Verfügbarkeit: Einstein manifolds

Einstein manifolds:

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1. Verfasser:	Besse, Arthur L. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer 2008
Ausgabe:	reprint of the 1987 ed.
Schriftenreihe:	Ergebnisse der Mathematik und ihrer Grenzgebiete Folge 3 ; 10
Schlagworte:	Einstein, Variétés d' Relativité (physique) Relativitätstheorie Einstein-Mannigfaltigkeit
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XII, 516 S. graph. Darst.
ISBN:	9783540741206

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Datensatz im Suchindex

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adam_text	Table of Contents Chapter 0. Introduction ........................................... 1 A. Brief Definitions and Motivation ................................ 1 B. Why Write a Book on Einstein Manifolds? ........................ 5 C. Existence .................................................... 6 D. Examples 1. Algebraic Examples ......................................... 6 2. Examples from Analysis ..................................... 7 3. Sporadic Examples ......................................... 8 E. Uniqueness and Moduli ....................................... 9 F. A Brief Survey of Chapter Contents .............................. 10 G. Leitfaden.................................................... 14 Η. Getting the Feel of Ricci Curvature .............................. 15 I. The Main Problems Today ..................................... 18 Chapter 1. Basic Material ......................................... 20 A. Introduction ................................................. 20 B. Linear Connections ........................................... 22 С Riemannian and Pseudo-Riemannian Manifolds ................... 29 D. Riemannian Manifolds as Metric Spaces .......................... 35 E. Riemannian Immersions, Isometries and Killing Vector Fields ....... 37 F. Einstein Manifolds ............................................ 41 G. Irreducible Decompositions of Algebraic Curvature Tensors ......... 45 H. Applications to Riemannian Geometry ........................... 48 I. Laplacians and Weitzenböck Formulas ........................... 52 J. Conformai Changes of Riemannian Metrics ....................... 58 K. First Variations of Curvature Tensor Fields ....................... 62 Chapter 2. Basic Material (Continued): Kahler Manifolds .............. 66 0. Introduction ................................................. 66 A. Almost Complex and Complex Manifolds ........................ 66 B. Hermitian and Kahler Metrics .................................. 69 С Ricci Tensor and Ricci Form ................................... 73 D. Holomorphic Sectional Curvature ............................... 75 E. Chern Classes ................................................ 78 viii Table of Contents F. The Ricci Form as the Curvature Form of a Line Bundle ............ 81 G. Hodge Theory ................................................ 83 H. Holomorphic Vector Fields and Infinitesimal Isometries ............ 86 I. The Calabi-Futaki Theorem .................................... 92 Chapter 3. Relativity ............................................. 94 A. Introduction ................................................. 94 B. Physical Interpretations ........................................ 94 С The Einstein Field Equation .................................... 96 D. Tidal Stresses ................................................ 97 E. Normal Forms for Curvature ................................... 98 F. The Schwarzschild Metric ...................................... 101 G. Planetary Orbits .............................................. 105 H. Perihelion Precession .......................................... 107 I. Geodesies in the Schwarzschild Universe ......................... 108 J. Bending of Light .............................................. 110 K. The Kruskal Extension ........................................ Ill L. How Completeness May Fail ................................... 113 M. Singularity Theorems .......................................... 