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Riemannian holonomy groups and calibrated geometry. (English) Zbl 1016.53041
Gross, Mark (ed.) et al., Calabi-Yau manifolds and related geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001. Berlin: Springer. Universitext. 1-68, 227-236 (2003).
These notes are an expanded version of a course of eight lectures given by the author at the summer school in Nordfjordeid, Norway, in June 2001. In accordance with the title of the course, and retaining the original basic format of the lectures, the author has divided this treatise into two principal parts. Out of twelve sections, altogether the first six sections discuss Riemannian holonomy groups, focussing in particular on Kähler and Calabi-Yau manifolds, whereas the remaining six sections provide an introduction to calibrated geometry and calibrated submanifolds of Riemannian manifolds with special holonomy, again with a particular view to Calabi-Yau manifolds.
The holonomy group \(\text{Hol} (g)\) of a Riemann \(n\)-manifold \((M,g)\) is a Lie subgroup of \(SO(n)\), which measures the constant tensors on the manifold. For generic metrics, one has \(\text{Hol}(g) =SO(n)\). However, metrics with special holonomy, that is metrics with \(\text{Hol} (g)\neq SO(n)\), are very interesting and important, too, as they include for instance, Kähler metrics with holonomy \(U(m)\), Calabi-Yau manifolds with holonomy \(SU(m)\), and hyper-Kähler manifolds with holonomy \(Sp(m)\).
The basics on Riemannian holonomy groups are discussed in §§1-6, which are arranged as follows. §1. General introduction; §2. Introduction to holonomy groups; §3. Berger’s classification of holonomy groups; §4. Kähler geometry and holonomy; §5 The Calabi conjecture; §6. The exceptional holonomy groups. Calibrated submanifolds are a class of \(k\)-dimensional submanifolds \(N\) of a Riemannian manifold \((M,g)\) defined using a closed \(k\)-form on \(M\) called a calibration. Manifolds with special holonomy generally come with one or more natural calibrations which then define particular classes of submanifolds in \(M\), e.g., the class of special Lagrangian submanifolds (SL \(m\)-folds) of Calabi-Yau manifolds. The geometry of SL \(m\)-folds in \(\mathbb{C}^m\) and Calabi-Yau \(m\)-folds is the main topic depicted in §§7-11. These sections contain the following material: §7. Introduction to calibrated geometry; §8. Calibrated submanifolds in \(\mathbb{R}^n\); §9. Constructions of SL \(m\)-folds in \(\mathbb{C}^m\); §10. Compact calibrated submanifolds; §11. Singularities of special Lagrangian \(m\)-folds.
Especially in this second part of the notes, the author has emphasized his own favours and research interests. This leads the reader to the forefront of current research in the field, and equally to an approach that could be very useful with regard to tackling the general mirror symmetry problem.
The final section (§12) briefly surveys the current research on the so-called SYZ Conjecture, which describes the mirror symmetry phenomenon between Calabi-Yau threefolds via special Lagrangian fibrations, and so leads over to the following lecture notes by M. Gross, entitled “Calabi-Yau manifolds and Mirror symmetry”, which form the second part of these summer school proceedings (see Zbl 1014.14019).
For the entire collection see [Zbl 1001.00028].

53C38 Calibrations and calibrated geometries
53C29 Issues of holonomy in differential geometry
53D12 Lagrangian submanifolds; Maslov index