The main topic treated in this book is the classical subject of Riemannian holonomy groups, together with a new theme of so-called calibrated submanifolds. The book is designed for graduate students of mathematics (and theoretical physics) and it introduces the reader to modern topics important for understanding new mathematics emerging from string theory.
The first part of the book is devoted to an explanation of holonomy groups in a Riemannian setting (including a short review of background material, connections, curvature, holonomy groups, G-structures, Riemannian symmetric spaces, the classification of Riemannian holonomy groups and Kähler manifolds). Special chapters are devoted to the Calabi conjecture and its proof, Calabi-Yau manifolds and their deformations, hyperkähler and quaternionic Kähler manifolds and the exceptional holonomy groups. Most of this material is related to a previous important monograph of the author, ‘Compact manifolds with special holonomy, Oxford University Press, 2000’. Completely new material is contained in chapters on calibrated submanifolds, special Lagrangian geometry, mirror symmetry and the Strominger-Yau-Zaslow conjecture. The book ends with a chapter on special types of calibrated submanifolds of manifolds with exceptional holonomy. The book is very well-written, it contains a wealth of important material and it should be on the shelf of everybody interested in the topic.