Einstein manifolds

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Einstein manifolds / Arthur L. Besse

資料種別:: 図書
出版情報:: Berlin ; New York : Springer-Verlag, c1987
形態:: xii, 510 p. ; 25 cm
シリーズ名:: Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 10 <BA00035552>
著者名:: Besse, Arthur L. <DA00505528>
ISBN:: 9783540741206 [3540741208] (: Berlin)
9780387152790 [0387152792] (: New York)
9783540741206 [3540741208] (: [pbk.])
書誌ID:: BA00248112

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注記:

Includes bibliographical references (p. [479]-499), notation index (p. [500]-504), and subject index (p. [505]-510)

主題:

Einstein manifolds; Relativity (Physics)

言語:

英語

目次情報:

Introduction / Chapter 0：

Brief Definitions and Motivation / A：

Why Write a Book on Einstein Manifolds? / B：

Existence / C：

Examples / D：

Algebraic Examples / 1：

Examples from Analysis / 2：

Sporadic Examples / 3：

Uniqueness and Moduli / E：

A Brief Survey of Chapter Contents / F：

Leitfaden / G：

Getting the Feel of Ricci Curvature / H：

The Main Problems Today / I：

Basic Material / Chapter 1：

Linear Connections

Riemannian and Pseudo-Riemannian Manifolds

Riemannian Manifolds as Metric Spaces

Riemannian Immersions, Isometries and Killing Vector Fields

Einstein Manifolds

Irreducible Decompositions of Algebraic Curvature Tensors

Applications to Riemannian Geometry

Laplacians and Weitzenbock Formulas

Conformal Changes of Riemannian Metrics / J：

First Variations of Curvature Tensor Fields / K：

Basic Material (Continued): Kahler Manifolds / Chapter 2：

Almost Complex and Complex Manifolds / 0：

Hermitian and Kahler Metrics

Ricci Tensor and Ricci Form

Holomorphic Sectional Curvature

Chern Classes

The Ricci Form as the Curvature Form of a Line Bundle

Hodge Theory

Holomorphic Vector Fields and Infinitesimal Isometries

The Calabi-Futaki Theorem

Relativity / Chapter 3：

Physical Interpretations

The Einstein Field Equation

Tidal Stresses

Normal Forms for Curvature

The Schwarzschild Metric

Planetary Orbits

Perihelion Precession

Geodesics in the Schwarzschild Universe

Bending of Light

The Kruskal Extension

How Completeness May Fail / L：

Singularity Theorems / M：

Riemannian Functionals / Chapter 4：

Basic Properties of Riemannian Functionals

The Total Scalar Curvature: First Order Properties

Existence of Metrics with Constant Scalar Curvature

The Image of the Scalar Curvature Map

The Manifold of Metrics with Constant Scalar Curvature

Back to the Total Scalar Curvature: Second Order Properties

Quadratic Functionals

Ricci Curvature as a Partial Differential Equation / Chapter 5：

Pointwise (Infinitesimal) Solvability

From Pointwise to Local Solvability: Obstructions

Local Solvability of Ric(g) = r for Nonsingular r

Local Construction of Einstein Metrics

Regularity of Metrics with Smooth Ricci Tensors

Analyticity of Einstein Metrics and Applications

Einstein Metrics on Three-Manifolds

A Uniqueness Theorem for Ricci Curvature

Global Non-Existence

Einstein Manifolds and Topology / Chapter 6：

Existence of Einstein Metrics in Dimension 2

The 3-Dimensional Case

The 4-Dimensional Case

Ricci Curvature and the Fundamental Group

Scalar Curvature and the Spinorial Obstruction

A Proof of the Cheeger-Gromoll Theorem on Complete Manifolds with Non-Negative Ricci Curvature