115 Chapter 4. Riemannian Functionals ................................ 116 A. Introduction ................................................. 116 B. Basic Properties of Riemannian Functionals ...................... 117 C. The Total Scalar Curvature: First Order Properties ................ 119 D. Existence of Metrics with Constant Scalar Curvature ............... 122 E. The Image of the Scalar Curvature Map .......................... 124 F. The Manifold of Metrics with Constant Scalar Curvature ........... 126 G. Back to the Total Scalar Curvature: Second Order Properties ........ 129 H. Quadratic Functionals ......................................... 133 Chapter 5. Ricci Curvature as a Partial Differential Equation ........... 137 A. Pointwise (Infinitesimal) Solvability .............................. 137 B. From Pointwise to Local Solvability: Obstructions ................. 138 С Local Solvability of Ric(^) = r for Nonsingular r ................... 140 D. Local Construction of Einstein Metrics ........................... 142 E. Regularity of Metrics with Smooth Ricci Tensors .................. 143 F. Analyticity of Einstein Metrics and Applications ................... 145 G. Einstein Metrics on Three-Manifolds ............................ 146 H, A Uniqueness Theorem for Ricci Curvature ....................... 152 I. Global Non-Existence ......................................... 153 Chapter 6. Einstein Manifolds and Topology ......................... 154 A. Introduction ................................................. 154 B. Existence of Einstein Metrics in Dimension 2...................... 155 С The 3-Dimensional Case ....................................... 157 Table of Contents ix D. The ^Dimensional Case ....................................... 161 E. Ricci Curvature and the Fundamental Group ..................... 165 F. Scalar Curvature and the Spinorial Obstruction ................... 169 G. A Proof of the Cheeger-Gromoll Theorem on Complete Manifolds with Non-Negative Ricci Curvature ............................. 171 Chapter 7. Homogeneous Riemannian Manifolds ..................... 177 A. Introduction ................................................. 177 B. Homogeneous Riemannian Manifolds ............................ 178 C. Curvature ................................................... 181 D. Some Examples of Homogeneous Einstein Manifolds ............... 186 E. General Results on Homogeneous Einstein Manifolds .............. 189 F. Symmetric Spaces ............................................. 191 G. Standard Homogeneous Riemannian Manifolds ................... 196 H. Tables ...................................................... 200 I. Remarks on Homogeneous Lorentz Manifolds .................... 205 Chapter 8. Compact Homogeneous Kahler Manifolds ................. 208 0. Introduction ................................................. 208 A. The Orbits of a Compact Lie Group for the Adjoint Representation .. 209 B. The Canonical Complex Structure ............................... 212 C. The G-Invariant Ricci Form .................................... 215 D. The Symplectic Structure of KiriUov-Kostant-Souriau .............. 220 E. The Invariant Kahler Metrics on the Orbits ....................... 221 F. Compact Homogeneous Kahler Manifolds ........................ 224 G. The Space of Orbits ........................................... 227 H. Examples .................................................... 229 Chapter 9. Riemannian Submersions ................................ 235 A. Introduction ...................................·.............. 235 B. Riemannian Submersions ...................................... 236 C. The Invariants A and T ........................................ 238 D. O Neill s Formulas for Curvature ................................ 241 E. Completeness and Connections ................................. 244 F. Riemannian Submersions with Totally Geodesic Fibres ............. 249 G. The Canonical Variation ....................................... 252 H. Applications to Homogeneous Einstein Manifolds ................. 256 1. Further Examples of Homogeneous Einstein Manifolds ............. 