Classics in Mathematics

By: Arthur L. Besse

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Paperback | 3 December 2007

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532 Pages

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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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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 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)

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Introduction	p. 1
Brief Definitions and Motivation	p. 1
Why Write a Book on Einstein Manifolds?	p. 5
Existence	p. 6
Examples
Algebraic Examples	p. 6
Examples from Analysis	p. 7
Sporadic Examples	p. 8
Uniqueness and Moduli	p. 9
A Brief Survey of Chapter Contents	p. 10
Leitfaden	p. 14
Getting the Feel of Ricci Curvature	p. 15
The Main Problems Today	p. 18
Basic Material	p. 20
Introduction	p. 20
Linear Connections	p. 22
Riemannian and Pseudo-Riemannian Manifolds	p. 29
Riemannian Manifolds as Metric Spaces	p. 35
Riemannian Immersions, Isometries and Killing Vector Fields	p. 37
Einstein Manifolds	p. 41
Irreducible Decompositions of Algebraic Curvature Tensors	p. 45
Applications to Riemannian Geometry	p. 48
Laplacians and Weitzenbock Formulas	p. 52
Conformal Changes of Riemannian Metrics	p. 58
First Variations of Curvature Tensor Fields	p. 62
Basic Material (Continued): Kahler Manifolds	p. 66
Introduction	p. 66
Almost Complex and Complex Manifolds	p. 66
Hermitian and Kahler Metrics	p. 69
Ricci Tensor and Ricci Form	p. 73
Holomorphic Sectional Curvature	p. 75
Chern Classes	p. 78
The Ricci Form as the Curvature Form of a Line Bundle	p. 81
Hodge Theory	p. 83
Holomorphic Vector Fields and Infinitesimal Isometries	p. 86
The Calabi-Futaki Theorem	p. 92
Relativity	p. 94
Introduction	p. 94
Physical Interpretations	p. 94
The Einstein Field Equation	p. 96
Tidal Stresses	p. 97
Normal Forms for Curvature	p. 98
The Schwarzschild Metric	p. 101
Planetary Orbits	p. 105
Perihelion Precession	p. 107
Geodesics in the Schwarzschild Universe	p. 108
Bending of Light	p. 110
The Kruskal Extension	p. 111
How Completeness May Fail	p. 113
Singularity Theorems	p. 115
Riemannian Functionals	p. 116
Introduction	p. 116
Basic Properties of Riemannian Functionals	p. 117
The Total Scalar Curvature: First Order Properties	p. 119
Existence of Metrics with Constant Scalar Curvature	p. 122
The Image of the Scalar Curvature Map	p. 124
The Manifold of Metrics with Constant Scalar Curvature	p. 126
Back to the Total Scalar Curvature: Second Order Properties	p. 129
Quadratic Functionals	p. 133
Ricci Curvature as a Partial Differential Equation	p. 137
Pointwise (Infinitesimal) Solvability	p. 137
From Pointwise to Local Solvability: Obstructions	p. 138
Local Solvability of Ric(g) = r for Nonsingular r	p. 140
Local Construction of Einstein Metrics	p. 142
Regularity of Metrics with Smooth Ricci Tensors	p. 143
Analyticity of Einstein Metrics and Applications	p. 145
Einstein Metrics on Three-Manifolds	p. 146
A Uniqueness Theorem for Ricci Curvature	p. 152
Global Non-Existence	p. 153
Einstein Manifolds and Topology	p. 154
Introduction	p. 154
Existence of Einstein Metrics in Dimension 2	p. 155
The 3-Dimensional Case	p. 157
The 4-Dimensional Case	p. 161
Ricci Curvature and the Fundamental Group	p. 165
Scalar Curvature and the Spinorial Obstruction	p. 169
A Proof of the Cheeger-Gromoll Theorem on Complete Manifolds with Non-Negative Ricci Curvature	p. 171
Homogeneous Riemannian Manifolds	p. 177
Introduction	p. 177
Homogeneous Riemannian Manifolds	p. 178
Curvature	p. 181
Some Examples of Homogeneous Einstein Manifolds	p. 186
General Results on Homogeneous Einstein Manifolds	p. 189
Symmetric Spaces	p. 191
Standard Homogeneous Riemannian Manifolds	p. 196
Tables	p. 200
Remarks on Homogeneous Lorentz Manifolds	p. 205
Compact Homogeneous Kahler Manifolds	p. 208
Introduction	p. 208
The Orbits of a Compact Lie Group for the Adjoint Representation	p. 209
The Canonical Complex Structure	p. 212
The G-Invariant Ricci Form	p. 215
The Symplectic Structure of Kirillov-Kostant-Souriau	p. 220
The Invariant Kahler Metrics on the Orbits	p. 221
Compact Homogeneous Kahler Manifolds	p. 224
The Space of Orbits	p. 227
