Differential geometry and symmetric spaces:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Acad. Pr.
1972
|
| Edition: | 5. print. |
| Series: | Pure and applied mathematics
12 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 486 S. |
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| 245 | 1 | 0 | |a Differential geometry and symmetric spaces |
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Record in the Search Index
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| adam_text | Titel: Differential geometry and symmetric spaces
Autor: Sigurður Helgason
Jahr: 1972
CONTENTS
Preface..............................vii
Suggestions to the Reader...........ix
CHAPTER I
Elementary Differential Geometry
1. Manifolds...............2
2. Tensor Fields..............8
1. Vector Fields and 1-Forms..........8
2. The Tensor Algebra............13
3. The Grassmann Algebra...........17
4. Exterior Differentiation...........19
3. Mappings...............22
1. The Interpretation of the Jacobian.........22
2. Transformation of Vector Fields.........24
3. Effect on Differential Forms..........25
4. Affine Connections.............26
5. Parallelism...............28
6. The Exponential Mapping...........32
7. Covariant Differentiation............40
8. The Structural Equations............43
9. The Riemannian Connection...........47
10. Complete Riemannian Manifolds..........55
11. Isometries...............60
12. Sectional Curvature.............64
13. Riemannian Manifolds of Negative Curvature.......70
14. Totally Geodesic Submanifolds..........78
Exercises...............82
Notes................85
CHAPTER II
Lie Groups and Lie Algebras
1. The Exponential Mapping...........88
1. The Lie Algebra of a Lie Group.........88
2. The Universal Enveloping Algebra.........90
3. Left Invariant Affine Connections.........92
4. Taylor s Formula and Applications.........94
2. Lie Subgroups and Subalgebras..........102
3. Lie Transformation Groups...........110
4. Coset Spaces and Homogeneous Spaces.........113
XI
CONTENTS
xn
• 116
5. The Adjoint Group............. 121
6. Semisimple Lie Groups • ... 125
Exercises ... 128
Notes ...•••¦-
CHAPTER 111
Structure of Semisimple Lie Algebras
. 130
1. Preliminaries ...... * 1^3
2. Theorems of Lie and Engel , . . .
3. Cartan Subalgebras............
4. Root Space Decomposition .... 146
5. Significance of the Root Pattern . -
6. Real Forms..........• .....
7. Cartan Decompositions.................
t. .... 160
Exercises .
161
Notes
CHAPTER IV
Symmetric Spaces
1. AfTme Locally Symmetric Spaces ... .....163
2. Groups of Isometries ... . . . . • 166
3. Ricmannian Globally Symmetric Spaces .....170
4. The Exponential Mapping and the Curvature .....179
5. Locally and Globally Symmetric Spaces .....183
6. Compact Lie Groups .... ..... 188
7. Totally Geodesic Submanifolds. Lie Triple Systems ...... 189
Exercises................. 191
Notes ........ ..... 191
CHAPTER V
Decomposition of Symmetric Spaces
1. Orthogonal Symmetric Lie Algebras.........193
2. The Duality................. 199
3. Sectional Curvature of Symmetric Spaces .......205
4. Symmetric Spaces with Semisimple Groups of Isometries.....207
5. Notational Conventions .... .....208
6. Rank of Symmetric Spaces...........209
Exercises...............213
Notes...........213
CHAPTER VI
Symmetric Spaces of the Noncompact Type
1. Decomposition of a Semisimple Lie Group .... 214
2. Maximal Compact Subgroups and Their Conjugacy .... 218
3. The Iwasawa Decomposition . 219
4. Nilpotent Lie Groups.................225
CONTENTS xiii
5. Global Decompositions ............234
6. The Complex Case.............237
Exercises...............239
Notes................240
CHAPTER VII
Symmetric Spaces of the Compact Type
1. The Contrast between the Compact Type and the Noncompact Type . . 241
2. The Weyl Group.............243
3. Conjugate Points. Singular Points. The Diagram.......250
4. Applications to Compact Groups..........254
5. Control over the Singular Set...........260
6. The Fundamental Group and the Center........264
7. Application to the Symmetric Space UjK........271
8. Classification of Locally Isometric Spaces........273
9. Appendix. Results from Dimension Theory........275
Exercises...............278
Notes................280
CHAPTER VIII
Hermitian Symmetric Spaces
