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.

Table of Contents

Chapter O. Introduction

Chapter 1. Basic Material

Chapter 2. Basic Material (Continued): Kähler Manifolds

Chapter 3. Relativity

Chapter 4. Riemannian Functionals

Chapter 5. Ricci Curvature as a Partial Differential Equation

Chapter 6. Einstein Manifolds and Topology

Chapter 7. Homogeneous Riemannian Manifolds

Chapter 8. Compaet Homogeneous Kähler Manifolds

Chapter 9. Riemannian Submersions

Chapter 10. Holonomy Groups

Chapter 11. Kähler-Einstein Metries and the Calabi Conjeeture

Chapter 12. The Moduli Spaee of Einstein Struetures

Chapter 13. Self-Duality

Chapter 14. Quaternion-Kähler Manifolds

Chapter 15. AReport on the Non-Compaet Case

Chapter 16. Generalizations of the Einstein Condition

Appendix. Sobolev Spaees and Elliptie Operators

Review

"[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."

S.M. Salamon in MathSciNet 1988

"It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."

T.J. Wilmore in Bulletin of the London Mathematical Society 1987

"Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008)

"The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)

About the Author

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.