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Compact manifolds with special holonomy. (English) Zbl 1027.53052
Oxford Mathematical Monographs. Oxford: Oxford University Press. xii, 436 p. (2000).
This book deals extensively with Riemannian manifolds having a special holonomy group, that is \(n\)-dimensional (oriented) Riemannian manifolds whose holonomy group is different from \(\text{SO}(n)\). In particular the holonomy groups \(\text{SU}(m)\), \(\text{Sp}(n)\), \(G_2\) and \(\text{Spin}(7)\) of Ricci-flat metrics are considered. At present this subject is attracting an increasing interest from a large audience and the book is intended both for mathematicians working in differential geometry and also for physicists working in String Theory, though the point of view is strictly mathematical. The author, which in the mid 1990’s produced the first examples of compact manifolds with exceptional holonomy \(G_2\) and \(\text{Spin}(7)\), is a very well acknowledged expert in this subject.
The book consists of two parts: a graduate textbook on Riemannian geometry and a research monograph on the exceptional holonomy groups \(G_2\) and \(\text{Spin}(7)\).
The first part is a very effective introduction to basic notions and results of modern differential geometry, such as connections, curvature, holonomy groups and their classification in the Riemannian case, complex and Kähler manifolds. Very recent results and topics, such as the proof of the Calabi conjecture and Calabi-Yau manifolds, hyper-Kähler manifolds, asymptotically locally Euclidean metrics and their generalizations are exhaustively treated. A very useful background material concerning the analysis of elliptic operators on manifolds and an introduction to complex algebraic geometry, which is needed to fully understand the statements and detailed proofs, is also presented. A very quick but suggestive introduction to String Theory, the branch of theoretical physics that involves Calabi-Yau manifolds, and to Mirror symmetry, a set of ideas and conjectures about Calabi-Yau manifolds which emerged from string theory, is given as well.
The second part places particular emphasis on constructions and existence theorems for compact manifolds with special holonomy. By using complex geometry and Calabi conjecture methods the author defines constructions of manifolds with exceptional holonomy more elaborated and powerful then those in his original papers and obtains many more examples. The cases of \(G_2\) and \(\text{Spin}(7)\) are extensively treated and several interesting examples are given in much detail. Aside from some technical differences, the construction of compact examples goes as follows. One begins with a torus \(T^k, k=7\) or \(8\), equipped with a flat \(G\)-structure of the considered type (\(G=G_2\) if \(k=7\), \(G=\text{Spin(7)}\) if \(k=8\)) and a finite group \(\Gamma\) of automorphisms of \(T^k\) preserving the \(G\)-structure. Then one resolves the singularities of \(T^k/\Gamma\) to get a compact \(k\)-manifold \(M\), and defines a 1-parameter family of \(G\)-structures on \(M\) depending on a real parameter \(t \in (0,\epsilon)\), such as the torsion of these structures is very ”little”. Then it is shown, using analysis, that when the parameter \(t\) is sufficiently small one can deform the \(G\)-structure depending on it to a nearby torsion-free \(G\)-structure on \(M\). If \(\pi_1(M)\) is finite then the holonomy group of such a \(G\)-structure is exactly \(G\), as required. At the end of the book there is given a description of a variation of this method for constructing compact 8-manifolds with holonomy \(\text{Spin}(7)\), in which one starts not with a torus \(T^8\) but with a Calabi-Yau 4-orbifold with a finite number of isolated singular points.
This book is highly recommended for people who are interested in the very recent developments of differential geometry and its relationships with present research in theoretical physics.

53C29 Issues of holonomy in differential geometry
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C27 Spin and Spin\({}^c\) geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry