Holonomy and special metrics

Course Overview


This graduate-level course offers an introduction to manifolds with special holonomy in Riemannian geometry.


The course is roughly organized into three parts (see syllabus below).


  1. Foundations of principal bundles and general G-structures, aiming to understand the role of torsion and its connection to the integrability problem for G-structures.

  2. A focus on Riemannian metrics, examining the consequences of the holonomy principle, the de Rham Splitting Theorem, and the theory of homogeneous and symmetric Riemannian manifolds, concluding with a proof of Berger's Classification Theorem.

  3. Various methods for constructing metrics with special holonomy, in both compact and non-compact settings.

If time allows, I will also discuss the moduli problem for special holonomy metrics.

Prerequisites: Differential geometry. Familiarity with Lie groups and basic Riemannian geometry, as well as some knowledge of PDEs, is desirable.

Logistics


The course will be held on Thursdays , from 10:40 to 12:15 and from 13:30 to 15:05.


The lectures will be held in person at YMSC, Shuangqing Building, in room A513 (TBC).


The course will be streamed via Zoom, password BIMSA.


There is an (unofficial) course Weixin group, for questions and exercise discussions. You can join via QR code.


Notes and references


I will mainly draw from


As well as several published research papers.


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


Syllabus (tentative)


  1. Pseudogroups and Lie groups and representations
  2. Bundles and more bundles
  3. Connections and more connections
  4. Integrability of G-structres
  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

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.