Einstein Manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
2008
|
| Edition: | Reprint of the 1987 edition |
| Series: | Classics in Mathematics
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Original erschien als Band 10 von "Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Serie" |
| Physical Description: | xiv, 516 Seiten |
| ISBN: | 9783540741206 3540741208 |
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Record in the Search Index
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| adam_text |
TABL
E OF CONTENT
S
CHAPTE
R 0. INTRODUCTIO
N 1
A. BRIEF DEFINITIONS AN
D MOTIVATIO
N 1
B
. WHY WRITE A BOOK O
N EINSTEIN MANIFOLDS? 5
C. EXISTENCE 6
D
. EXAMPLES
1. ALGEBRAIC EXAMPLES 6
2. EXAMPLES FROM ANALYSIS 7
3. SPORADIC EXAMPLES 8
E. UNIQUENESS AN
D MODUL
I 9
F
. A BRIEF SURVEY OF CHAPTE
R CONTENT
S 10
G. LEITFADEN 14
H
. GETTING THE FEEL OF RICCI CURVATUR
E 15
I. TH
E MAIN PROBLEM
S TODA
Y 18
CHAPTE
R 1. BASIC MATERIA
L 20
A. INTRODUCTIO
N 20
B. LINEAR CONNECTIONS 22
C. RIEMANNIAN AN
D PSEUDO-RIEMANNIA
N MANIFOLDS 29
D
. RIEMANNIAN MANIFOLDS AS METRIC SPACES 35
E. RIEMANNIAN IMMERSIONS, ISOMETRIES AN
D KILLING VECTOR FIELDS 37
F
. EINSTEIN MANIFOLDS 41
G. IRREDUCIBLE DECOMPOSITIONS OF ALGEBRAIC CURVATUR
E TENSORS 4
5
H
. APPLICATIONS T
O RIEMANNIAN GEOMETR
Y 48
I. LAPLACIANS AN
D WEITZENBOECK FORMULA
S 52
J. CONFORMAL CHANGES OF RIEMANNIAN METRICS 58
K
. FIRS
T VARIATION
S OF CURVATUR
E TENSO
R FIELDS 62
CHAPTE
R 2. BASIC MATERIA
L (CONTINUED): KAHLE
R MANIFOLDS 66
0. INTRODUCTIO
N 66
A. ALMOST COMPLEX AN
D COMPLEX MANIFOLDS 66
B. HERMITIAN AN
D KAHLE
R METRICS 69
C. RICCI TENSO
R AN
D RICCI FOR
M 73
D
. HOLOMORPHI
C SECTIONAL CURVATUR
E 75
E. CHERN CLASSES 78
GESCANNT DURCH
BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/986835870
DIGITALISIERT DURCH
VIII TABLEOF CONTENTS
F
. THE RICCI FORM AS THE CURVATURE FORM OF A LINE BUENDLE 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
C. 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. GEODESICS IN THE SCHWARZSCHILD UNIVERSE 108
J. BENDING OF LIGHT 110
K. THE KRUSKAL EXTENSION 111
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 MA
P 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
C. LOCAL SOLVABILITY OF RIC(G) = R FOR NONSINGULAR R 140
D. LOCAL CONSTRUCTION OF EINSTEIN METRICS 1*2
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
C. THE 3-DIMENSIONAL CASE 157
TABLEOF CONTENTS IX
D
. THE 4-DIMENSIONAL CASE 161
E. RICCI CURVATUR
E AN
D THE FUNDAMENTA
L GROU
P 165
F
. SCALAR CURVATUR
E AN
D THE SPINORIAL OBSTRUCTIO
N 169
G. A PROOF OF THE CHEEGER-GROMOLL THEOREM ON COMPLETE MANIFOLDS
WITH NON-NEGATIVE RICCI CURVATUR
E 171
CHAPTE
R 7. HOMOGENEOUS RIEMANNIAN MANIFOLDS 177
A. INTRODUCTIO
N 177
B. HOMOGENEOUS RIEMANNIAN MANIFOLDS 178
C. CURVATUR
E 181
D
. SOME EXAMPLES OF HOMOGENEOUS EINSTEIN MANIFOLDS 186
E. GENERAL RESULTS ON HOMOGENEOUS EINSTEIN MANIFOLDS 189
F
. SYMMETRIE SPACES 191
G. STANDAR
D HOMOGENEOU
S RIEMANNIAN MANIFOLDS 196
H. TABLES 200
I. REMARKS ON HOMOGENEOU
S LORENTZ MANIFOLDS 205
CHAPTE
R 8. COMPAC
T HOMOGENEOUS KAHLE
R MANIFOLDS 208
0. INTRODUCTIO
N 208
A. THE ORBIT
S OF A COMPAC
T LIE GROU
P FOR THE ADJOINT REPRESENTATION .
