The only other textbook on calibrated geometry that the reviewer is aware of is by R. Harvey [Spinors and calibrations, Academic Press, Boston, MA, 1990]. That book is an excellent introduction to the subject at the linear algebra level, including some of the earliest explicit examples in Euclidean spaces. However, it is now very much out of date, given the veritable explosion of research in this area since the early 1990's, motivated by the interest in calibrated submanifolds by physicists working in string theory and M-theory. There are now a great many more explicit examples of calibrated submanifolds known, both in Euclidean spaces and in complete non-compact manifolds, and even some in compact manifolds. In addition, a lot of analytic results concerning calibrated submanifolds have been obtained recently, many of them by the author and his students, concerning calibrated submanifolds with conical singularities, asymptotically conical calibrated submanifolds, and the deformation theories and desingularizations of both.

The book presupposes a working knowledge of differential geometry, and although it discusses Riemannian metrics, Riemannian curvature, de Rham cohomology, and Hodge theory, it does so essentially to fix notations, and does not claim to give an exhaustive treatment of those topics. In addition, the first chapter gives a quick introduction (or review) of topics in analysis that are used later. Specifically, the discussion concerns elliptic partial differential equations on manifolds and the elliptic regularity results associated to such equations. This includes the definitions and properties of the appropriate Banach spaces in which one studies such problems, notably Sobolev spaces and Hölder spaces. Previous exposure to such topics would be helpful to the reader but not strictly essential. These ideas were used heavily in the second half of [D. D. Joyce, op. cit.], but in the present book they appear mainly in the proof of the Calabi conjecture, and as such readers who are not interested in the proof of that theorem can safely avoid the analysis. It should be stressed that a great many of the results described in Chapter 8, on special Lagrangian geometry, and in the chapters on exceptional holonomy groups and their calibrated submanifolds, are proved using these elliptic PDE techniques. However, in the present book, these results are explained but proofs are, by necessity of space, omitted.

Chapter 2 discusses connections, their curvature, and their holonomy for connections on both vector bundles and principal bundles. The relationship between the two viewpoints is also carefully explained. The discussion then specializes to connections on the tangent bundle of a manifold, where a new invariant, the torsion, can be defined.

In Chapter 3 the author quickly reviews Riemannian geometry, and
defines the Riemannian holonomy Hol_{g}(M) of a Riemannian
manifold (M,g) to be the holonomy group of the Levi-Civita connection,
the unique torsion-free connection on TM compatible with the metric.
This is followed by a discussion of the classification theorem (of
Berger) of the possible Riemannian holonomy groups of an irreducible,
nonsymmetric, simply connected Riemannian manifold. Berger's
classification is not proved here (see [S. M. Salamon, op. cit.] for a
proof), but the idea of the proof is clearly explained. This chapter
ends with a look at the relationship between Hol_{g}(M) and
spinors.

Chapter 4 is a short introduction to calibrated geometry in general, including the proof of the fundamental theorem of calibrated geometry of Harvey and Lawson. There is also an interesting section about classification of possible constant coefficient calibrations in Euclidean space, which is material that is not very well known. The chapter concludes with a discussion of calibrated currents, in the sense of geometric measure theory, and stresses that in order to understand the possible singularities that can occur in calibrated submanifolds, which is important for applications to physics, it is natural to begin by studying conical singularities.

The next three chapters, 5, 6, and 7, are about Riemannian manifolds with holonomy U(m) or SU(m), which are Kahler and Calabi-Yau manifolds. There is a thorough treatment of the differential geometry of Kahler manifolds, including the curvature properties of such, and the extremely important d d-bar-lemma. The Hodge theorem for Kahler manifolds is stated, without proof, and its consequences are examined. There is a very detailed and essentially complete proof of the Calabi conjecture, which aside from Yau's original paper, is difficult to find in the literature (although [T. Aubin, Some nonlinear problems in Riemannian geometry, Springer, Berlin, 1998; G. Tian, Canonical metrics in Kahler geometry, Birkhäuser, Basel, 2000] are two other sources for this.) The proof is broken down into several steps, and the technical analytic results are separated into four key theorems, which makes the idea of the proof easy to understand, even if the reader decides to skip the more technical aspects. There is also some talk about the algebro-geometric techniques used to study Kahler and Calabi-Yau manifolds, especially about methods of resolving singularities which are useful for proving existence results for special holonomy metrics in glueing constructions.

