Verfügbarkeit: Differential geometry and symmetric spaces

Differential geometry and symmetric spaces:

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Bibliographische Detailangaben
Beteilige Person:	Sigurður Helgason 1927-2023 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York [u.a.] Acad. Pr. 1972
Ausgabe:	5. print.
Schriftenreihe:	Pure and applied mathematics 12
Schlagwörter:	Differentialgeometrie Symmetrischer Raum
Links:	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
Umfang:	XIV, 486 S.

Internformat

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Datensatz im Suchindex

DE-BY-TUM_call_number	0102 MAT 530f 2001 A 16718
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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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author	Sigurður Helgason 1927-2023
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series	Pure and applied mathematics
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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
url	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
volume_link	(DE-604)BV010177228
work_keys_str_mv	AT sigurðurhelgason differentialgeometryandsymmetricspaces

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