
Differential Geometry, Lie Groups, and Symmetric Spaces
Hardcover | 1 January 2001
At a Glance
641 Pages
New edition
26.67 x 19.05 x 3.81
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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. |
ISBN: 9780821828489
ISBN-10: 0821828487
Series: Graduate Studies in Mathematics
Published: 1st January 2001
Format: Hardcover
Language: English
Number of Pages: 641
Audience: Professional and Scholarly
Publisher: American Mathematical Society
Country of Publication: US
Edition Type: New edition
Dimensions (cm): 26.67 x 19.05 x 3.81
Weight (kg): 1.38
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