1. Almost Complex Manifolds...........281
2. Complex Tensor Fields. The Ricci Curvature.......285
3. Bounded Domains. The Kernel Function........293
4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type 301
5. Irreducible Orthogonal Symmetric Lie Algebras.......306
6. Irreducible Hermitian Symmetric Spaces........310
7. Bounded Symmetric Domains...........311
Exercises...............322
Notes................325
CHAPTER IX
On the Classification of Symmetric Spaces
1. Reduction of the Problem...........326
2. Automorphisms..............331
3. Involutive Automorphisms...........334
4. E. Cartan s List of Irreducible Ricmannian Globally Symmetric Spaces . . 339
1. Some Matrix Groups and Their Lie Algebras.......339
2. The Simple Lie Algebras over C and Their Compact Real Forms. The
Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV 346
3. The Involutive Automorphisms of Simple Compact Lie Algebras. The
Irreducible Globally Symmetric Spaces of Type I and Type III . . . 347
4. Irreducible Hermitian Symmetric Spaces.......354
5. Two-Point Homogeneous Spaces. Symmetric Spaces of Rank One. Closed
Geodesies..............355
Exercises...............358
Notes................359
XIV
CONTENTS
CHAPTER X
Functions on Symmetric Spaces
1. Integral Formulas ....•¦••••
1. Generalities ....¦¦¦•¦•
2. Invariant Measures on Coset Spaces.....
3. Some Integral Formulas for Semisimple Lie Groups
4. Integral Formulas for the Cartan Decomposition .
5. The Compact Case .........
2. Invariant Differential Operators.......
1. Generalities. The Laplace-Beltrami Operator .
2. Invariant Differential Operators on Reductive Coset Spaces .
3. The Case of a Symmetric Space......
3. Spherical Functions. Definition and Examples ....
4. Elementary Properties of Spherical Functions .
5. Some Algebraic Tools ........
6. The Formula for the Spherical Function
1. The Euclidean Type .........
2. The Compact Type .........
3. The Noncompact Type •........
7. Mean Value Theorems .........
1. The Mean Value Operators .......
2. Approximations by Analytic Functions.....
3. The Darboux Equation in a Symmetric Space
4. Poisson s Equation in a Two-Point Homogeneous Space .
Exercises........
Notes...........
Bibliography........
List of Notational Conventions
Symbols Frequently Used.....
Author Index......
Subject Index.....
361
361
367
372
379
382
385
385
389
396
398
408
418
422
422
423
427
435
435
440
442
444
449
454
457
473
476
479
482
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| any_adam_object | 1 |
| author | Sigurður Helgason 1927-2023 |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:48:57Z |
| institution | BVB |
| language | English |
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| publisher | Acad. Pr. |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Sigurður Helgason 1927-2023 Verfasser (DE-588)123045762 aut Differential geometry and symmetric spaces 5. print. New York [u.a.] Acad. Pr. 1972 XIV, 486 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 12 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Symmetrischer Raum (DE-588)4184206-6 gnd rswk-swf Symmetrischer Raum (DE-588)4184206-6 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Pure and applied mathematics 12 (DE-604)BV010177228 12 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001753177&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sigurður Helgason 1927-2023 Differential geometry and symmetric spaces Pure and applied mathematics Differentialgeometrie (DE-588)4012248-7 gnd Symmetrischer Raum (DE-588)4184206-6 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4184206-6 |
| title | Differential geometry and symmetric spaces |
| title_auth | Differential geometry and symmetric spaces |
| title_exact_search | Differential geometry and symmetric spaces |
| title_full | Differential geometry and symmetric spaces |
| title_fullStr | Differential geometry and symmetric spaces |
| title_full_unstemmed | Differential geometry and symmetric spaces |
| title_short | Differential geometry and symmetric spaces |
| title_sort | differential geometry and symmetric spaces |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd Symmetrischer Raum (DE-588)4184206-6 gnd |
| topic_facet | Differentialgeometrie Symmetrischer Raum |
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