Symmetric Spaces Spring 2023

Lecturer
Alessandra Iozzi
Coordinator
Lisa Ricci

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.
  • Geometry at infinity.

    • Goal

    Learn the basics of symmetric spaces.

    Lecture and exercise class

    The lecture will start on Wednesday, 22nd of February 2023 and takes place every week on

  • Wed. 10-12 in HG G 19.1, and
  • Th. 12-14 in HG G 19.1

    • Here there are the Lie groups lecture notes of Prof A. Iozzi (HS 21).

    Exercise classes

    Exercise sessions take place on Thursdays every second week in HG G 19.1 from 12-14. The first exercise session will be on Thursday the 2nd of March.

  • Thursday 30.03.2023: Lecture
  • Thursday 6.04.2023: Exercise class
  • Thursday 20.04.2023: Exercise class

    • Exercise Sheets

    The exercise sheet will be uploaded on this page. The exercise sheet k will be uploaded on Friday of week 2k - 1 and it will be due on Thursday 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_Surname_Name.pdf.

    exercise sheet due by upload link solutions comments
    Exercise sheet 1 9.03.22 upload ex sheet 1 Solution 1
    Exercise sheet 2 23.03.22 upload ex sheet 2 Solution 2
    Exercise sheet 3 06.04.23 upload ex sheet 3 Solution 3
    Exercise sheet 4 27.04.23 upload ex sheet 4 Solution 4
    Exercise sheet 5 11.05.22 upload ex sheet 5 Solution 5
    Exercise sheet 6 upload ex sheet 6

    Literature

    Here are some lecture notes (FS 18).

    • W. Boothby, An introduction to differentiable manifolds and Riemannian geometry (Academic Press, 2002)
    • M. do Carmo, Riemannian geometry (Birkhäuser, 1992)
    • M. Berget, A panoramic view of Riemannian geometry (Springer, 2003)
    • S. Helgason, Differential geometry, Lie groups and symmetric spaces (Academic Press, 1978)
    • G. Link, An introduction to globally symmetric spaces, pdf
    • P. Eberlein, Geometry of nonpositively curved manifolds (Chicago University Press, 1996)
    • J. Maubon, Riemannian symmetric spaces of the non-compact type: differential geometry, pdf