I spent a few weeks on the USTC campus, in Hefei, China, in the summer of 2011, where I was invited to give some lectures in the annual summer school there. I gave 10 lectures on the topics related to F-structures and collapsing Riemiann manifolds.
Lecture 1 | Riemannian geometry. Examples of the 3-sphere and warped products. Connections and curvature. Why study collapsing? |
Lecture 2 | Differential geometry, the Lie derivative, and Lie groups. |
Lecture 3 | Interaction between Lie group structure and Riemannian geometry. Example: SU(2). The Hopf action and the Hopffibration. The Heisenberg group and its quotients. |
Lecture 4 | The fundamental examples of collapse: The Berger spheres, collapsing warped products, collapse along a torus action, collapse of nilmaifolds. |
Lecture 5 | The Gromov-Hausdorff distance and its properties. Covergence of Riemannian manifolds. |
Lecture 6 | Basic definitions surrounding F-structures. |
Lecture 7 | Additional F-structure definitions. Examples of F-structures. Basic Theorems. |
Lecture 8 | Non-polarizable F-structures, complete flat manifolds, and elementary F-strucutures |
Lecture 9 | How various types of F-structures imply collapse with bounded curvature. |
Lecture 10 | How collapse with bounded curvature implies collapse |
Addendum: References and additional concepts.