An introduction to globally symmetric spaces (Q2895505)

Symmetric spaces are very interesting objects in geometry, because they can be described from various viewpoints and since these descriptions give us important relations. In particular, we obtain relations between a symmetric space \(S\) with geodesic symmetry \(s_o\) at \(o\in S\) and compact subgroup \(K\) of the isometry group \(G\) of \(S\) which is stable with respect to an involution \(\sigma\in\Aut(G)\). This is also reflected in the Cartan involution of the Lie algebra \({\mathfrak{g}}\) of \(G\). Obviously, algebraic tools on \({\mathfrak{g}}\) are very useful for the study of the geometry of \(S\), for example, in the study of relations between 1-parameter groups of isometries in \(G\) and geodesics in \(S\), for describing the curvature of \(S\), and so on.NEWLINENEWLINEThe first part of the present paper is an excellent survey on these fundamental constructions. The exposition is straightforward, using statements from essential monographs. The constructions are well illustrated with important examples. It can be recommended to the readers as a basic source about the subject.NEWLINENEWLINESymmetric spaces are either of compact type, or of non-compact type, or of Euclidean type. In the second part of the paper, those of non-compact type are studied in more detail. Rank of a symmetric space and roots are defined and, again, illustrated with examples. Further, Iwasawa decomposition, Weyl group and Cartan decomposition are studied.NEWLINENEWLINEIn the last part of the paper, the geometry at infinity of a globally symmetric spaces \(S\) of non-compact type is studied. All these spaces are homeomorphic to \({\mathbb{R}}^{n}\), and they can be compactified by attaching its geometrical boundary. This boundary can be described using the rich algebraic structure on \(S\), and the Furstenberg boundary with a natural differentiable structure can be defined. Further, pairs of points in the geometric boundary which can be joined by a geodesic are studied. The Bruhat decomposition allows to describe the pairs of points with this property. Finally, Busemann functions are introduced, which serve as a tool in the construction of \(G\)-invariant Finsler pseudo-distances on \(S\).NEWLINENEWLINEFor the entire collection see [Zbl 1230.00040].