Lie groups 25
4. The tangent space of a Lie group - Lie algebras 29
5. One-parameter subgroups 32
6. The Campbell-Baker-Hausdorff formula 37
7. Lie's theorems 38
Chapter 2. Maximal Tori and the Classification Theorem 40
1. Representation theory: elementary concepts 41
2. The adjoint representation 45
3. The Killing form 49
4. Maximal tori 53
5. The classification of compact and connected Lie groups 56
6. Complex semisimple Lie algebras 58
Chapter 3. The Geometry of a Compact Lie Group 68
1. Riemannian manifolds: A review 68
2. Left-invariant and bi-invariant metrics 76
3. Geometrical aspects of a compact Lie group 78
Chapter 4. Homogeneous Spaces 82
1. Coset manifolds 82
2. Reductive homogeneous spaces 88
3. The isotropy representation 89
Chapter 5. The Geometry of a Reductive Homogeneous Space 94
1. G-invariant metrics 94
2. The Riemannian connection 96
3. Curvature 97
Chapter 6. Symmetric Spaces 104
1. Introduction 104
2. The structure of a symmetric space 105
3. The geometry of a symmetric space 108
4. Duality 109
Chapter 7. Generalized Flag Manifolds 112
1. Introduction 112
2. Generalized flag manifolds as adjoint orbits 113
3. Lie theoretic description of a generalized flag manifold 115
4. Painted Dynkin diagrams 115
5. T-roots and the isotropy representation 117
6. G-invariant Riemannian metrics 120
7. G-invariant complex structures and Kahler metrics 122
8. G-invariant Kahler-Einstein metrics 125
9. Generalized flag manifolds as complex manifolds 128
Chapter 8. Advanced topics 130
1. Einstein metrics on homogeneous spaces 130
2. Homogeneous spaces in symplectic geometry 135
3. Homogeneous geodesies in homogeneous spaces 140
Bibliography 146
Index 156
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