. 209
B. TH
E CANONICAL COMPLEX STRUCTURE 212
C. THE G-INVARIANT RICCI FOR
M 215
D
. TH
E SYMPLECTIC STRUCTUR
E OF KIRILLOV-KOSTANT-SOURIAU 220
E. TH
E INVARIAN
T KAHLE
R METRICS ON TH
E ORBIT
S 221
F
. COMPAC
T HOMOGENEOU
S KAHLER MANIFOLDS 224
G. TH
E SPACE OF ORBIT
S 227
H
. EXAMPLES 229
CHAPTE
R 9. RIEMANNIAN SUBMERSIONS 235
A. INTRODUCTIO
N 235
B. RIEMANNIAN SUBMERSIONS 236
C. TH
E INVARIANTS
A
AN
D
T
238
D
. O'NEILL'S FORMULA
S FOR CURVATUR
E 241
E. COMPLETENESS AN
D CONNECTIONS 244
F
. RIEMANNIAN SUBMERSIONS WITH TOTALLY GEODESIC FIBRES 249
G. TH
E CANONICAL VARIATIO
N 252
H
. APPLICATIONS T
O HOMOGENEOU
S EINSTEIN MANIFOLDS 256
1. FURTHE
R EXAMPLES OF HOMOGENEOU
S EINSTEIN MANIFOLDS 263
J. WARPED PRODUCT
S 265
K. EXAMPLES OF NON-HOMOGENEOU
S COMPAC
T EINSTEIN MANIFOLDS WITH
POSITIVE SCALAR CURVATUR
E 272
CHAPTE
R 10. HOLONOM
Y GROUP
S 278
A. INTRODUCTIO
N 278
B. DEFMITIONS 280
X TABLEOF 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. SYMMETRIE SPACES; THEIR HOLONOMY 294
H. STRUCTURE II 300
I. THE NON-SIMPLY CONNECTED CASE 307
J. LORENTZIAN MANIFOLDS 309
K. TABLES 311
CHAPTER 11. KAEHLER-EINSTEIN METRICS AND THE CALABI CONJECTURE 318
A. KAEHLER-EINSTEIN METRICS 318
B. THE RESOLUTION OF THE CALABI CONJECTURE AN
D ITS CONSEQUENCES 322
C. 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 FIAT MANIFOLDS 342
C. 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 KAEHLER-EINSTEIN METRICS 361
K. THE MODULI SPACE OF THE UNDERLYING MANIFOLD OF K
3 SURFACES 365
CHAPTER 13. SELF-DUALITY 369
A. INTRODUCTION 369
B. SELF-DUALITY 370
C. HALF-CONFORMALLY FIAT MANIFOLDS 372
D
. THE PENROSE CONSTRUCTION 379
E. THE REVERSE PENROSE CONSTRUCTION 385
F
. APPLICATION TO THE CONSTRUCTION OF HALF-CONFORMALLY FIAT EINSTEIN
MANIFOLDS 390
CHAPTER 14. QUATERNION-KAEHLER MANIFOLDS 396
A. INTRODUCTION 396
B. HYPERKAEHLERIAN MANIFOLDS 398
C. EXAMPLES OF HYPERKAEHLERIAN MANIFOLDS 400
TABLE OF CONTENTS XI
D. QUATERNION-KAEHLE
R MANIFOLDS 402
E. SYMMETRIE QUATERNION-KAEHLE
R MANIFOLDS 408
F
. QUATERNIONI
C MANIFOLDS 410
G. TH
E TWISTOR SPACE OF A QUATERNIONI
C MANIFOLD 412
H. APPLICATIONS OF THE TWISTOR SPACE THEORY 415
I. EXAMPLES OF NON-SYMMETRI
C QUATERNION-KAEHLE
R MANIFOLDS 419
CHAPTE
R 15. A REPOR
T O
N THE NON-COMPAC
T CASE 422
A. INTRODUCTIO
N 422
B. A CONSTRUCTIO
N OF NONHOMOGENEOU
S EINSTEIN METRICS 423
C. BUENDLE CONSTRUCTION
S 424
D
. BOUNDED DOMAIN
S OF HOLOMORPH
Y 428
CHAPTE
R 16. GENERALIZATIONS OF THE EINSTEIN CONDITIO
N 432
A. INTRODUCTIO
N 432
B. NATURA
L LINEAR CONDITION
S ON
DR
433
C. CODAZZI TENSOR
S 436
D
. TH
E CASE
DREC'IQ @ S):
RIEMANNIAN MANIFOLDS WITH HARMONI
E
WEYL TENSOR 440
E. CONDITIO
N
DREC(S):
RIEMANNIAN MANIFOLDS WITH HARMONI
E
CURVATUR
E 443
F
. TH
E CASE DR 6 C(Q) 447
G. CONDITIO
N
DR E
C(/L): RIEMANNIAN MANIFOLDS SUCH THA
T
(D
X
R)(X,X)
= 0 FOR ALL TANGEN
T VECTORS
X
450
H. ORIENTED RIEMANNIAN 4-MANIFOLDS WITH
OEW
+
= 0 451
APPENDIX. SOBOLEV SPACES AN
D ELLIPTIC OPERATOR
S 456
A. HOLDER SPACES 456
B. SOBOLEV SPACES 457
C. EMBEDDING THEOREM
S 457
D
. DIFFERENTIAL OPERATOR
S 459
E. ADJOINT 460
F
