Lecturer

Prof. Dr. Marc Burger

Coordinator

Raphael Appenzeller, (mail)

Content

Possible topics include:

• Locally and globally Symmetric Spaces
• Symmetric Spaces from the differential geometric viewpoint
• Symmetric Spaces from the Lie groups viewpoint
• Cartan decomposition
• Decomposition in compact/non-compact/Euclidean factors
• Duality theory
• Flats and rank
• Root space decomposition
• Iwasawa decomposition
• Examples

Vorlesungsverzeichnis

Coronavirus-Precautions

To allow a smooth continuation we provide you with the following ressources:

• The lectures will be recorded. You can find the link to the respective lectures in the overview-table. Note that it may take a few days for the video to be uploaded.
• Some handwritten notes of the course are available here: approximately Week 1, Week 2, Week 3, Week 4, Week 5, Week 6a, Week 6b, Week 7a, Week 7b, Week 8, Week 10a, Week 10b, Week 11, Week 12a, Week 12b (with some extra on cohomology starting on page 71), Week 13a, Week 13b, Week 13c, Week 14a, Week 14b.
• The course follows relatively closely the script by Prof. Alessandra Iozzi.
• We have a forum where you can ask questions and see what questions the other people asked. The forum requires a separate registration. Despite it's appearance it is not an official website of ETH, rather it is a forum set up by Prof. Will Merry.

Prerequisites

Some basic knowledge of differential geometry and Lie groups.

Exercises

New exercises and solutions will appear here approximately every two weeks. If you hand in your solution, you will get the corrected exercises back. Please scan your solutions and send them via mail to "raphael.appenzeller@math.ethz.ch". Doing exercises is an essential part of any lecture.

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exercise sheet	due by	solutions
Exercise sheet 1	Friday 13.3.2020	Solutions 1
Exercise sheet 2	Friday 27.3.2020	Solutions 2
Exercise sheet 3	Friday 17.4.2020	Solutions 3
Exercise sheet 4	Friday 1.5.2020	Solutions 4
Exercise sheet 5	Friday 21.5.2020	Solutions 5
Exercise sheet 6	Friday 5.6.2020	Solutions 6

Schedule

Lectures (V) and exercise classes (U) take place on Tuesday and Thursday.

Tuesday 10-12 a.m. HG F26.5, HG G 43 (Hermann Weyl Zimmer) at home	Thursday 8-10 a.m. HG F26.5 HG G 43 (Hermann Weyl Zimmer) at home
18.2. V	20.2. V
25.2. V	27.2. V
3.3. V	5.3. U
10.3. V	12.3. V - Video recording
17.3. V - Video recording	19.3. U - Video recording
24.3. V - Video recording	26.3. V - Video recording
31.3. V - Video recording	2.4. U - Recording and notes
7.4. V - Video recording	9.4. V - Video recording
break	break
21.4. V - Video recording	23.4. U V - Video recording
28.4. V - technical problems (no recording)	30.4. V - Video recording
5.5. V - Video recording	7.5. U V - Video recording
12.5. V U - Recording and notes	11.5. V - Video recording; 12.5. V - Video recording
18.5. V - Video recording	22.5. V - Video recording
26.5. Recap with RA, Notes and recording	28.5. Recap with RA, Notes and recording

Literature

References for Riemannian geometry:

• M. Berger, "A panoramic view of Riemannian geometry", Springer, 2003
• W.M. Boothby, "An Introduction to Differentiable manifolds and Riemannian Geometry', Elsevier 1986
• M.P. do Carmo, "Riemannian Geomtrey", Birkhauser, 1992

References for symmetric spaces:

• S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Academic press, 1979
• E. Cartan, "Les expaces riemanniens symmetriques", ICM 1932, pdf
• P.B. Eberlein, "Geometry of Nonpositively Curved Manifolds", Chicago Lectures in Mathematics, 1996
• J. Maubon, "Riemannian symmetric spaces of the non-compact type: differential geometry", 26 pages, pdf
• A. Borel, "Semi-simple Lie groups and Riemannian symmetric spaces", 1998, Springer link
• M. Burger, "Introduction to the study of riemmanian symmetric spaces" MSRI Lectures, January 2015, 1, 2, 3, 4.
• A. Iozzi, "Symmetric Spaces", ETH, 2014. pdf
• S. Sternberg, "Lie Algebras", 2004. pdf
• M. R. Bridson, A. Haefliger, "Metric Spaces of Non-Positive Curvature", 1999, Springer link