**Featured Review.**

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 G_{2} 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
*T _{x}M* induced by parallel translation around loops
based at

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

(i) SO(*n*),

(ii) U(*n*),

(iii) SU(*n*),

(iv) Sp(*n*),

(v) Sp(*n*)Sp(1),

(vi) G_{2} or

(vii) Spin(7).

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
*c*_{1}=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 **C**^{n}/*G*
where *G* is a finite subgroup of SU(*n*);
motivating examples are the Eguchi-Hanson metric on
*T ^{*}*

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
**C**^{n}/*G* x **R**^{m};
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 **C**^{n}/*G*: the
group *G* is now allowed to act non-freely on the unit sphere in
**C**^{n} 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 G_{2} 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