Symmetric Spaces

Prerequisites: A basic course in differential geometry.

Contents: This course will be devoted to the theory of symmetric spaces. We will study their Riemannian geometry as well as their intimate connection to the theory of semisimple Lie groups. Here is a rough syllabus of the course: Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples. Symmetric spaces of non-compact type: flat subspaces and the notion of rank, roots and root space decomposition. Iwasawa decomposition, Weyl group, Cartan decomposition. Geometry at infinity: geometric boundary, Furstenberg boundary, Bruhat decomposition, visibility at infinity, Busemann functions.

Lecturer: Prof. Dr. Urs Lang
Assistant: Alexandru Gabriel Sava

Lecture

On Fridays: 10.00-12.00 a.m. at HG G26.5, starting 27.02. Lectures and exercise sessions alternate weekly.

Exercises

On Wednesdays, every second week: 10.00-12.00 a.m. at HG E3, starting on 26.02.

Exercise sheets will be published here as the course progresses, allowing the students to deepen their understanding of the theory. There will be no Testat condition.

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Literature

The following items may be useful.

Werner Ballmann: Lectures on spaces of nonpositive curvature.
Armand Borel: Semisimple groups and Riemannian symmetric spaces.
Martin Bridson and André Haefliger: Metric spaces of non-positive curvature.
Patrick B. Eberlein: Geometry of nonpositively curved manifolds.
Sigurdur Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces.
Shoshichi Kobayashi and Katsumi Nomizu: Foundations of differential geometry. Vol. II.
Joseph A. Wolf: Spaces of constant curvature.