You are here

Einstein Manifolds

Arthur L. Besse
Publication Date: 
Number of Pages: 
Classics in Mathematics
[Reviewed by
Michael Berg
, on

For better or for worse (well, I claim it’s without question the former) it has come to me to review in this column a number of works by French mathematicians with a perverse sense of humor. Of course, it’s Bourbaki I refer to here, and I’ve even had occasion to  review a book about Bourbaki as a quasi-secretsociety . But Bourbaki’s humor certainly masks a serious purpose, i.e., uniformizing the teaching of higher mathematics in France and beyond, and the approach taken by these scholars is by no means accepted without challenge. Notably, even in France itself a rather vocal opposition can be identified, including very prominent figures indeed. I accordingly wonder if Marcel Berger would identify himself as a member of this faithful opposition.

The question arises because the “author” Arthur L. Besse, of Einstein Manifolds, the book under review, is also, like Bourbaki, a nom de plume of a group of scholars. This group has Berger at its center, and the last line or two of the biographical sketch, on the book’s fly leaf, suggests at least a mild desire to distance Besse from Bourbaki philosophically: “What do Bourbaki and Besse have in common? Hardly anything. Simply that both are mathematicians, of course, and share a taste for working in pleasant and quiet places.”

Well, we also learn that Besse-en-Chendesse was the locale for the first gathering, in 1975, of Marcel Berger and his students in a “Table Ronde” forum, to use the CNRS vernacular, to study manifolds all of whose geodesics are closed: hence the regal choice of Arthur as Christian name and Besse as family name. It is also mentioned that Besse’s first publication, Manifolds All of Whose Geodesics Are Closed, appeared in 1978, as number 93 in Springer’s Ergebnisse series.

Einstein Manifolds is accordingly described as Besse’s second book: “The experience [of writing the first book] was so enjoyable that Arthur did not stop there, and settled down to write another book. [¶] A preliminary workshop took place in another village even lovelier than the first: Espalion, in the South-West of France. The second book, Einstein Manifolds, was eventually published in 1987.” Parenthetically, to quote Einstein himself, “aber ist dass wirklich so?”: we find in the book’s bibliography the entry Géométrie Riemannienne en Dimension 4, dating to 1981: is this an “intermediate” book? Or maybe just a set of lecture notes? Well, it matters not: if anything, it’s a minor oversight. À propos, the book under review ends with six pages of errata, but, given that we’re dealing with over 500 pages of deep, difficult, and beautiful mathematics, that makes for a pretty good batting average.

Yes, there is no doubt that Einstein Manifolds is a magnificent work of mathematical scholarship. Here are, respectively, S. M. Salamon (MathSciNet, 1988) and T. J. Wilmore (Bull. LMS, 1987) on the present book(see the back-cover): “[T]he reader will have no difficulty in getting the feel of [the book’s] contents and in discovering excellent examples of all intersections of geometry with partial differential equations, topology, and Lie groups…” and “It seemed likely to anyone who read the previous [?] book by the same author, namely Manifolds All of Whose Geodesics Are Closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is fulfilled.”

And it really is a phenomenal achievement. A terrific and very readable Introduction of nearly twenty pages (in which the phrase, “The book we present here is intended to be a complete reference book,” is found on p. 5) is followed by sixteen chapters, an appendix on PDE (“Sobolev Spaces and Elliptic Operators”) and an addendum. Each chapter but two starts with an introduction of its own, the book itself starts with (a lot!) of “Basic Material” (including an extensive discussion of Kähler manifolds), relativity is covered early on, and then it’s a feast of Riemannian geometry.

It is worth noting that, on p. 6, the authors observe that “things look completely different when n = 4 and when n ≥ 5. Ridiculous as it may seem, when n ≥ 5 we do not [even] know the answer to the simple question, ‘Does every compact manifold carry at least one Einstein metric?’” One is reminded of Einstein’s famous remark that he sought to find out “God’s thoughts, the rest [being] just details:” our 4-dimensional universe is indeed very special, even mathematically, and this provides a profound philosophical justification for the excellent book under review.

All this having been said, it is fitting and proper that Einstein Manifolds, originally number 10 in the 3rd Ergebnisse series (in 1987), be reissued today in Springer’s Classics in Mathematics series. It is truly a seminal work on an incomparably fascinating and important subject.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.

Introduction.- Basic Material.- Basic Material: Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler-Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.- Appendix. Sobolev Spaces and Elliptic Operators.- Addendum.- Bibliography.- Notation Index.- Subject Index.