Homogeneous Riemannian Manifolds / Chapter 7：

Curvature

Some Examples of Homogeneous Einstein Manifolds

General Results on Homogeneous Einstein Manifolds

Symmetric Spaces

Standard Homogeneous Riemannian Manifolds

Tables

Remarks on Homogeneous Lorentz Manifolds

Compact Homogeneous Kahler Manifolds / Chapter 8：

The Orbits of a Compact Lie Group for the Adjoint Representation

The Canonical Complex Structure

The G-Invariant Ricci Form

The Symplectic Structure of Kirillov-Kostant-Souriau

The Invariant Kahler Metrics on the Orbits

The Space of Orbits

Riemannian Submersions / Chapter 9：

The Invariants A and T

O'Neill's Formulas for Curvature

Completeness and Connections

Riemannian Submersions with Totally Geodesic Fibres

The Canonical Variation

Applications to Homogeneous Einstein Manifolds

Further Examples of Homogeneous Einstein Manifolds

Warped Products

Examples of Non-Homogeneous Compact Einstein Manifolds with Positive Scalar Curvature

Holonomy Groups / Chapter 10：

Definitions

Covariant Derivative Vanishing Versus Holonomy Invariance. Examples

Riemannian Products Versus Holonomy

Structure I

Holonomy and Curvature

Symmetric Spaces; Their Holonomy

Structure II

The Non-Simply Connected Case

Lorentzian Manifolds

Kahler-Einstein Metrics and the Calabi Conjecture / Chapter 11：

Kahler-Einstein Metrics

The Resolution of the Calabi Conjecture and its Consequences

A Brief Outline of the Proofs of the Aubin-Calabi-Yau Theorems

Compact Complex Manifolds with Positive First Chern Class

Extremal Metrics

The Moduli Space of Einstein Structures / Chapter 12：

Typical Examples: Surfaces and Flat Manifolds

Basic Tools

Infinitesimal Einstein Deformations

Formal Integrability

Structure of the Premoduli Spaces

The Set of Einstein Constants

Rigidity of Einstein Structures

Dimension of the Moduli Space

Deformations of Kahler-Einstein Metrics

The Moduli Space of the Underlying Manifold of K3 Surfaces

Self-Duality / Chapter 13：

Half-Conformally Flat Manifolds

The Penrose Construction

The Reverse Penrose Construction

Application to the Construction of Half-Conformally Flat Einstein Manifolds

Quaternion-Kahler Manifolds / Chapter 14：

Hyperkahlerian Manifolds

Examples of Hyperkahlerian Manifolds

Symmetric Quaternion-Kahler Manifolds

Quaternionic Manifolds

The Twistor Space of a Quaternionic Manifold

Applications of the Twistor Space Theory

Examples of Non-Symmetric Quaternion-Kahler Manifolds

A Report on the Non-Compact Case / Chapter 15：

A Construction of Nonhomogeneous Einstein Metrics

Bundle Constructions

Bounded Domains of Holomorphy

Generalizations of the Einstein Condition / Chapter 16：

Natural Linear Conditions on Dr

Codazzi Tensors

The Case Dr [set membership] C[superscript [infinity]] (Q [plus sign in circle] S): Riemannian Manifolds with Harmonic Weyl Tensor

Condition Dr [set membership] C[superscript [infinity]] (S): Riemannian Manifolds with Harmonic Curvature

The Case Dr [set membership] C[superscript [infinity]] (Q)

Condition Dr [set membership] C[superscript [infinity]] (A): Riemannian Manifolds such that (D[subscript x]r)(X, X) = 0 for all Tangent Vectors X

Oriented Riemannian 4-Manifolds with [delta]W[superscript +] = 0

Sobolev Spaces and Elliptic Operators / Appendix：

Holder Spaces

Sobolev Spaces

Embedding Theorems

Differential Operators

Adjoint

Principal Symbol

Elliptic Operators

Schauder and L[superscript p] Estimates for Linear Elliptic Operators

Existence for Linear Elliptic Equations

Regularity of Solutions for Elliptic Equations

Existence for Nonlinear Elliptic Equations

Addendum

Infinitely Many Einstein Constants on S[superscript 2] x S[superscript 2m+1]

Explicit Metrics with Holonomy G[subscript 2] and Spin(7)

Inhomogeneous Kahler-Einstein Metrics with Positive Scalar Curvature

Uniqueness of Kahler-Einstein Metrics with Positive Scalar Curvature

Hyperkahlerian Quotients

Bibliography

Notation Index

Subject Index

Errata

Introduction / Chapter 0：

Brief Definitions and Motivation / A：

Why Write a Book on Einstein Manifolds? / B：

Existence / C：

Examples / D：

Algebraic Examples / 1：

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