. PRINCIPAL SYMBOL 460
G. ELLIPTIC OPERATOR
S 461
H. SCHAUDER AN
D
L"
ESTIMATES FOR LINEAR ELLIPTIC OPERATOR
S 463
I. EXISTENCE FOR LINEA
R ELLIPTIC EQUATION
S 464
J. REGULARITY OF SOLUTION
S FOR ELLIPTIC EQUATION
S 466
K. EXISTENCE FOR NONLINEA
R ELLIPTIC EQUATION
S 467
ADDENDUM 471
A. INFMITELY MAN
Y EINSTEIN CONSTANT
S ON
S
2
X S
2M+
1
471
B. EXPLICIT METRICS WITH HOLONOM
Y
G
2
AN
D SPIN(7) 472
C. INHOMOGENEOU
S KAEHLER-EINSTEIN METRICS WITH POSITIVE SCALAR
CURVATUR
E 474
XII TABL
E OF CONTENTS
D
. UNIQUENESS OF KAEHLER-EINSTEIN METRICS WITH POSITIVE SCALAR
CURVATURE 475
E. HYPERKAEHLERIAN QUOTIENTS 477
BIBLIOGRAPHY 479
NOTATION INDEX 500
SUBJECT INDEX 505
ERRAT
A 511 |
| any_adam_object | 1 |
| author | Berger, Marcel 1927-2016 |
| author_GND | (DE-588)12047672X |
| author_facet | Berger, Marcel 1927-2016 |
| author_role | aut |
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| building | Verbundindex |
| bvnumber | BV025491064 |
| classification_rvk | SK 350 |
| ctrlnum | (OCoLC)723810183 (DE-599)BVBBV025491064 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Reprint of the 1987 edition |
| format | Book |
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| id | DE-604.BV025491064 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-20T10:33:46Z |
| institution | BVB |
| isbn | 9783540741206 3540741208 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020103746 |
| oclc_num | 723810183 |
| open_access_boolean | |
| owner | DE-11 DE-188 |
| owner_facet | DE-11 DE-188 |
| physical | xiv, 516 Seiten |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series2 | Classics in Mathematics |
| spelling | Berger, Marcel 1927-2016 Verfasser (DE-588)12047672X aut Einstein Manifolds Arthur L. Besse Reprint of the 1987 edition Berlin ; Heidelberg Springer 2008 xiv, 516 Seiten txt rdacontent n rdamedia nc rdacarrier Classics in Mathematics Original erschien als Band 10 von "Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Serie" Einstein-Mannigfaltigkeit (DE-588)4113398-5 gnd rswk-swf Relativitätstheorie (DE-588)4049363-5 gnd rswk-swf Einstein-Mannigfaltigkeit (DE-588)4113398-5 s Relativitätstheorie (DE-588)4049363-5 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-74311-8 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3043895&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020103746&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berger, Marcel 1927-2016 Einstein Manifolds Einstein-Mannigfaltigkeit (DE-588)4113398-5 gnd Relativitätstheorie (DE-588)4049363-5 gnd |
| subject_GND | (DE-588)4113398-5 (DE-588)4049363-5 |
| title | Einstein Manifolds |
| title_auth | Einstein Manifolds |
| title_exact_search | 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-Mannigfaltigkeit (DE-588)4113398-5 gnd Relativitätstheorie (DE-588)4049363-5 gnd |
| topic_facet | Einstein-Mannigfaltigkeit Relativitätstheorie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3043895&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020103746&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bergermarcel einsteinmanifolds |