Einstein manifolds
type: Item
instance of: scholarly article
title: Einstein manifolds
zbMATH Open document ID: 1147.53001
author: Arthur L. Besse
published in: Classics in Mathematics
publication date: 2007-12-12T00:00:00Z
review text: The monograph under review is a reprint of the 1987 Edition [Berlin etc.: Springer (1987; Zbl 0613.53001)]. The author decided to write a monograph on Einstein manifold in September 1979, at a symposium on Einstein manifolds held at Espalion, France, because most basic questions were still open, but good progress had been made, due in particular to the solution of Calabi's conjecture by T. Aubin and S. T. Yau and to N. Koiso's results on the moduli space. Einstein manifolds are not only interesting in themselves, but are also related to many important topics of Riemannian geometry, for example, Riemannian submersions, homogeneous Riemannian spaces, Riemannian functionals and their critical points, Yang-Mills theory, holonomy groups, algebraic geometry via \(K3\) surfaces. The present book is intended to be a complete reference book. It is structured into 17 chapters, an appendix on Sobolev spaces and elliptic operators and an addendum of results on five topics, too recent to include them into the main text. The fact that four of these topics are related to Kähler-Einstein manifolds is indicative of the permanently-developing links between complex and Riemannian geometry. The various chapters are self-contained. The introduction starts with brief definitions and motivation for the study of Einstein manifolds. Interesting standard examples of Einstein manifolds are given. Let \((M,g)\) be a Riemannian manifold. The curvature of \(g\) determines a symmetric bilinear form on the tangent space \(T_pM\), \(p\in M\), namely, the Ricci tensor \(r\). The metric \(g\) is said to be Einstein when \(r\) is a constant multiple of \(g\). Chapters 1--3 deal with basic material on Riemannian and pseudo-Riemannian manifolds, Kähler manifolds and relativity. Riemannian functionals are studied in Chapter 4. Since the work of Lagrange it is well-known that the equations of classical mechanics can be obtained as solutions of a variational problem, using a suitable functional. Then, a na…
reviewed by: Adela-Gabriela Mihai
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5219762
zbMATH Keywords: Codazzi tensor
zbMATH Keywords: algebraic curvature tensors
zbMATH Keywords: conformal changes
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: Einstein manifolds
zbMATH Keywords: Riemannian submersions
zbMATH Keywords: warped products
zbMATH Keywords: holonomy groups
zbMATH Keywords: self-duality
zbMATH Keywords: Calabi's conjecture.
MaRDI profile type: Publication
description: scientific article