The main subject of this book is a study of compact Riemannian manifolds with special holonomy groups: particular attention is devoted to the holonomy groups SU(m), Sp(m), G2 and Spin(7) (the corresponding metrics are Ricci-flat in all these cases). Examples of such manifolds are hard to find, but the reader can learn many important constructions and examples of such manifolds here.

The book has two parts. The first reviews the basic tools needed for later constructions. It is a very nice introduction to the field, covering many important topics (connections, their curvatures and holonomy groups, both in the principal bundle and vector bundle cases; connections on tangent bundles and their torsions; G-structures on manifolds; Riemannian holonomy groups and their classification; Kähler manifolds and their curvature, exterior algebra of Kähler manifolds; some important facts about complex algebraic varieties, line bundles and divisors; the Calabi-Yau conjecture and its full proof; Calabi-Yau manifolds, orbifolds and their resolutions; hyperkähler manifolds and other quaternionic geometries). Some parts of this material will be useful when preparing courses for graduate or postgraduate students. The second part contains a wealth of new research material concerning constructions of compact manifolds with special (resp. exceptional) holonomy and computation of their Betti numbers. The SU(m) and Sp(m) holonomy are treated first, and manifolds with G2 holonomy and Spin(7) holonomy follow. The proofs here are involved and difficult, and are placed at the end of individual chapters so that one can postpone reading them until later.

The book is written in a very clear and understandable way, with careful explanation of the main ideas and many remarks and comments, and it includes systematic suggestions for further reading. The topic of the book has been inspired by the recent intensive interaction between theoretical physics and mathematics, and the book is really outstanding. It can be warmly recommended to mathematicians (in geometry and global analysis, in particular) as well as to physicists interested in string theory.

Reviewer:

vs