# Symmetric Spaces Spring 2018

Lecturer
Prof. Dr. Alessandra Iozzi
Coordinator
Yannick Krifka (E-mail)

## 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.

The exercise sheets will be posted below.

Exercise Sheet Solution Comment
Exercise Sheet 1 Solution 1
Exercise Sheet 2 Solution 2
Exercise Sheet 3 Solution 3
Exercise Sheet 4 Solution 4
Exercise Sheet 5 Solution 5
timeroom
Tue 10-12 am HG D 5.2
Thu 08-10 am HG G 5

About every second week there will be an exercise class led by Yannick Krifka instead of a lecture on Thursday.

However, there will be some irregularities that are being displayed in the following colour coded table:

Tuesday Thursday
Feb 20 Feb 22
Feb 27 Mar 1
Mar 6 Mar 8
Mar 13 Mar 15
Mar 20 Mar 22
Mar 27 Mar 29
Apr 3 Apr 5
Apr 10 Apr 12
Apr 17 Apr 19
Apr 24 Apr 26
May 1 May 3
May 8 May 10
May 15 May 17
May 22 May 24
May 29 May 31

Legend: lecture, exercise class, holiday.

You can find notes on a previous course on symmetric spaces here.

Notes on this course will be uploaded here.

Apart from these notes the following books might be of interest to you.

• 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" (BirkhĂ¤user, 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)