263 J. Warped Products ............................................. 265 K. Examples of Non-Homogeneous Compact Einstein Manifolds with Positive Scalar Curvature ..................................___ 272 Chapter 10. Holonomy Groups .................................... 278 A. Introduction ................................................. 278 B. Definitions ................................................... 280 x Table of Contents C. Covariant Derivative Vanishing Versus Holonomy Invariance. Examples .................................................... 282 D. Riemannian Products Versus Holonomy ......................... 285 E. Structure 1 ................................................... 288 F. Holonomy and Curvature ...................................... 290 G. Symmetric Spaces; Their Holonomy ............................. 294 H. Structure II .................................................. 300 I. The Non-Simply Connected Case ............................... 307 J. Lorentzian Manifolds ......................................... 309 K. Tables ...................................................... 311 Chapter 11. Kähler-Einstein Metrics and the Calabi Conjecture ......... 318 A. Kähler-Einstein Metrics ....................................... 318 B. The Resolution of the Calabi Conjecture and its Consequences ....... 322 С A Brief Outline of the Proofs of the Aubin-Calabi-Yau Theorems ..... 326 D. Compact Complex Manifolds with Positive First Chern Class ....... 329 E. Extremal Metrics ............................................. 333 Chapter 12. The Moduli Space of Einstein Structures .................. 340 A. Introduction ................................................. 340 B. Typical Examples: Surfaces and Flat Manifolds .................... 342 С Basic Tools .................................................. 345 D. Infinitesimal Einstein Deformations .............................. 346 E. Formal Integrability .......................................... 348 F. Structure of the Premoduli Spaces ............................... 351 G. The Set of Einstein Constants ................................... 352 H. Rigidity of Einstein Structures .................................. 355 I. Dimension of the Moduli Space ................................. 358 J. Deformations of Kähler-Einstein Metrics ......................... 361 K. The Moduli Space of the Underlying Manifold of КЗ Surfaces ....... 365 Chapter 13. Self-Duality .......................................... 369 A. Introduction ................................................. 369 B. Self-Duality .................................................. 370 С Half-Conformally Flat Manifolds ............................... 372 D. The Penrose Construction ...................................... 379 E. The Reverse Penrose Construction .............................. 385 F. Application to the Construction of Half-Conformally Flat Einstein Manifolds ................................................... 390 Chapter 14. Quaternion-Kähler Manifolds ........................... 396 A. Introduction ................................................. 396 B. Hyperkählerian Manifolds ..................................... 398 C. Examples of Hyperkählerian Manifolds .......................... 400 Table of Contents xi D. Quaternion-Kähler Manifolds ................................. 402 E. Symmetric Quaternion-Kähler Manifolds ......................... 408 F. Quaternionic Manifolds....................................... 410 G. The Twistor Space of a Quaternionic Manifold .................... 412 H. Applications of the Twistor Space Theory ........................ 415 I. Examples of Non-Symmetric Quaternion-Kähler Manifolds ......... 419 Chapter 15. A Report on the Non-Compact Case ..................... 422 A. Introduction ................................................. 422 B. A Construction of Nonhomogeneous Einstein Metrics .............. 423 С Bundle Constructions ......................................... 424 D. Bounded Domains of Holomorphy .............................. 428 Chapter 16. Generalizations of the Einstein Condition ................. 432 A. Introduction ................................................. 