Einstein Manifolds

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Springer Science & Business Media, 03.12.2007 - 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

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
Notation Index
500
Errata
511
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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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