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

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.

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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