Inhaltspezifische Aktionen

Introduction to Symmetric Spaces

Tobias Hartnick (Gießen)


02.05.2019 von 16:00 bis 18:00 (Europe/Berlin / UTC200)


Hörsaal des Mathematischen Instituts, Arndtstraße 2, 1. Stock, Raum 111

Termin zum Kalender hinzufügen



For every point in the Euclidean plane one can define a point reflection at this point, which is an isometry and acts with eigenvalue -1 on any line through the given point. A metric space which admits such “geodesic point reflections” is called a Riemannian symmetric space. In dimension 2 there are only two other examples: The hyperbolic plane (non-compact, negative curvature) and the sphere (compact, positive curvature).


Starting from these examples we will develop a general axiomatic framework of "reflection spaces”. We will the focus on the class of non-positively curved non-compact Riemannian symmetric spaces, which generalize the Euclidean and hyperbolic plane (but not the sphere) to higher dimensions. Throughout we will focus on examples. All these examples can be realized as spaces of positive-definite symmetric matrices, hence they are rather concrete.


We will explain their axiomatic characterization among reflection spaces (due to Loos), their classification (due to Cartan) and some of their main properties. In particular, we will explain how lattices in reductive Lie groups act on such spaces and their boundaries (the so called “building at infinity”).