Einstein ManifoldsSpringer, 12.11.2007 - 510 Seiten Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. Recently, it has produced several striking results, which have been of great interest also to physicists. This Ergebnisse volume is the first book which presents an up-to-date overview of the state of the art in this field. "Einstein Manifold"s is a successful attempt to organize the abundant literature, with emphasis on examples. Parts of it can be used separately as introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals. |
Inhalt
| 1 | |
| 7 | |
| 14 | |
| 20 | |
Riemannian and PseudoRiemannian Manifolds | 29 |
Riemannian Manifolds as Metric Spaces | 35 |
F Einstein Manifolds | 41 |
H Applications to Riemannian Geometry | 48 |
H Applications to Homogeneous Einstein Manifolds | 256 |
Further Examples of Homogeneous Einstein Manifolds | 263 |
K Examples of NonHomogeneous Compact Einstein Manifolds with | 272 |
Holonomy Groups | 278 |
Covariant Derivative Vanishing Versus Holonomy Invariance | 282 |
E Structure I | 288 |
G Symmetric Spaces Their Holonomy | 294 |
H Structure II | 300 |
J Conformal Changes of Riemannian Metrics | 58 |
Kähler Manifolds | 66 |
Ricci Tensor and Ricci Form | 73 |
F The Ricci Form as the Curvature Form of a Line Bundle | 81 |
The CalabiFutaki Theorem | 92 |
E Normal Forms for Curvature | 98 |
29 | 102 |
G Planetary Orbits | 105 |
48 | 108 |
K The Kruskal Extension | 111 |
62 | 115 |
B Basic Properties of Riemannian Functionals | 117 |
E The Image of the Scalar Curvature | 124 |
66 | 133 |
Local Solvability of Ricg r for Nonsingular | 140 |
G Einstein Metrics on ThreeManifolds | 146 |
H A Uniqueness Theorem for Ricci Curvature | 152 |
The 4Dimensional Case | 161 |
F Scalar Curvature and the Spinorial Obstruction | 169 |
Homogeneous Riemannian Manifolds | 177 |
Some Examples of Homogeneous Einstein Manifolds | 186 |
G Standard Homogeneous Riemannian Manifolds | 197 |
Remarks on Homogeneous Lorentz Manifolds | 205 |
B The Canonical Complex Structure | 212 |
The Symplectic Structure of KirillovKostantSouriau | 220 |
G The Space of Orbits | 227 |
Riemannian Submersions | 235 |
ONeills Formulas for Curvature | 241 |
F Riemannian Submersions with Totally Geodesic Fibres | 249 |
The NonSimply Connected Case | 307 |
KählerEinstein Metrics and the Calabi Conjecture | 318 |
A Brief Outline of the Proofs of the AubinCalabiYau Theorems | 326 |
E Extremal Metrics | 333 |
The Moduli Space of Einstein Structures | 340 |
Infinitesimal Einstein Deformations | 346 |
G The Set of Einstein Constants | 352 |
Dimension of the Moduli Space | 358 |
K The Moduli Space of the Underlying Manifold of K3 Surfaces | 365 |
HalfConformally Flat Manifolds | 372 |
The Penrose Construction | 379 |
E The Reverse Penrose Construction | 385 |
QuaternionKähler Manifolds | 396 |
QuaternionKähler Manifolds | 402 |
E Symmetric QuaternionKähler Manifolds | 408 |
H Applications of the Twistor Space Theory | 415 |
A Report on the NonCompact Case | 422 |
Bounded Domains of Holomorphy | 428 |
Codazzi Tensors | 436 |
Riemannian Manifolds with Harmonic | 443 |
Riemannian Manifolds such that | 450 |
Appendix Sobolev Spaces and Elliptic Operators | 456 |
H Schauder and LP Estimates for Linear Elliptic Operators | 463 |
Addendum | 471 |
Uniqueness of KählerEinstein Metrics with Positive Scalar | 475 |
| 481 | |
| 496 | |
| 500 | |
Andere Ausgaben - Alle anzeigen
Häufige Begriffe und Wortgruppen
admits automorphism canonical Chapter Chern class compact manifold complex manifold complex structure computation conformally flat constant curvature Corollary covariant derivative CP¹ curvature tensor decomposition defined Definition denote diffeomorphism differential dimension eigenvalue Einstein manifolds Einstein metric elliptic equation examples exists finite formula function G-invariant g₁ G₂ harmonic hence Hermitian holomorphic holonomy group holonomy representation hyperkählerian invariant isometry isomorphic isotropy Kähler form Kähler manifold Kähler metric Kähler-Einstein metric Lemma Lie algebra Lie group linear M₁ metric g Moduli Space non-compact orbit orthogonal orthonormal positive scalar curvature proof Proposition quaternion-Kähler manifold quaternionic quotient resp Ricci curvature Ricci form Ricci-flat Riemannian manifold Riemannian metric Riemannian submersion satisfies scalar curvature sectional curvature simply connected SO(n solution Sp(n subgroup symmetric spaces tangent Theorem trivial vanishes vector bundle vector field warped product zero