Differential Geometry, Lie Groups, and Symmetric Spaces

By: Sigurdur Helgason

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Hardcover | 1 January 2001

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The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason'sclassic Differential Geometry, Lie Groups, and Symmetric Spaces has been-and continues to be-the standard source for this material.Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebrasover $\mathbf{R}$.The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39.

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Groups & Group Theory Calculus & Mathematical Analysis Complex Analysis Topology Algebraic Topology

Preface	p. xiii
Preface to the 2001 Printing	p. xvii
Suggestions to the Reader	p. xix
Sequel to the Present Volume	p. xxi
Groups and Geometric Analysis Contents	p. xxiii
Geometric Analysis on Symmetric Spaces Contents	p. xxv
Elementary Differential Geometry
Manifolds	p. 2
Tensor Fields	p. 8
Vector Fields and 1-Forms	p. 8
Tensor Algebra	p. 13
The Grassman Algebra	p. 17
Exterior Differentiation	p. 19
Mappings	p. 22
The Interpretation of the Jacobian	p. 22
Transformation of Vector Fields	p. 24
Effect on Differential Forms	p. 25
Affine Connections	p. 26
Parallelism	p. 28
The Exponential Mapping	p. 32
Covariant Differentiation	p. 40
The Structural Equations	p. 43
The Riemannian Connection	p. 47
Complete Riemannian Manifolds	p. 55
Isometries	p. 60
Sectional Curvature	p. 64
Riemannian Manifolds of Negative Curvature	p. 70
Totally Geodesic Submanifolds	p. 78
Appendix	p. 82
Topology	p. 82
Mappings of Constant Rank	p. 86
Exercises and Further Results	p. 88
Notes	p. 95
Lie Groups and Lie Algebras
The Exponential Mapping	p. 98
The Lie Algebra of a Lie Group	p. 98
The Universal Enveloping Algebra	p. 100
Left Invariant Affine Connections	p. 102
Taylor's Formula and the Differential of the Exponential Mapping	p. 104
Lie Subgroups and Subalgebras	p. 112
Lie Transformation Groups	p. 120
Coset Spaces and Homogeneous Spaces	p. 123
The Adjoint Group	p. 126
Semisimple Lie Groups	p. 131
Invariant Differential Forms	p. 135
Perspectives	p. 144
Exercises and Further Results	p. 147
Notes	p. 153
Structure of Semisimple Lie Algebras
Preliminaries	p. 155
Theorems of Lie and Engel	p. 158
Cartan Subalgebras	p. 162
Root Space Decomposition	p. 165
Significance of the Root Pattern	p. 171
Real Forms	p. 178
Cartan Decompositions	p. 182
Examples. The Complex Classical Lie Algebras	p. 186
Exercises and Further Results	p. 191
Notes	p. 196
Symmetric Spaces
Affine Locally Symmetric Spaces	p. 198
Groups of Isometries	p. 201
Riemannian Globally Symmetric Spaces	p. 205
The Exponential Mapping and the Curvature	p. 214
Locally and Globally Symmetric Spaces	p. 218
Compact Lie Groups	p. 223
Totally Geodesic Submanifolds. Lie Triple Systems	p. 224
Exercises and Further Results	p. 226
Notes	p. 227
Decomposition of Symmetric Spaces
Orthogonal Symmetric Lie Algebras	p. 229
The Duality	p. 235
Sectional Curvature of Symmetric Spaces	p. 241
Symmetric Spaces with Semisimple Groups of Isometries	p. 243
Notational Conventions	p. 244
Rank of Symmetric Spaces	p. 245
Exercises and Further Results	p. 249
Notes	p. 251
Symmetric Spaces of the Noncompact Type
Decomposition of a Semisimple Lie Group	p. 252
Maximal Compact Subgroups and Their Conjugacy	p. 256
The Iwasawa Decomposition	p. 257
Nilpotent Lie Groups	p. 264
Global Decompositions	p. 270
The Complex Case	p. 273
Exercises and Further Results	p. 275
Notes	p. 279
Symmetric Spaces of the Compact Type
The Contrast between the Compact Type and the Noncompact Type	p. 281
The Weyl Group and the Restricted Roots	p. 283
Conjugate Points. Singular Points. The Diagram	p. 293
Applications to Compact Groups	p. 297
Control over the Singular Set	p. 303
The Fundamental Group and the Center	p. 307
The Affine Weyl Group	p. 314
Application to the Symmetric Space U/K	p. 318
Classification of Locally Isometric Spaces	p. 325
Geometry of U/K. Symmetric Spaces of Rank One	p. 327
Shortest Geodesics and Minimal Totally Geodesic Spheres	p. 334
Appendix. Results from Dimension Theory	p. 344
Exercises and Further Results	p. 347
Notes	p. 350
Hermitian Symmetric Spaces
Almost Complex Manifolds	p. 352
Complex Tensor Fields. The Ricci Curvature	p. 356
Bounded Domains. The Kernel Function	p. 364
Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type	p. 372
Irreducible Orthogonal Symmetric Lie Algebras	p. 377
Irreducible Hermitian Symmetric Spaces	p. 381
Bounded Symmetric Domains	p. 382
Exercises and Further Results	p. 396
Notes	p. 400
Structure of Semisimple Lie Groups
Cartan, Iwasawa, and Bruhat Decompositions	p. 401
The Rank-One Reduction	p. 407
The SU(2, 1) Reduction	p. 409
Cartan Subalgebras	p. 418
Automorphisms	p. 421
The Multiplicities	p. 428
Jordan Decompositions	p. 430
Exercises and Further Results	p. 434
Notes	p. 436
The Classification of Simple Lie Algebras and of Symmetric Spaces
Reduction of the Problem	p. 438
The Classical Groups and Their Cartan Involutions	p. 444
Some Matrix Groups and Their Lie Algebras	p. 444
Connectivity Properties	p. 447
The Involutive Automorphisms of the Classical Compact Lie Algebras	p. 451
Root Systems	p. 455
Generalities	p. 455
Reduced Root Systems	p. 459
Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams	p. 461
The Nonreduced Root Systems	p. 474
The Highest Root	p. 475
Outer Automorphisms and the Covering Index	p. 478
The Classification of Simple Lie Algebras over C	p. 481
Automorphisms of Finite Order of Semisimple Lie Algebras	p. 490
The Classifications	p. 515
The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV	p. 515
The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV	p. 517
Irreducible Hermitian Symmetric Spaces	p. 518
Coincidences between Different Classes. Special Isomorphisms	p. 518
Exercises and Further Results	p. 520
Notes	p. 535
Solutions to Exercises	p. 538
Some Details	p. 586
Bibliography	p. 599
List of Notational Conventions	p. 629
Symbols Frequently Used	p. 632
Index	p. 635
Reviews for the First Edition	p. 641
Table of Contents provided by Syndetics. All Rights Reserved.

Differential Geometry, Lie Groups, and Symmetric Spaces

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