Up to date lecture notes.
We also have a forum, which you can use to ask questions, point out typos, etc.Learn the basics of Symmetric Spaces.
Basic concepts about Lie Groups are useful, but may be picked up during the course. The notes Useful facts about Lie Groups contain all the required concepts (even if you haven't taken a course on Lie groups), and so you are encouraged to skim through them and attempt the exercises.
These notes of Prof. A. Iozzi from HS21 contain a complete reference of all that is needed (and much more). Basics of Riemannian Geometry are also useful, but will be introduced fully where required.
| Lecture number | Date | Summary | Notes |
|---|---|---|---|
| 1 | 18.09 | Broad overview of Symmetric Spaces | |
| 2 | 19.09 | Riemannian Geometry background, Hopf-Rinow, transitivity of \(\operatorname{Iso}(M)^\circ\) on \(M\) | |
| 3 | 25.09 | Symmetric spaces as quotients of their isometry group, connections and parallel transport | |
| Ex. Cl. 1 | 26.09 | Recap of material, geometric picture of the Iwasawa decomposition, homogeneous spaces | |
| 4 | 02.10 | Transvections and Parallel transport | |
| 5 | 03.10 | Recap of Lie Groups prerequisites | |
| 6 | 09.10 | The algebraic point of view | |
| 7 | 10.10 | The algebraic point of view (continued) | |
| 8 | 16.10 | Exponential map and geodesics | |
| Ex. Cl. 2 | 17.10 | Recap of the algebraic point of view, and worked through examples of RSSs | |
| 9 | 23.10 | Proof of the rellationship between Lie Triple Systems and totally geodesic submanifolds | |
| Ex. Cl. 3 | 24.10 | Recap of the previous lecture. The Siegel upper half plane as the symmetric space of \(\operatorname{SP}(2n,\mathbb{R})\), and the exceptional isogeny between \(\operatorname{SL}(2,\mathbb{R})\) and \(\operatorname{SO}(2,1)\). | |
| 10 | 30.10 | The Riemannian Symmetric Pair \((\operatorname{SL}(n,\mathbb{R}), \operatorname{SO}(n,\mathbb{R})) \) and introduction to Orthogonal Symmetric Lie Algebras (OSLAs). | |
| 11 | 31.10 | Decomposition Theorem for OSLAs | |
| 12 | 06.11 | Curvature of Symmetric spaces of (non)compact/Euclidean type. | |
| Ex. Cl. 4 | 07.11 | Detailed review of Killing forms, Dieudonné's Theorem, decomposition of OSLAs. | |
| 13 | 13.11 | Duality of symmetric spaces of (non)compact type. | |
| 14 | 14.11 | \(G\)-invariant differential forms on symmetric spaces of non-compact type. Introduction to CAT(0) spaces. | |
| 15 | 20.11 | Symmetric Spaces of non compact type are CAT(0) | |
| Ex. Cl. 5 | 21.11 | More on CAT(0) spaces. Introduction to de Rham cohomology. | |
| 16 | 27.11 | Flats and Rank | |
| 17 | 28.11 | Roots and root spaces | |
| 18 | 04.12 | The reduced root system is indeed a root system | Ex. Cl. 6 | 05.12 | Summary of roots and the Jacobson-Morozov Theorem | 19 | 11.12 | Abstract root systems | 20 | 12.12 | Iwasawa decomposition | 21 | 18.12 | Linear algebraic groups and the Borel Density Theorem | Ex. Cl. 7 | 19.12 | Abstract Root Systems and Finite reflection Groups |
The exercise sheet will be uploaded on this page. The exercise sheet k will be uploaded on Friday of week 2k - 1.
Please, upload your solution via the SAM upload tool.
In order to access the website you will need a NETHZ-account and you will have to be connected to the ETH-network. From outside the ETH network you can connect to the ETH network via VPN. Here are instructions on how to do that.
Make sure that your solution is one PDF file and that its file name is formatted in the following way:
solution_<number of exercise sheet>_<your last name>_<your first name>.pdf
Example: solution_2_GonenCohen_Segev.pdf
If there are any issues, or you would like to hand in your solutions on paper, please contact Segev Gonen Cohen.
Exercise Sheet 0 is a recap of the useful concepts from Lie groups. It will only be discussed in Exercise Class if there is sufficient demand, please contact Segev if this is the case for you.
Here are some lecture notes (FS 18).