Holonomy and special metrics

Course Overview

This graduate-level course offers an introduction to principal bundles, connections, and holonomy. We begin with the foundations of principal bundles and G-structures, and introduce the notions of curvature and torsion.

We then explore Riemannian products, homogeneous and symmetric spaces, and culminate with Berger’s classification of Riemannian holonomy groups, followed by a study of the special geometries appearing on Berger’s list. The course concludes with the deformation theory of these structures and a brief introduction to spinors and their relation to special holonomy.

Prerequisites: Differential geometry. Familiarity with Lie groups and basic Riemannian geometry is desirable.

Notes and references

I will mainly draw from

• S. Kobayashi and K. Nomizu Foundations of Differential Geometry I & II,


• D. Joyce, Complex Analytic and Differential Geometry,


• S. Salamon, Riemannian geometry and holonomy groups.

I will (try to) keep up-to-date lecture notes as the term progresses.

Syllabus (tentative)

0. Overview


1. Principal bundles and G-structures


2. Flatnesss, curvature and torsion


3. Holonomy and the holonomy principle


4. Products and the deRham's splitting principle


5. Homogeneous and symmetric spaces


6. Berger holonomy classification


7. Kähler metrics and the Calabi-Yau theorem


8. HyperKähler, Quaternionic Kähler and twistor spaces


9. The exceptional cases: G2 and Spin(7)


10. The deformation problem


11. Spinors


12. Functionals and critical points

Exercises

There will be a list of exercises published weekly. You are encouraged to work through them and discuss them with your colleagues (e.g. on the WeChat group). No official soluion will be provided.