Holonomy and special metrics
This graduate-level course offers an introduction to manifolds with special holonomy in Riemannian geometry.
The course is roughly divided into three parts, (cf. syllabus below). We begin with the foundations of principal bundles and general G-structures, with the goal of understanding the significance of torsion and its relationship to the integrability problem for G-structures.
We then focus on Riemannian metrics, exploring the consequences of the holonomy principle, the de Rham Splitting Theorem, and the theory of homogeneous and symmetric Riemannian manifolds, culminating in a proof of the Berger Classification Theorem.
In the third and final part of the course, we will discuss various construction techniques for special holonomy metrics, in both compact and non-compact settings. Time permitting, I will also discuss the moduli problem for special holonomy metrics
The course is roughly divided into three parts, (cf. syllabus below). We begin with the foundations of principal bundles and general G-structures, with the goal of understanding the significance of torsion and its relationship to the integrability problem for G-structures.
We then focus on Riemannian metrics, exploring the consequences of the holonomy principle, the de Rham Splitting Theorem, and the theory of homogeneous and symmetric Riemannian manifolds, culminating in a proof of the Berger Classification Theorem.
In the third and final part of the course, we will discuss various construction techniques for special holonomy metrics, in both compact and non-compact settings. Time permitting, I will also discuss the moduli problem for special holonomy metrics
Lecturer
Date
30th March ~ 29th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 10:40 - 12:15 | Shuangqing | ZOOM 05 | 293 812 9202 | BIMSA |
| Monday | 13:30 - 15:05 | Shuangqing | ZOOM 05 | 293 812 9202 | BIMSA |
Website
Prerequisite
Differential geometry. Familiarity with Lie groups and basic Riemannian geometry, as well as some knowledge of PDEs, is desirable.
Syllabus
1. Pseudogroups and Lie groups
2. Bundles and more bundles
3. Connections and more connections
4. Integrability of G-structures
5. Riemannian geometry recap
6. The holonomy principle and its consequences
7. Products and the de Rham Splitting Theorem
8. Homogeneous and symmetric spaces
9. Berger’s classification theorem
10. Special metrics in non-compact manifolds
11. Special metrics on compact manifolds
12. The moduli problem
2. Bundles and more bundles
3. Connections and more connections
4. Integrability of G-structures
5. Riemannian geometry recap
6. The holonomy principle and its consequences
7. Products and the de Rham Splitting Theorem
8. Homogeneous and symmetric spaces
9. Berger’s classification theorem
10. Special metrics in non-compact manifolds
11. Special metrics on compact manifolds
12. The moduli problem
Reference
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.
D. Joyce, Complex Analytic and Differential Geometry,
S. Salamon, Riemannian geometry and holonomy groups.
Audience
Graduate
, Postdoc
, Advanced Undergraduate
Video Public
Yes
Notes Public
Yes
Language
English