VI
Contents
6. Complex semisimple Lie algebras 41
Chapter 3. The Geometry of a Compact Lie Group 51
1. Riemannian manifolds: A review 51
2. Left-invariant and bi-invariant metrics 59
3. Geometrical aspects of a compact Lie group 61
Chapter 4. Homogeneous Spaces 65
1. Coset manifolds 65
2. Reductive homogeneous spaces 71
3. The isotropy representation 72
Chapter 5. The Geometry of a Reductive Homogeneous Space 77
1. G-invariant metrics 77
2. The Riemannian connection 79
3. Curvature 80
Chapter 6. Symmetric Spaces 87
1. Introduction 87
2. The structure of a symmetric space 88
3. The geometry of a symmetric space 91
4. Duality 92
Chapter 7. Generalized Flag Manifolds 95
1. Introduction 95
2. Generalized flag manifolds as adjoint orbits 96
3. Lie theoretic description of a generalized flag mani-
fold 98
4. Painted Dynkin diagrams 98
5. T-roots and the isotropy representation 100
6. G-invariant Riemannian metrics 103
7. G-invariant complex structures and Kahler metrics 105
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