# Symmetric Spaces Spring 2021

Lecturer
Alessandra Iozzi
Coordinator
Yannick Krifka

## Content

### Description

• Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples.
• Symmetric spaces of non-compact type: flats and rank, roots and root spaces.
• Iwasawa decomposition, Weyl group, Cartan decomposition.
• Hints of the geometry at infinity of $$\text{SL}(n,\mathbb{R})/\text{SO}(n,\mathbb{R})$$.

### Goal

Learn the basics of symmetric spaces.

## Exercises

The exercise sheets will be released below. More challenging exercises are indicated by $$\dagger$$.

Exercise Sheet Solution Comment
Exercise Sheet 1 Solution Exercise Sheet 1
Exercise Sheet 2 Solution Exercise Sheet 2
Exercise Sheet 3 Solution Exercise Sheet 3 Added hint to exercise 1 b). Added references to solution.
Exercise Sheet 4 Solution Exercise Sheet 4
Exercise Sheet 5 Solution Exercise Sheet 5 A mistake in exercise 1 b) has been fixed.

The general rule is that exercise sheet $$k$$ will be released on Thursday of week $$2k -1$$ and it will be due on Thursday of week $$2k+1$$.

It is not mandatory to hand-in solutions and your solutions do not contribute in any way to your final grade. However, it is an opportunity to get some feedback on your understanding of the material covered in class.

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

For example: If your first name is Alice, your last name is Miller, and you want to hand-in your solution to exercise sheet number 2, then you will have to upload your solution as one PDF file with the file name  solution_2_Miller_Alice.pdf.

## Lectures and Exercise Classes

The course will start on Wednesday, 24th of February 2021.

TimeRoomZoom
Wed 08-10 HG F 26.5 Wed Zoom
Thu 08-10HG F 26.5 Thu Zoom

According to the ETH Covid-19 regulations the semester will start online. All lectures and exercise classes will be held as Zoom meetings until further notice.

Every fourth time there will be an exercise class held by Yannick Krifka instead of a lecture. For a detailed schedule see below.

Date Event Recording Notes
Wed 24/02/21 Lecture Recording Notes
Thu 25/02/21 Lecture Recording Notes
Wed 03/03/21 Lecture Recording Notes
Thu 04/03/21 Exercise Class Recording Notes
Wed 10/03/21 Lecture Recording Notes
Thu 11/03/21 Lecture Recording Notes
Wed 17/03/21 Lecture Recording Notes
Thu 18/03/21 Exercise Class Recording Notes
Wed 24/03/21 Lecture Recording Notes
Thu 25/03/21 Lecture Recording Notes
Wed 31/03/21 Lecture Recording Notes
Thu 01/04/21 Exercise Class Recording Notes
Wed 14/04/21 Lecture Recording Notes
Thu 15/04/21 Lecture Recording Notes
Wed 21/04/21 Lecture Recording Notes
Thu 22/04/21 Exercise Class Recording Notes
Wed 28/04/21 Lecture Recording Notes
Thu 29/04/21 Lecture Recording Notes
Wed 05/05/21 Lecture Recording Notes
Thu 06/05/21 Exercise Class Recording Notes
Wed 12/05/21 Lecture Recording Notes
Wed 19/05/21 Lecture Recording Notes
Thu 20/05/21 Exercise Class Recording Notes
Wed 26/05/21 Lecture Recording Notes
Thu 27/05/21 Lecture Recording (1st half), Recording (2nd half) Notes
Wed 02/06/21 Lecture Recording Notes
Thu 03/06/21 Lecture Recording Notes

## Forum

This course has its own subforum in the D-MATH forum. Feel free to ask questions and discuss about the lecture and exercises there.

## Literature

There are some tentative notes for this course available here.

• A. Borel: "Semisimple groups and Riemannian symmetric spaces" (Springer, 1998)
• M. Bridson, A. Haefliger: "Metric spaces of non-positive curvature" (Springer, 1999)
• M. do Carmo: "Riemannian geometry" (Birkhäuser, 1992)
• P. Eberlein: "Geometry of nonpositively curved manifolds" (Chicago University Press, 1996)
• S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, 1978)
• S. Kobayashi, K. Nomizu: "Foundations of differential geometry. Vol. II" (Wiley, 1996)