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Tables of Contents for Differential Geometry and Symmetric Spaces
Chapter/Section Title
Page #
Page Count
Preface to the New Printing
ix
 
Preface
xi
 
Suggestions to the Reader
xiii
 
Elementary Differential Geometry
Manifolds
2
6
Tensor Fields
8
14
Vector Fields and 1-Forms
8
5
The Tensor Algebra
13
4
The Grassmann Algebra
17
2
Exterior Differentiation
19
3
Mappings
22
4
The Interpretation of the Jacobian
22
2
Transformation of Vector Fields
24
1
Effect on Differential Forms
25
1
Affine Connections
26
2
Parallelism
28
4
The Exponential Mapping
32
8
Covariant Differentiation
40
3
The Structural Equations
43
4
The Riemannian Connection
47
8
Complete Riemannian Manifolds
55
5
Isometries
60
4
Sectional Curvature
64
6
Riemannian Manifolds of Negative Curvature
70
8
Totally Geodesic Submanifolds
78
10
Exercises
82
3
Notes
85
3
Lie Groups and Lie Algebras
The Exponential Mapping
88
14
The Lie Algebra of a Lie Group
88
2
The Universal Enveloping Algebra
90
2
Left Invariant Affine Connections
92
2
Taylor's Formula and Applications
94
8
Lie Subgroups and Subalgebras
102
8
Lie Transformation Groups
110
3
Coset Spaces and Homogeneous Spaces
113
3
The Adjoint Group
116
5
Semisimple Lie Groups
121
9
Exercises
125
3
Notes
128
2
Structure of Semisimple Lie Algebras
Preliminaries
130
3
Theorems of Lie and Engel
133
4
Cartan Subalgebras
137
3
Root Space Decomposition
140
6
Significance of the Root Pattern
146
6
Real Forms
152
4
Cartan Decompositions
156
7
Exercises
160
1
Notes
161
2
Symmetric Spaces
Affine Locally Symmetric Spaces
163
3
Groups of Isometries
166
4
Riemannian Globally Symmetric Spaces
170
9
The Exponential Mapping and the Curvature
179
4
Locally and Globally Symmetric Spaces
183
5
Compact Lie Groups
188
1
Totally Geodesic Submanifolds. Lie Triple Systems
189
4
Exercises
191
1
Notes
191
2
Decomposition of Symmetric Spaces
Orthogonal Symmetric Lie Algebras
193
6
The Duality
199
6
Sectional Curvature of Symmetric Spaces
205
2
Symmtric Spaces with Semisimple Groups of Isometries
207
1
Notational Conventions
208
1
Rank of Symmetric Spaces
209
5
Exercises
213
1
Notes
213
1
Symmetric Spaces of the Noncompact Type
Decomposition of a Semisimple Lie Group
214
4
Maximal Compact Subgroups and Their Conjugacy
218
1
The Iwasawa Decomposition
219
6
Nilpotent Lie Groups
225
9
Global Decompositions
234
3
The Complex Case
237
4
Exercises
239
1
Notes
240
1
Symmetric Spaces of the Compact Type
The Contrast between the Compact Type and the Noncompact Type
241
2
The Weyl Group
243
7
Conjugate Points. Singular Points. The Diagram
250
4
Applications to Compact Groups
254
6
Control over the Singular Set
260
4
The Fundamental Group and the Center
264
7
Application to the Symmetric Space U/K
271
2
Classification of Locally Isometric Spaces
273
2
Appendix. Results from Dimension Theory
275
6
Exercises
278
2
Notes
280
1
Hermitian Symmetric Spaces
Almost Complex Manifolds
281
4
Complex Tensor Fields. The Ricci Curvature
285
8
Bounded Domains. The Kernel Function
293
8
Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
301
5
Irreducible Orthogonal Symmetric Lie Algebras
306
4
Irreducible Hermitian Symmetric Spaces
310
1
Bounded Symmetric Domains
311
15
Exercises
322
3
Notes
325
1
On the Classification of Symmetric Spaces
Reduction of the Problem
326
5
Automorphisms
331
3
Involutive Automorphisms
334
5
E.Cartan's List of Irreducible Riemannian Globally Symmetric Spaces
339
16
Some Matrix Groups and Their Lie Algebras
339
7
The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV
346
1
The Involutive Automorphisms of Simple Compact Lie Algebras. The Irreducible Globally Symmetric Spaces of Type I and Type III
347
7
Irreducible Hermitian Symmetric Spaces
354
1
Two-Point Homogeneous Spaces. Symmetric Spaces of Rank One. Closed Geodesics
355
6
Exercises
358
1
Notes
359
2
Functions on Symmetric Spaces
Integral Formulas
361
24
Generalities
361
6
Invariant Measures on Coset Spaces
367
5
Some Integral Formulas for Semisimple Lie Groups
372
7
Integral Formulas for the Cartan Decomposition
379
3
The Compact Case
382
3
Invariant Differential Operators
385
13
Generalities. The Laplace-Beltrami Operator
385
4
Invariant Differential Operators on Reductive Coset Spaces
389
7
The Case of a Symmetric Space
396
2
Spherical Functions. Definition and Examples
398
10
Elementary Properties of Spherical Functions
408
10
Some Algebraic Tools
418
4
The Formula for the Spherical Function
422
13
The Euclidean Type
422
1
The Compact Type
423
4
The Noncompact Type
427
8
Mean Value Theorems
435
22
The Mean Value Operators
435
5
Approximations by Analytic Functions
440
2
The Darboux Equation in a Symmetric Space
442
2
Poisson's Equation in a Two-Point Homogeneous Space
444
5
Exercises
449
5
Notes
454
3
Bibliography
457
16
List of Notational Conventions
473
3
Symbols Frequently Used
476
3
Author Index
479
3
Subject Index
482
5
Reviews For The First Edition
487