對齊性空間的研究使我們對微分幾何和李群有了
更深的了解。例如，在幾何中一般性的定理和性質對
於齊性空間也成立，並且在這個架構上通常更容易理
解和證明。對於李群，相當多的分析或者開始於或者
歸結到齊性空間(通常是對稱空間)上。多年來，對很
多數學家來說，這本經典著作已經是、也會繼續是這
方面資料的標準來源。
《微分幾何李群和對稱空間(英文版)(精)》作者
西於聚爾·黑爾加松首先對微分幾何做了一個簡潔、
自足的介紹，然後細心處理了李群的理論基礎，其陳
述方式自1962年以來成為許多後續作者所采用的標準
方式。這為引進和研究對稱空間創造了條件，而這正
是本書的核心部分。本書的結尾則按照Victor Kac的
方法，通過e上單李代數的Killing—Cartan分類和R
上單李代數的Cartan分類，對對稱空間進行了分類。
本書每章後面都配有豐富且實用的習題，且書後
附有全部問題的解答或提示。在這一版中，作者做了
一些修正，並添加了一些有益的注記和有用的參考文
獻。
Sigurdur Helgason因本書和Groups and
Geometric Analysis而獲Steele獎。

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
1.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 III Structure of Semisimple Lie Algebras
1. Preliminaries
2. Theorems of Lie and Engel
3. Cartan Subalgebras
4. Root Space Decomposition
5. Significance of the Root Pattern
6. Real Forms
7. Cartan Decompositions
8. Examples. The Complex Classical Lie Algebras
Exercises and Further Results
Notes
CHAPTER IV Symmetric Spaces
1. Affine Locally Symmetric Spaces
2. Groups of Isometries
3. Riemannian Globally Symmetric Spaces
4. The Exponential Mapping and the Curvature
5. Locally and Globally Symmetric Spaces
6. Compact Lie Groups
7. Totally Geodesic Submanifolds. Lie Triple Systems
Exercises and Further Results
Notes
CHAPTER V Decomposition of Symmetric Spaces
1. Orthogonal Symmetric Lie Algebras
2. The Duality
3. Sectional Curvature of Symmetric Spaces
4. Symmetric Spaces with Semisimple Groups of Isometries
5. Notational Conventions
6. Rank of Symmetric Spaces
Exercises and Further Results
Notes
CHAPTER VI Symmetric Spaces of the Noncompact Type
1. Decomposition of a Semisimple Lie Group
2. Maximal Compact Subgroups and Their Conjugacy
3. The Iwasawa Decomposition
4. Nilpotent Lie Groups
5. Global Decompositions
6. The Complex Case
Exercises and Further Results
Notes
CHAPTER VII Symmetric Spaces of the Compact Type
1. The Contrast between the Compact Type and the Noncompact Type
2. The Weyl Group and the Restricted Roots
3. Conjugate Points. Singular Points. The Diagram
4. Applications to Compact Groups
5. Control over the Singular Set
6. The Fundamental Group and the Center
7. The Aiffne Weyl Group
8. Application to the Symmetric Space U/K
9. Classification of Locally Isometric Spaces
10. Geometry of U/K. Symmetric Spaces of Rank One
11. Shortest Geodesics and Minimal Totally Geodesic Spheres
12. Appendix. Results from Dimension Theory
Exercises and Further Results
Notes
CHAPTER VIII Hermitian Symmetric Spaces
1. Almost Complex Manifolds
2. Complex Tensor Fields. The Ricci Curvature
3. Bounded Domains. The Kernel Function
4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
5. Irreducible Orthogonal Symmetric Lie Algebras
6. Irreducible Hermitian Symmetric Spaces
7. Bounded Symmetric Domains
Exercises and Further Results
Notes
CHAPTER IX Structure of Semisimple Lie Groups
1. Caftan, Iwasawa, and Bruhat Decompositions
2. The Rank-One Reduction
3. The SU(2, 1) Reduction
4. Cartan Subalgebras
5. Automorphisms
6. The Multiplicities
7. Jordan Decompositions
Exercises and Further Result
Notes
CHAPTER X The Classification of Simple Lie Algebras and of Symmetric Spaces
1. Reduction of the Problem
2. The Classical Groups and Their Cartan Involutions
1.Some Matrix Groups and Their Lie Algebras
2.Connectivity Properties
3.The Involutive /lutomorphisms of the Classical Compact Lie Al&ebras
3. Root Systems
1.Generalities
2.Reduced Root Systems
3.Classification of Reduced Root Systems. Coxeter Graphs and Dynln'n
Diagrams
4.The Nonreduced Root Systems
5.The Highest Root
6.Outer Automorphirms and the Covering Index
4. The Classification of Simple Lie Algebras over C
5. Automorphisms of Finite Order of Semisimple Lie Algebras
6. The Classifications
1.The Simple Lie Algebras ever C and Their Compact Real Forms. The
Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV
2.The Real Forms of Simple Lie Algebras ~oer C. Irreducible Riemannian
Globally Symmetric Spaces of Type I and Type IF\"
3.Irreducible Hermitian Symmetric Spaces
4.Coincidences between Different Classes. Special lsomorphisms
Exercises and Further Results
Notes
SOLUTIONS TO EXERCISES
SOME DETAILS
SUPPLEMENTARY NOTES
ERRATA
BIBLIOGRAPHY
LIST OF NOTATIONAL CONVENTIONS
SYMBOLS FREQUENTLY USED
INDEX