432 B. Natural Linear Conditions on Dr ................................ 433 C. Codazzi Tensors .............................................. 436 D. The Case DreC^iQ ® S): Riemannian Manifolds with Harmonic Weyl Tensor ................................................. 440 E. Condition Dr є C^iS): Riemannian Manifolds with Harmonic Curvature ................................................... 443 F. The Case DreC^iQ) .......................................... 447 G. Condition Dr є С^А): Riemannian Manifolds such that (Dxr)(X,X) = 0 for all Tangent Vectors X ........................ 450 H. Oriented Riemannian 4-Manifolds with 6W+ = 0.................. 451 Appendix. Sobolev Spaces and Elliptic Operators ..................... 456 A. Holder Spaces ................................................ 456 B. Sobolev Spaces ............................................... 457 С Embedding Theorems ......................................... 457 D. Differential Operators ......................................... 459 E. Adjoint ..................................................... 460 F. Principal Symbol ............................................. 460 G. Elliptic Operators ............................................. 461 H. Schauder and Lp Estimates for Linear Elliptic Operators ............ 463 I. Existence for Linear Elliptic Equations ........................... 464 J. Regularity of Solutions for Elliptic Equations ..................... 466 K. Existence for Nonlinear Elliptic Equations ........................ 467 Addendum ...................................................... 471 A. Infinitely Many Einstein Constants on S2 x S2m+1................. 471 B. Explicit Metrics with Holonomy G2 and Spin(7) ................... 472 С Inhomogeneous Kähler-Einstein Metrics with Positive Scalar Curvature ................................................... 474 xii Table of Contents D. Uniqueness of Kähler-Einstein Metrics with Positive Scalar Curvature ................................................... 475 E. Hyperkählerian Quotients ...................................... 477 Bibliography .................................................... 479 Notation Index .................................................. 500 Subject Index .................................................... 505 Errata ......................................................... 511
adam_txt	Table of Contents Chapter 0. Introduction . 1 A. Brief Definitions and Motivation . 1 B. Why Write a Book on Einstein Manifolds? . 5 C. Existence . 6 D. Examples 1. Algebraic Examples . 6 2. Examples from Analysis . 7 3. Sporadic Examples . 8 E. Uniqueness and Moduli . 9 F. A Brief Survey of Chapter Contents . 10 G. Leitfaden. 14 Η. Getting the Feel of Ricci Curvature . 15 I. The Main Problems Today . 18 Chapter 1. Basic Material . 20 A. Introduction . 20 B. Linear Connections . 22 С Riemannian and Pseudo-Riemannian Manifolds . 29 D. Riemannian Manifolds as Metric Spaces . 35 E. Riemannian Immersions, Isometries and Killing Vector Fields . 37 F. Einstein Manifolds . 41 G. Irreducible Decompositions of Algebraic Curvature Tensors . 45 H. Applications to Riemannian Geometry . 48 I. Laplacians and Weitzenböck Formulas . 52 J. Conformai Changes of Riemannian Metrics . 58 K. First Variations of Curvature Tensor Fields . 62 Chapter 2. Basic Material (Continued): Kahler Manifolds . 66 0. Introduction . 66 A. Almost Complex and Complex Manifolds . 66 B. Hermitian and Kahler Metrics . 69 С Ricci Tensor and Ricci Form . 73 D. Holomorphic Sectional Curvature . 75 E. Chern Classes . 78 viii Table of Contents F. The Ricci Form as the Curvature Form of a Line Bundle . 81 G. Hodge Theory . 83 H. Holomorphic Vector Fields and Infinitesimal Isometries . 86 I. The Calabi-Futaki Theorem . 92 Chapter 3. Relativity . 94 A. Introduction . 94 B. Physical Interpretations . 94 С The Einstein Field Equation . 96 D. Tidal Stresses . 97 E. Normal Forms for Curvature . 98 F. The Schwarzschild Metric . 101 G. Planetary Orbits . 105 H. Perihelion Precession . 107 I. Geodesies in the Schwarzschild Universe . 108 J. Bending of Light . 