Chapter 8 is essentially a survey of the plethora of results on
special Lagrangian submanifolds (SL m-folds) of Calabi-Yau manifolds
which have appeared in the last decade. It should be required reading
for anyone looking to begin research in this area. Since this section
tries to mention as many of the known results as possible, proofs are
omitted. However, extensive references are provided for all results
which are mentioned. Some of the topics discussed in this chapter
include: SL m-folds with large symmetry groups; evolution equations for
constructing SL m-folds; ruled SL m-folds; relation to integrable
systems; explicit smooth and singular examples in **C**^{n};
SL cones and asymptotically conical SL m-folds; the deformation theory
of SL m-folds; SL m-folds with isolated conical singularities; and the
desingularization of such. A first reading of this chapter will
probably make the reader feel exhausted. At this time, there is not a
more complete discussion of special Lagrangian geometry that is
available anywhere.

Let us skip Chapter 9 for the moment. Chapter 10 is a brief treatment of hyper-Kahler and quaternionic-Kahler manifolds, the two remaining ``non-exceptional'' holonomies on Berger's list. The simplest example, the K3 surface, is examined in detail. Fewer examples are known of these manifolds, and it is still an open problem whether or not there exist any compact quaternionic-Kahler manifolds with positive scalar curvature which are not symmetric spaces.

The final two chapters close the book with a discussion of the two
exceptional holonomies, G_{2} and Spin(7), and their associated
calibrated submanifolds: associative, coassociative, and Cayley
submanifolds. The similarities with and the differences from the
special Lagrangian case are stressed. The treatment here is briefer,
partly due to the fact that these geometries have been studied less,
perhaps because the associated PDE's are now systems rather than scalar
equations, and hence more difficult to analyze.

Finally we come to Chapter 9: Mirror symmetry and the SYZ
conjecture. Although the author himself warns the reader that "this is
the most flawed and unsatisfactory chapter in the book", it is the
prediction of this reviewer that it will be the most read chapter, and
it is certainly the most entertaining. The main reason that there has
been a huge increase in attention to metrics with special holonomy and
their calibrated submanifolds is due to the role they are expected to
play in superstring theory and M-theory. Modern physical theories need,
for their formulation, a compact Riemannian manifold M of dimension 6,
7, or 8, admitting a parallel spinor (which arises from considerations
of supersymmetry). This means that M must have holonomy SU(3)
(Calabi-Yau), or G_{2}, or Spin(7). Furthermore, there are
additional special objects of interest called instantons and branes,
and after translation to mathematical language, these turn out to be
calibrated submanifolds, together with a special connection on a bundle
over them. The idea of mirror symmetry of Calabi-Yau 3-folds (and
potentially of exceptional holonomy manifolds as well) was introduced
in the early 1990's by string theorists. While the actual statement of
mirror symmetry in precise mathematical terms is continually evolving,
there have been many rigorous theorems proved and progress continues to
be made. Two useful textbooks on mirror symmetry are [D. A. Cox and S.
H. Katz, Mirror symmetry and algebraic geometry, Amer. Math. Soc.,
Providence, RI, 1999; K. Hori et al., Mirror symmetry, Amer. Math.
Soc., Providence, RI, 2003].

In Chapter 9 the author begins with a mathematician's general introduction to string theory, expressing the frustration that many of us have in trying to understand the physics and the physicists. In their defence, they often turn out to be right, or at least approximately right, after we have succeeded in deciphering what they really intended to say. Regardless of the eventual outcome for string theory as a successful theory of physics, it is indisputable that it has led to many startling and amazing advances in differential geometry, which might have been left undiscovered without the physicists to point us in the right direction. The reviewer shares the opinion of the author, who says "underlying string theory is some very major area of not yet understood mathematics, which one could call 'quantum geometry', that string theorists grasp at a heuristic, intuitive level". He goes on to briefly describe the two main formulations of mirror symmetry in mathematical terms: the homological mirror symmetry programme begun by Kontsevich, and the SYZ conjecture of Strominger-Yau-Zaslow. The author discusses what is known in both cases, what was thought to be true but has since been proven cannot possibly hold as originally expected, and what modifications should probably be made so that some form of precise statement has a chance to be true. These final remarks are by the author's own admission somewhat speculative, but certainly no more so than what is usual concerning this topic.

Overall, the present book is an excellent introduction to Riemannian holonomy groups and calibrated geometry. The author writes in a very clear, somewhat informal style, which makes reading the book enjoyable. This is an impressive achievement, considering the technical complexity of some of the topics discussed, notably the proof of the Calabi conjecture. The book is a much needed and welcome addition to the literature on differential geometry that is certain to be the standard reference on these topics for many years to come.

**Reviewed by Spiro Karigiannis.** Back