## 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

20 | |

Kähler Manifolds | 66 |

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

Relativity | 94 |

H Perihelion Precession | 107 |

Existence of Metrics with Constant Scalar Curvature | 122 |

Ricci Curvature as a Partial Differential Equation | 137 |

H A Uniqueness Theorem for Ricci Curvature | 152 |

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 |

14 | 396 |

Homogeneous Riemannian Manifolds | 177 |

Compact Homogeneous Kähler Manifolds | 208 |

Riemannian Submersions | 235 |

Holonomy Groups | 278 |

Covariant Derivative Vanishing Versus Holonomy Invariance | 282 |

E Structure I | 288 |

G Symmetric Spaces Their Holonomy | 294 |

H Structure II | 300 |

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 |

QuaternionKähler Manifolds | 402 |

15 | 422 |

35 | 428 |

The Case Dre CQ S Riemannian Manifolds with Harmonic | 440 |

F The Case Dre CQ | 447 |

48 | 448 |

H Oriented Riemannian 4Manifolds with 6W 0 | 453 |

Addendum | 471 |

Uniqueness of KählerEinstein Metrics with Positive Scalar | 475 |

483 | |

500 | |

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admits automorphism canonical Chapter Chern class compact manifold complex manifold complex structure computation condition conformally flat constant curvature coordinates Corollary covariant derivative curvature tensor decomposition defined Definition denote diffeomorphism differential dimension eigenvalue Einstein manifolds Einstein metric Einstein structures elliptic equation examples exists finite formula function G-invariant geometry harmonic hence Hermitian holomorphic holonomy group holonomy representation hyperkählerian implies invariant isometry isomorphic isotropy Kähler form Kähler manifold Kähler metric Kähler-Einstein metric Lemma Lie algebra Lie group linear Math metric g Moduli Space negative non-compact obtain orbit orthogonal orthonormal positive scalar curvature proof Proposition quaternion-Kähler manifold quaternionic quotient resp Ric(g Ricci curvature Ricci form Ricci tensor Ricci-flat Riemannian manifold Riemannian metric Riemannian submersion satisfies scalar curvature sectional curvature simply connected SO(n solution Sp(n subgroup tangent Theorem trivial vanishes vector bundle vector field warped product zero