110 K. The Kruskal Extension . Ill L. How Completeness May Fail . 113 M. Singularity Theorems . 115 Chapter 4. Riemannian Functionals . 116 A. Introduction . 116 B. Basic Properties of Riemannian Functionals . 117 C. The Total Scalar Curvature: First Order Properties . 119 D. Existence of Metrics with Constant Scalar Curvature . 122 E. The Image of the Scalar Curvature Map . 124 F. The Manifold of Metrics with Constant Scalar Curvature . 126 G. Back to the Total Scalar Curvature: Second Order Properties . 129 H. Quadratic Functionals . 133 Chapter 5. Ricci Curvature as a Partial Differential Equation . 137 A. Pointwise (Infinitesimal) Solvability . 137 B. From Pointwise to Local Solvability: Obstructions . 138 С Local Solvability of Ric(^) = r for Nonsingular r . 140 D. Local Construction of Einstein Metrics . 142 E. Regularity of Metrics with Smooth Ricci Tensors . 143 F. Analyticity of Einstein Metrics and Applications . 145 G. Einstein Metrics on Three-Manifolds . 146 H, A Uniqueness Theorem for Ricci Curvature . 152 I. Global Non-Existence . 153 Chapter 6. Einstein Manifolds and Topology . 154 A. Introduction . 154 B. Existence of Einstein Metrics in Dimension 2. 155 С The 3-Dimensional Case . 157 Table of Contents ix D. The ^Dimensional Case . 161 E. Ricci Curvature and the Fundamental Group . 165 F. Scalar Curvature and the Spinorial Obstruction . 169 G. A Proof of the Cheeger-Gromoll Theorem on Complete Manifolds with Non-Negative Ricci Curvature . 171 Chapter 7. Homogeneous Riemannian Manifolds . 177 A. Introduction . 177 B. Homogeneous Riemannian Manifolds . 178 C. Curvature . 181 D. Some Examples of Homogeneous Einstein Manifolds . 186 E. General Results on Homogeneous Einstein Manifolds . 189 F. Symmetric Spaces . 191 G. Standard Homogeneous Riemannian Manifolds . 196 H. Tables . 200 I. Remarks on Homogeneous Lorentz Manifolds . 205 Chapter 8. Compact Homogeneous Kahler Manifolds . 208 0. Introduction . 208 A. The Orbits of a Compact Lie Group for the Adjoint Representation . 209 B. The Canonical Complex Structure . 212 C. The G-Invariant Ricci Form . 215 D. The Symplectic Structure of KiriUov-Kostant-Souriau . 220 E. The Invariant Kahler Metrics on the Orbits . 221 F. Compact Homogeneous Kahler Manifolds . 224 G. The Space of Orbits . 227 H. Examples . 229 Chapter 9. Riemannian Submersions . 235 A. Introduction .·. 235 B. Riemannian Submersions . 236 C. The Invariants A and T . 238 D. O'Neill's Formulas for Curvature . 241 E. Completeness and Connections . 244 F. Riemannian Submersions with Totally Geodesic Fibres . 249 G. The Canonical Variation . 252 H. Applications to Homogeneous Einstein Manifolds . 256 1. Further Examples of Homogeneous Einstein Manifolds . 263 J. Warped Products . 265 K. Examples of Non-Homogeneous Compact Einstein Manifolds with Positive Scalar Curvature ._ 272 Chapter 10. Holonomy Groups . 278 A. Introduction . 278 B. Definitions . 280 x Table of Contents C. Covariant Derivative Vanishing Versus Holonomy Invariance. Examples . 282 D. Riemannian Products Versus Holonomy . 285 E. Structure 1 . 288 F. Holonomy and Curvature . 290 G. Symmetric Spaces; Their Holonomy . 294 H. Structure II . 300 I. The Non-Simply Connected Case . 307 J. Lorentzian Manifolds . 309 K. Tables . 311 Chapter 11. Kähler-Einstein Metrics and the Calabi Conjecture . 318 A. Kähler-Einstein Metrics . 318 B. The Resolution of the Calabi Conjecture and its Consequences . 322 С A Brief Outline of the Proofs of the Aubin-Calabi-Yau Theorems . 326 D. Compact Complex Manifolds with Positive First Chern Class . 329 E. Extremal Metrics . 333 Chapter 12. The Moduli Space of Einstein Structures . 340 A. Introduction . 340 B. Typical Examples: Surfaces and Flat Manifolds . 342 С Basic Tools . 345 D. Infinitesimal Einstein Deformations . 346 E. Formal Integrability . 348 F. Structure of the Premoduli Spaces . 351 G. The Set of Einstein Constants . 352 H. Rigidity of Einstein Structures . 355 I. Dimension of the Moduli Space . 358 J. Deformations of Kähler-Einstein Metrics . 361 K. The Moduli Space of the Underlying Manifold of КЗ Surfaces . 365 Chapter 13. Self-Duality . 369 A. Introduction . 369 B. Self-Duality . 