对齐性空间的研究使我们对微分几何和李群有了更深的了解。例如，在几何方面，一般性的定理和性质对于齐性空间也都成立，并且在这个架构上通常更容易理解和证明。在李群方面，相当多的分析或者开始于或者归结到齐性空间（通常是对称空间）上。多年来，对很多数学家来说，这本经典著作已经是、也会继续是这方面资料的标准来源。 作者从对微分几何的一个简洁、自足的介绍开始，然后是对李群理论基础的细心处理，其陈述方式自1962年以来成为许多后续作者所采用的标准方式。这为引进和研究对称空间创造了条件，这是本书的核心部分。本书的结尾则利用C上单纯李代数的 Killing-Cartan 分类和 R上单纯李代数的 Cartan分类，并按照 Victor Kac 的方法对对称空间进行了分类。 每章后都配有内容广泛的非常有用的习题，且书后附有全部问题的解答或提示。本书中，作者做了一些修正，并添加了一些有益的注记和有用的参考文献。
CONTENTS
PREFACE
PREFACE To THE 2001 PRINTING
SUGGESTIONS To THE READER
SEQUEL To THE PRESENT VOLUME
GROUPS AND GEOMETRIC ANAI VSIS CONTENTS
GEOMETRIC ANALYSIS ON SYMMETRIC SPACES CONTENTs
CHAPTER I
Elementary Differential Geometry
1. Manifolds
2. Tensor Fields
1.Vector Fields and 1- Forms
2.Tensor Algebra
3.The Grassman Algebra
4.Exterior Differentiation
3. Mappings
l.The Interpretation of the Jacobian
2.Transformation of Vector Fields
3.Effect on Differential Forms
4. Afine Connections
5. Parallelism
6. The Exponential Mapping
7. Covariant Diferentiation
8. The Structural Equations
9. The Riemannian Connection
10. Complete Riemannian Manifolds
11. Isometries
12. Sectional Curvature
13. Riemannian Manifolds of Negative Curvature
14. Totally Geodesic Submanifolds
15. Appendix
1.Topology
2.Mappings of Constant RankExercises and Further ResultsNotes
CHAPTER II
Lie Groups and Lie Algebras
1. The Exponential Mapping
1.The Lie Algebra of a Lie Group
2.The Universal Enceloping Algebra
3.Left Inuariant Affine Commectins
4.Taylor's Formula and the Differential of the Expomential Mapping J
2. Lie Subgroups and Subalgebras
3. Lie Tranfomation Groups
4. Coset Spaces and Homogeneous Spaces
5. The Adjoint Group
6. Semisimple Lie Groups Forms
7. Invariant Diferential Forms
8. Perspectives
Exercises and Further Results
Notes
......
CHAPTER II
Structure of Semisimple Lie Algebras
CHAPTER lV
Symmetric Spaces
CHAPTER V
Decomposition of Symmetric Spaces
CHAPTER VI
Symmetric Spaces of the Noncompact Type
CHAPTER VII
Symmetric Spaces of the Compact Type
CHAPTER VIII
Hermitian Symmetric Spaces
CHAPTER IX
Stucture of Semisimple Lie Groups
CHAPTER X
The Classification of Simple Lie Algebras and of Symmetr
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