## Einstein ManifoldsEinstein'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. |

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### Inhalt

369? | 1 |

Introduction | 18 |

Riemannian Manifolds as Metric Spaces | 35 |

Applications to Riemannian Geometry | 48 |

First Variations of Curvature Tensor Fields | 62 |

F The Ricci Form as the Curvature Form of a Line Bundle | 81 |

Relativity | 94 |

H Perihelion Precession | 107 |

The NonSimply Connected Case | 307 |

KéihlerEinstein 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 |

5 | 343 |

Inﬁnitesimal Einstein Deformations | 346 |

G The Set of Einstein Constants | 352 |

Existence of Metrics with Constant Scalar Curvature | 122 |

Ricci Curvature as a Partial Differential Equation | 137 |

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 |

F Compact Homogeneous Kéihler Manifolds | 226 |

Riemannian Submersions | 235 |

ONeills Formulas for Curvature | 241 |

F Riemannian Submersions with Totally Geodesic Fibres | 249 |

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 |

73 | 357 |

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 |

QuaternionKahler Manifolds | 396 |

QuaternionKahler Manifolds | 402 |

E Symmetric QuaternionKahler 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 L1 Estimates for Linear Elliptic Operators | 463 |

Addendum | 471 |

Uniqueness of KahlerEinstein Metrics with Positive Scalar | 475 |

500 | |

Errata | 511 |

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