370 С Half-Conformally Flat Manifolds . 372 D. The Penrose Construction . 379 E. The Reverse Penrose Construction . 385 F. Application to the Construction of Half-Conformally Flat Einstein Manifolds . 390 Chapter 14. Quaternion-Kähler Manifolds . 396 A. Introduction . 396 B. Hyperkählerian Manifolds . 398 C. Examples of Hyperkählerian Manifolds . 400 Table of Contents xi D. Quaternion-Kähler Manifolds . 402 E. Symmetric Quaternion-Kähler Manifolds . 408 F. Quaternionic Manifolds. 410 G. The Twistor Space of a Quaternionic Manifold . 412 H. Applications of the Twistor Space Theory . 415 I. Examples of Non-Symmetric Quaternion-Kähler Manifolds . 419 Chapter 15. A Report on the Non-Compact Case . 422 A. Introduction . 422 B. A Construction of Nonhomogeneous Einstein Metrics . 423 С Bundle Constructions . 424 D. Bounded Domains of Holomorphy . 428 Chapter 16. Generalizations of the Einstein Condition . 432 A. Introduction . 432 B. Natural Linear Conditions on Dr . 433 C. Codazzi Tensors . 436 D. The Case DreC^iQ ® S): Riemannian Manifolds with Harmonic Weyl Tensor . 440 E. Condition Dr є C^iS): Riemannian Manifolds with Harmonic Curvature . 443 F. The Case DreC^iQ) . 447 G. Condition Dr є С^А): Riemannian Manifolds such that (Dxr)(X,X) = 0 for all Tangent Vectors X . 450 H. Oriented Riemannian 4-Manifolds with 6W+ = 0. 451 Appendix. Sobolev Spaces and Elliptic Operators . 456 A. Holder Spaces . 456 B. Sobolev Spaces . 457 С Embedding Theorems . 457 D. Differential Operators . 459 E. Adjoint . 460 F. Principal Symbol . 460 G. Elliptic Operators . 461 H. Schauder and Lp Estimates for Linear Elliptic Operators . 463 I. Existence for Linear Elliptic Equations . 464 J. Regularity of Solutions for Elliptic Equations . 466 K. Existence for Nonlinear Elliptic Equations . 467 Addendum . 471 A. Infinitely Many Einstein Constants on S2 x S2m+1. 471 B. Explicit Metrics with Holonomy G2 and Spin(7) . 472 С Inhomogeneous Kähler-Einstein Metrics with Positive Scalar Curvature . 474 xii Table of Contents D. Uniqueness of Kähler-Einstein Metrics with Positive Scalar Curvature . 475 E. Hyperkählerian Quotients . 477 Bibliography . 479 Notation Index . 500 Subject Index . 505 Errata . 511
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series	Ergebnisse der Mathematik und ihrer Grenzgebiete
series2	Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 Classics in mathematics
spelling	Besse, Arthur L. Verfasser aut Einstein manifolds Arthur L. Besse reprint of the 1987 ed. Berlin [u.a.] Springer 2008 XII, 516 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 10 Classics in mathematics Einstein, Variétés d' ram Relativité (physique) ram Relativitätstheorie (DE-588)4049363-5 gnd rswk-swf Einstein-Mannigfaltigkeit (DE-588)4113398-5 gnd rswk-swf Einstein-Mannigfaltigkeit (DE-588)4113398-5 s Relativitätstheorie (DE-588)4049363-5 s DE-604 Ergebnisse der Mathematik und ihrer Grenzgebiete Folge 3 ; 10 (DE-604)BV000899194 10 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016539814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Besse, Arthur L. Einstein manifolds Ergebnisse der Mathematik und ihrer Grenzgebiete Einstein, Variétés d' ram Relativité (physique) ram Relativitätstheorie (DE-588)4049363-5 gnd Einstein-Mannigfaltigkeit (DE-588)4113398-5 gnd
subject_GND	(DE-588)4049363-5 (DE-588)4113398-5
title	Einstein manifolds
title_auth	Einstein manifolds
title_exact_search	Einstein manifolds
title_exact_search_txtP	Einstein manifolds
title_full	Einstein manifolds Arthur L. Besse
title_fullStr	Einstein manifolds Arthur L. Besse
title_full_unstemmed	Einstein manifolds Arthur L. Besse
title_short	Einstein manifolds
title_sort	einstein manifolds
topic	Einstein, Variétés d' ram Relativité (physique) ram Relativitätstheorie (DE-588)4049363-5 gnd Einstein-Mannigfaltigkeit (DE-588)4113398-5 gnd
topic_facet	Einstein, Variétés d' Relativité (physique) Relativitätstheorie Einstein-Mannigfaltigkeit
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016539814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000899194
work_keys_str_mv	AT bessearthurl einsteinmanifolds

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