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

time	room
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" (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)