This impressive book is in two parts: the first is an advanced textbook on Riemannian geometry and holonomy; the second is a research monograph on the construction of compact examples in dimensions 7 and 8 with the exceptional holonomies G2 and Spin(7). The intended audience consists of students and researchers in differential geometry and theoretical physicists in string theory, but the approach is mathematical throughout.
If (M,g) is a Riemannian manifold, then its holonomy group is defined to be the group of linear transformations of the tangent space TxM induced by parallel translation around loops based at x. This captures important curvature information about the manifold; for example, restricting the holonomy group often forces g to be an Einstein metric.
In 1955, M. Berger proved an important classification theorem
[Bull. Soc. Math. France 83 (1955), 279-330].
The modern version of his result says that manifolds that are not
locally Riemannian products are either locally isometric to a
symmetric space, classified by Cartan, or their (restricted)
holonomy group is one of
(vi) G2 or
Case (i) is the generic situation, whilst the other holonomy groups imply the existence of special geometric structures on M. For (ii) and (iii) we have Kahler geometry; in case (iii) the metric is also Ricci-flat. Cases (iv) and (v) are hyper-Kahler and quaternionic Kahler geometry; (vi) and (vii) are the exceptional cases; they can occur only in dimensions 7 and 8 respectively, and are both Ricci-flat.
Finding compact examples of manifolds with holonomy in classes (iii)-(vii) is no easy task. For Ricci-flat Kahler manifolds (class (iii)), many manifolds are given by S.-T. Yau's solution of the Calabi conjecture [S.-T. Yau, Comm. Pure Appl. Math. 31 (1978), 339-411]: any complex algebraic manifold with c1=0 admits such a metric. In consequence these spaces are often referred to as Calabi-Yau manifolds, particularly in the physics literature where these structures have much interest in relation to string theory. Note, however, that Yau's result only gives existence of such metrics; explicit compact examples have a tendency to be flat.
Hyper-Kahler manifolds are particular examples of Calabi-Yau manifolds, and it was not long after Yau's work that A. Beauville was able to give a limited number of compact examples [J. Diff. Geom. 18 (1983), 755-782].
The remaining cases (v)-(vii) prove to be more stubborn. It was therefore a major breakthrough when the author of this book constructed compact manifolds with exceptional holonomy in the mid 1990's. The impact of this work is witnessed by the fact that manifolds with exceptional holonomy have acquired the name Joyce manifolds in the physics literature.
As mentioned above, the first half of this book is a course in Riemannian geometry and holonomy. The emphasis is on discussing the Ricci-flat geometries that occur, rather than deriving the holonomy classification. There is detailed information about the use of analysis for showing existence of solutions to nonlinear elliptic partial differential equations on manifolds, one highlight being an account of the proof of the Calabi conjecture. Complete metrics with special holonomy on non-compact manifolds are also discussed in detail. A particularly important family of examples is constructed on resolutions of Cn/G where G is a finite subgroup of SU(n); motivating examples are the Eguchi-Hanson metric on T*CP1 and P. B. Kronheimer's asymptotically Euclidean metrics in dimension four [J. Diff. Geom. 29 (1989), 665-683]. There is also some mathematical material on the relation of these topics to string theory and the mirror conjecture, with valuable pointers to the literature.
The second half of the book concentrates on the construction of compact examples with exceptional holonomy. The basic idea is as follows: take a torus T of the appropriate dimension; form a finite quotient where the singularities are modelled on Cn/G x Rm; resolve the singularities and glue in the earlier constructed special holonomy metrics on these resolutions to get a metric that fails to have the right holonomy group, but where the failure is controlled; finally, show that this metric can be deformed to a smooth metric with exceptional holonomy. In dimension four, this gives the so-called Kummer construction of K3 surfaces.
The constructions of metrics with exceptional holonomy given in this book are more general than those in Joyce's original papers. This is made possible by allowing and finding new types of metrics with special holonomy on resolutions of Cn/G: the group G is now allowed to act non-freely on the unit sphere in Cn and the metrics are only required to be Quasi Asymptotically Locally Euclidean. This gives rise to new examples exhibiting a richer variety of topological properties, but also makes the proofs technically more complicated. However, the author takes great pains to describe the general outline and ideas and to guide the reader through the difficulties. The book contains many explicit examples of the construction and takes time to show how Betti numbers may be found in many of these cases. The cases of G2 and Spin(7) are treated separately; technical differences in the behaviour of these geometries affect many details of the proofs and constructions. However, one feature that should be emphasised is that the author studies both geometries via the parallel differential forms that they carry, rather than fixing on the metric itself. The final chapter ends with a variation on the above construction in the Spin(7) case, where the starting point is a Calabi-Yau four-fold instead of a torus. Interesting examples starting from hypersurfaces in weighted projective spaces are given.
The book is well written, with many references to the literature, helpful guidance to the reader, discussion of relationships to physics and pointers to directions for future research. The constructed manifolds provide valuable new examples of solutions of the Einstein equations and are already playing a role in the physics literature of string theory. This book is an important reference in the study of Ricci-flat geometry.
Reviewed by Andrew Swann. Back