An introduction to globally symmetric spaces

Abstract

These notes provide an introduction to globally symmetric spaces with an emphasis on those of non-compact type. We intend to give a guideline through a part of the landscape rather than proving every detail; however, where proofs are omitted, the reader gets a precise reference.

The first part of the text gives an overview on the geometry and algebraic coding of arbitrary globally symmetric spaces. We then restrict ourselves to symmetric spaces of non-compact type and describe the Iwasawa and Cartan decomposition in the second part. The last part is devoted to the study of the geometry at infinity of globally symmetric spaces of non-compact type. We give descriptions of the geometric boundary, the Furstenberg boundary and the Bruhat decomposition, which help to identify the pairs of boundary points which can be joined by a geodesic or flat. Finally we show how invariant Finsler distances on the globally symmetric space can be constructed with the help of Busemann functions.