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Einstein manifolds, by Arthur L. Besse. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987, xii + 510 pp., $89.00. ISBN 3-54015279-2 Even when speaking to a group of differential geometers one cannot safely assume that everyone knows what an Einstein metric is. Why then would A. L. Besse write a 500 page book on the subject? With characteristic frankness he addresses that question in §B of the excellent nineteen page introduction (an impressive mathematical essay in its own right).

local criteria as to when there exists a local coordinate system with respect to which g = Y^idx1)2. Namely, this is possible if and only if R = 0, in which case we say that g is flat. Being a tensor, Rijki = 0 with respect to some coordinate system if and only if they are zero with respect to any local coordinate system. The big problem with Riemann's curvature tensor is its enormous number of components R^ki- Some relief is provided by the familiar symmetries Rtjki = -Rjiki = –Rijik – Rkiij (which mean that at a point p in M, R is a symmetric linear transformation on the space of skew-symmetric 2-tensors at p) together with the Bianchi identities RijM + Rujk + Rikij = 0.

This still leaves n2(n - 1)(« +1)/12 components (see §1.108). When « is 2 the one component is i*i2i2> which is related to the Gaussian curvature K by K = Run/ dctg, where detg = gug22 - g\2- That's encouraging, but when n = 3 there are already six components, and when n = 4 there are 20.

Contraction of a tensor (taking its trace) produces a tensor with two fewer indices. The only nontrivial way (up to sign) to contract R is by its first and third indices, which produces the Ricci tensor r, in local coordinates rij = YlgklRkUj – r)i-> a symmetric 2-tensor, thus of the same type as the metric g itself (here gij is the inverse matrix of gy).

given metric g in order to deform it to an Einstein metric. Considering the heat type equation dg/dt = 2pg/n - 2rg, where p is a constant equal to the average of the scalar curvature s of g, he proves that if g is a metric on the compact connected 3-manifold M such that rg is positive definite at every point, then the solution gt of the above equation goes to an Einstein metric as t – oo. About ten years earlier, T. Aubin had proved that if rg > 0 · at every point of M, and is positive definite at some point, then there exists a Riemannian metric on M with positive definite Ricci curvature everywhere. So the approach gets intriguingly close, but... still no cigar. Einstein Manifolds is not one of those expository surveys which tells you about a great variety of results, but sends you to the research literature for any details, proofs or real understanding. Topics are covered in detail with proofs and commentary, what might be called annotated proofs. Five pages of Chapter 5, §G, are devoted to Hamilton's proof. Any compact surface admits a metric of constant Gaussian curvature. If a compact «-manifold admits a metric of constant sectional curvature, then its universal cover is diffeomorphic to R" or the «-sphere. Thus S2 x Sl does not admit a metric of constant sectional curvature. Do you know where to find a good brief account of compact 3-manifolds of constant negative curvature? Try Chapter 6, §C. One point made is that constant sectional curvature is not the correct generalization for higher dimensions of constant Gaussian curvature for 2-manifolds. It is too strong. Einstein metrics may be the correct generalization. In dimension 4 we find Einstein metrics which do not have constant sectional curvature, for example the complex projective plane CP2 with its Fubini-Study metric (see §9.77). The sectional curvatures of this metric fill out the interval [1,4] at every point. In addition, this manifold is complex and this metric is Kaehlerian. It is an example of the important class of Kaehler-Einstein spaces (see Chapter 2).

The Ricci tensor of Kaehler spaces gives rise to a closed type (1,1) form, the Ricci form of course, which represents the first Chern class of the manifold. This topological property of the Ricci tensor on a Kaehler manifold is very important. Let g and h be Kaehler metrics on the complex manifold M, and let pg and Ph denote their Ricci forms. Then their volume forms are related by pg = fju^, where > 0 everywhere, and the Ricci forms satisfy (1) pg = ph-id'd"\ogf The famous Calabi problem is, given g and > 0, does there exists a Kaehler metric g on M whose Ricci form is given by (1)? The answer (due to S. T. Yau and T. Aubin) is yes if M is compact (see §2.101 and Chapter 11). Solution of this problem involved deep and significant research into PDE of Monge-Ampère type. Thus it is that the study of Einstein metrics leads into significant problems, this time in PDE. The Einstein space approach to the Poincaré conjecture is based on the supposition that there are no topological restrictions imposed by the existence of an Einstein metric on a compact simply connected 3-manifold (except that the sign of the Einstein constant must be positive).

topological restrictions are known. The result for 4-manifolds, due to N. Hitchin and J. Thorpe (see §6.35), is that the Euler characteristic % and signature T of a compact oriented 4-dimensional Einstein manifold must satisfy the inequality x > §M« This result follows from the important role played by Einstein metrics in the wonderful world of 4-dimensional Riemannian geometry (see §6D), where the Hodge * operator creates the beautiful properties of self-duality, and where we meet the marvelous Penrose twistor spaces (see Chapter 13).

This material, by the way, is given a masterly presentation in §§1G and 1H and in Chapter 13.

The book was written to be read piecemeal. (Who really reads mathematics any other way?) The author does not hesitate to recapitulate on p. 370 what he wrote on p. 51 if he thinks it will help the reader who has just jumped into Chapter 13.

Needless to say, this kind of style is essential for a really useful reference book. Their name suggests that Einstein metrics have something to do with general relativity. Most of the relationship is analogy, but analogy is important and the author includes Chapter 3 on Relativity to explain it all. Other physics in which Einstein spaces play a more direct role includes KaluzaKlein theory (see §9.63) and Yang-Mills theory (see §9.35). These arise in the midst of a general discourse on Riemannian submersions (Chapter 9) applied to special homogeneous Riemannian manifolds (Chapter 7).