Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals. Chapter 0. Introduction -- Chapter 1. Basic Material -- Chapter 2. Basic Materisl (continued) : Kahler 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. Compact Homogeneous Kahler Manifolds -- Chapter 9. Riemannian Submersions -- Chapter 10. Holonomy Groups -- Chapter 11. Kahler-einstein Metrics And The Calabi Conjecture -- Chapter 12. The Moduli Space Of Einstein Structures -- Chapter 13. Self-duality -- Chapter 14. Quaternion-kahler Manifolds -- Chapter 15. A Report On The Non-compact Case -- Chapter 16. Generalization Of The Einstein Condition -- Appendix. Sobolev Spaces And Elliptic Operators -- Addendum. Arthur L. Besse. Originally Issued As Vol. 10 Of The Ergebnisse Der Mathematik Und Ihrer Grenzgebiete, 3rd Series--p. [iii]. Includes Bibliographical References (p. [479]-499) And Index. From the reviews: "[...] 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 equations, 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'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.
