Einstein Manifolds

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Springer, 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.

 

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Inhalt

Basic Material
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
Introduction
483
Almost Complex and Complex Manifolds
500
Urheberrecht

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Über den Autor (2007)

Arthur L. Besse

Besides his personal work in Riemannian Geometry, Marcel Berger is well known for his persistent and untiring propaganda for the problems he considers to be (and which actually are) natural and fundamental.

In 1975, he convinced his students to organise a workshop about one of his favorite problems, namely to understand manifolds, all of whose geodesics are closed. The workshop took place in Besse-en-Chandesse, a very pleasant village in the centre of France, and turned out to be so successful that a consensus emerged to write a book about this topic. Arthur Besse was born.

At that time, such a first name seemed old-fashioned and funny in France. But why not ? Besides, the association with King Arthur could not be overlooked, since this type of meeting was denoted, by the CNRS, as a "Table Ronde" !

The experience was so enjoyable that Arthur did not stop there, and settled down to write another book.

A preliminary workshop took place in another village, even lovelier than the first: Espalion, in the South-West of France. This second book, Einstein Manifolds, was eventually published in 1987.

Years have passed. Arthur's friends (the list of which can be found in the beginning of his books) haved scattered to various places.

For Arthur himself, who never laid any claim to immortality, it may be time for retirement.

One FAQ. What do Bourbaki and Besse have in common? Hardly anything. Simply that both are mathematicians, of course, and share a taste for working in pleasant and quiet places.

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