##
**Einstein manifolds.**
*(English)*
Zbl 0613.53001

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 10. Berlin etc.: Springer-Verlag. XII, 510 p. (1987).

This book is a self-contained presentation of numerous topics and results in Riemannian geometry centered around, but far from being exclusively devoted to its main theme, the compact Einstein manifolds.

The first chapter covers introductory material and may independently serve as a reference for topics like algebraic curvature tensors, Weitzenböck formulas, conformal changes of metric and first variations of curvature. Chapter 2 is a similar introduction to the geometry of Kähler manifolds.

Most other chapters are devoted to single, broad areas of global differential geometry, reaching far beyond Einstein manifolds and can be read for their own sake. Their topics include general relativity with a detailed account of the Schwarzschild model and Kruskal’s extension (Chapter 3), various natural functionals in the space of Riemannian metrics (Chapter 4), the question of realizability of 2-tensor fields as Ricci tensors (Chapter 5), homogeneous Riemannian, or compact Kähler manifolds (Chapter 7 and 8), Riemannian submersions, including warped products (Chapter 9), holonomy groups (Chapter 10 and Addendum) and self- duality (Chapter 13).

The Einstein manifolds are primarily dealt with in Chapters 6, 11, 12, 14 and 15 where, respectively, the known topological obstructions, Calabi’s conjecture, moduli of Einstein structures, quaternion-Kähler manifolds and constructions of non-compact Kähler-Einstein manifolds are discussed. Chapter 16 is devoted to some generalizations of Einstein manifolds.

The first chapter covers introductory material and may independently serve as a reference for topics like algebraic curvature tensors, Weitzenböck formulas, conformal changes of metric and first variations of curvature. Chapter 2 is a similar introduction to the geometry of Kähler manifolds.

Most other chapters are devoted to single, broad areas of global differential geometry, reaching far beyond Einstein manifolds and can be read for their own sake. Their topics include general relativity with a detailed account of the Schwarzschild model and Kruskal’s extension (Chapter 3), various natural functionals in the space of Riemannian metrics (Chapter 4), the question of realizability of 2-tensor fields as Ricci tensors (Chapter 5), homogeneous Riemannian, or compact Kähler manifolds (Chapter 7 and 8), Riemannian submersions, including warped products (Chapter 9), holonomy groups (Chapter 10 and Addendum) and self- duality (Chapter 13).

The Einstein manifolds are primarily dealt with in Chapters 6, 11, 12, 14 and 15 where, respectively, the known topological obstructions, Calabi’s conjecture, moduli of Einstein structures, quaternion-Kähler manifolds and constructions of non-compact Kähler-Einstein manifolds are discussed. Chapter 16 is devoted to some generalizations of Einstein manifolds.

Reviewer: A.Derdzinski

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### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C30 | Differential geometry of homogeneous manifolds |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

53C20 | Global Riemannian geometry, including pinching |