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Riemannian holonomy groups and calibrated geometry. (English) Zbl 1200.53003
Oxford Graduate Texts in Mathematics 12. Oxford: Oxford University Press (ISBN 978-0-19-921559-1/pbk; 978-0-19-921560-7/hbk). ix, 303 p. (2007).
This book, consisting of 12 chapters, is concerned with Riemannian holonomy groups and calibrated geometry. As is well known, calibrated submanifolds form a natural class of subobjects of a Riemannian manifold \(\left( M,g\right) \), and they are significant in string theory as branes. The book is addressed to graduates and researchers in differential geometry as well as string theorists. The prerequisites are a good knowledge of topology, differential geometry and Lie groups.
A little more than half of the book is a revised version of the author’s [Compact manifolds with special holonomy. Oxford: Oxford University Press (2000; Zbl 1027.53052)], whose main objective was to explain an extended research project on compact manifolds with holonomy \(G_{2}\) and \(\mathrm{Spin}\left( 7\right)\). Chapters 1, 2, 3, 5, 6, 7 and 10 of this book stand for rewritten versions of Chapters 1–7 of [Zbl 1027.53052] in order. They form the core of the Riemannian holonomy material. Chapter 11 of this book is concerned with the exceptional holonomy groups, summarizing Chapters 10–15 of [Zbl 1027.53052] and subsequent developments. Chapters 4, 8, 9 and 12 of this book are completely new, dealing with calibrated geometry.
Discussing minimal and calibrated submanifolds, Chapter 4 gives an account of the relation between calibrated geometry and holonomy groups, which is the central to this book. Therein the author considers the problem of classifying constant calibration in \(\mathbb{R}^{n}\) as well as geometric measure theory.
Special Lagrangian submanifolds were invented in [R. Harvey and H. B. Lawson, Acta Math. 148, 47–157 (1982; Zbl 0584.53021)]. In 1996, Strominger, Yau and Zaslow [A. Strominger et al., Nucl. Phys., B 479, No. 1–2, 243–259 (1996; Zbl 0896.14024)] announced the famous SYZ conjecture concerning mirror symmetry of Calabi-Yau-\(3\)-folds in terms of fibrations by special Lagrangian \(3\)-folds including singular fibers, which inspired considerable interest in special Lagrangian geometry. Therefore, Chapter 8 is concerned with special Lagrangian geometry in \(\mathbb{C}^{m}\), covering the basic theory and constructions of examples.
Mirror symmetry is an esoteric concept concerning the relationship between pairs of Calabi-Yau \(3\)-folds. It was coined by physicists in the 1990s in the midst of endeavor to quantize gravity by modelling particles not as points but as \(1\)-dimensional loops of string. Chapter 9 is concerned with mirror symmetry and the SYZ conjecture. As the author admits, this chapter is “the most flawed and unsatisfactory chapter in the book”, because mirror symmetry is only “a swamp of complicated, continuously evolving conjectures, which are slowly being turned into theorems”. The chapter begins with an introduction to string theory and mirror symmetry, describing the first form of mirror symmetry, namely, the occurrence of Calabi-Yau-\(3\)-folds in pairs \(X,\widehat{X}\) with Hodge numbers abiding by
\[ \begin{aligned} h^{p,q}\left( X\right) & =h^{3-p,q}\left( \widehat{X}\right) \\ & \text{where, the author wrongly writes }\\ h^{p,q}\left( X\right) & =h^{q,p}\left( \widehat{X}\right) \text{ consistently} \end{aligned} \]
and the prediction of numbers of rational curves on \(\widehat{X}\) in terms of the complex structure moduli space of its mirror \(X\). The author then moves to the homological mirror symmetry and the SYZ conjecture.
Chapter 12 applies calibrations and calibrated submanifolds to manifolds with exceptional holonomy, namely, calibrated submanifolds in \(7\)-manifolds with holonomy \(G_{2}\), called associative \(3\)-folds and coassociative \(4\)-folds, and calibrated submanifolds in \(8\)-manifolds with holonomy \(\mathrm{Spin}\left( 7\right) \) called Cayley \(4\)-folds.
All in all, the book is well written and recommendable.

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C29 Issues of holonomy in differential geometry
53C80 Applications of global differential geometry to the sciences