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
日期
2026年03月30日 至 06月29日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一 | 10:40 - 12:15 | Shuangqing | ZOOM 05 | 293 812 9202 | BIMSA |
| 周一 | 13:30 - 15:05 | Shuangqing | ZOOM 05 | 293 812 9202 | BIMSA |
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修课要求
Differential geometry. Familiarity with Lie groups and basic Riemannian geometry, as well as some knowledge of PDEs, is desirable.
课程大纲
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
参考资料
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.
听众
Graduate
, 博士后
, Advanced Undergraduate
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笔记公开
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语言
英文