Examples	p. 229
Riemannian Submersions	p. 235
Introduction	p. 235
Riemannian Submersions	p. 236
The Invariants A and T	p. 238
O'Neill's Formulas for Curvature	p. 241
Completeness and Connections	p. 244
Riemannian Submersions with Totally Geodesic Fibres	p. 249
The Canonical Variation	p. 252
Applications to Homogeneous Einstein Manifolds	p. 256
Further Examples of Homogeneous Einstein Manifolds	p. 263
Warped Products	p. 265
Examples of Non-Homogeneous Compact Einstein Manifolds with Positive Scalar Curvature	p. 272
Holonomy Groups	p. 278
Introduction	p. 278
Definitions	p. 280
Covariant Derivative Vanishing Versus Holonomy Invariance. Examples	p. 282
Riemannian Products Versus Holonomy	p. 285
Structure I	p. 288
Holonomy and Curvature	p. 290
Symmetric Spaces; Their Holonomy	p. 294
Structure II	p. 300
The Non-Simply Connected Case	p. 307
Lorentzian Manifolds	p. 309
Tables	p. 311
Kahler-Einstein Metrics and the Calabi Conjecture	p. 318
Kahler-Einstein Metrics	p. 318
The Resolution of the Calabi Conjecture and its Consequences	p. 322
A Brief Outline of the Proofs of the Aubin-Calabi-Yau Theorems	p. 326
Compact Complex Manifolds with Positive First Chern Class	p. 329
Extremal Metrics	p. 333
The Moduli Space of Einstein Structures	p. 340
Introduction	p. 340
Typical Examples: Surfaces and Flat Manifolds	p. 342
Basic Tools	p. 345
Infinitesimal Einstein Deformations	p. 346
Formal Integrability	p. 348
Structure of the Premoduli Spaces	p. 351
The Set of Einstein Constants	p. 352
Rigidity of Einstein Structures	p. 355
Dimension of the Moduli Space	p. 358
Deformations of Kahler-Einstein Metrics	p. 361
The Moduli Space of the Underlying Manifold of K3 Surfaces	p. 365
Self-Duality	p. 369
Introduction	p. 369
Self-Duality	p. 370
Half-Conformally Flat Manifolds	p. 372
The Penrose Construction	p. 379
The Reverse Penrose Construction	p. 385
Application to the Construction of Half-Conformally Flat Einstein Manifolds	p. 390
Quaternion-Kahler Manifolds	p. 396
Introduction	p. 396
Hyperkahlerian Manifolds	p. 398
Examples of Hyperkahlerian Manifolds	p. 400
Quaternion-Kahler Manifolds	p. 402
Symmetric Quaternion-Kahler Manifolds	p. 408
Quaternionic Manifolds	p. 410
The Twistor Space of a Quaternionic Manifold	p. 412
Applications of the Twistor Space Theory	p. 415
Examples of Non-Symmetric Quaternion-Kahler Manifolds	p. 419
A Report on the Non-Compact Case	p. 422
Introduction	p. 422
A Construction of Nonhomogeneous Einstein Metrics	p. 423
Bundle Constructions	p. 424
Bounded Domains of Holomorphy	p. 428
Generalizations of the Einstein Condition	p. 432
Introduction	p. 432
Natural Linear Conditions on Dr	p. 433
Codazzi Tensors	p. 436
The Case Dr [set membership] C[superscript [infinity]] (Q [plus sign in circle] S): Riemannian Manifolds with Harmonic Weyl Tensor	p. 440
Condition Dr [set membership] C[superscript [infinity]] (S): Riemannian Manifolds with Harmonic Curvature	p. 443
The Case Dr [set membership] C[superscript [infinity]] (Q)	p. 447
Condition Dr [set membership] C[superscript [infinity]] (A): Riemannian Manifolds such that (D[subscript x]r)(X, X) = 0 for all Tangent Vectors X	p. 450
Oriented Riemannian 4-Manifolds with [delta]W[superscript +] = 0	p. 451
Sobolev Spaces and Elliptic Operators	p. 456
Holder Spaces	p. 456
Sobolev Spaces	p. 457
Embedding Theorems	p. 457
Differential Operators	p. 459
Adjoint	p. 460
Principal Symbol	p. 460
Elliptic Operators	p. 461
Schauder and L[superscript p] Estimates for Linear Elliptic Operators	p. 463
Existence for Linear Elliptic Equations	p. 464
Regularity of Solutions for Elliptic Equations	p. 466
Existence for Nonlinear Elliptic Equations	p. 467
Addendum	p. 471
Infinitely Many Einstein Constants on S[superscript 2] x S[superscript 2m+1]	p. 471
Explicit Metrics with Holonomy G[subscript 2] and Spin(7)	p. 472
Inhomogeneous Kahler-Einstein Metrics with Positive Scalar Curvature	p. 474
Uniqueness of Kahler-Einstein Metrics with Positive Scalar Curvature	p. 475
Hyperkahlerian Quotients	p. 477
Bibliography	p. 479
Notation Index	p. 500
Subject Index	p. 505
Errata	p. 511
Table of Contents provided by Ingram. All Rights Reserved.

Classics in Mathematics

At a Glance

Paperback

Industry Reviews

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