Masterclass on 'Riemannian holonomy groups', given at the Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark, August 2018

Overview
Let (X,g) be a connected, simply-connected Riemannian n-manifold. The holonomy group Hol(g) is a subgroup of SO(n) which measures the tensors T on X constant under the Levi-Civita connection ∇ of g. Berger (1955) showed that if g is irreducible and nonsymmetric then Hol(g) is one of:
(i) SO(n) (the boring case, for generic g);
(ii) U(m) for n=2m (Kähler metrics);
(iii) SU(m) for n=2m (Calabi-Yau metrics);
(iv) Sp(m) for n=4m (hyperkähler metrics);
(v) Sp(m)Sp(1) for n=4m (quaternionic Kähler metrics);
(vi) G2 for n=7; and (vii) Spin(7) for n=8 (the exceptional holonomy groups).
Each of classes (ii)-(vii) have their own interesting geometry.
We will review the theory of holonomy groups and Berger's classification, and discuss Riemannian manifolds in classes (ii)-(vii), especially compact manifolds. If time allows we will also discuss calibrated submanifolds, distinguished classes of minimal submanifolds in Riemannian manifolds with special holonomy.

Reading:
D. Joyce, 'Riemannian holonomy groups and calibrated geometry', 303 pages, Oxford Graduate Texts in Mathematics 12, OUP, 2007.
-- Buy it on on the Web from OUP or Amazon.

D. Joyce, 'Compact manifolds with special holonomy', 436 pages, Oxford Mathematical Monographs series, OUP, 2000.
-- Buy it on on the Web from OUP or Amazon.

PDF files to download:
Here are some scruffy handwritten lecture notes.

Notes for lecture 1

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Notes for lecture 15