Riemannian Symmetric Spaces and Bounded Domains in C

This paper is to serve as an introduction to the study of symmetric spaces, with the goal of describing Hermitian symmetric spaces of noncompact type. There are three basic types of symmetric space: compact, noncompact, and Euclidean, defined in terms of properties of g, the Lie algebra of its group of isometries. It turns out that a simply connected symmetric space can be described completely in terms of the pair (g, s), where s is an involutive automorphism of g. All symmetric spaces of noncompact and Euclidean type are simply connected, but the compact case is quite different and a full understanding is beyond the scope of this paper. Our main goal is to discuss the Hermitian symmetric spaces which are all simply connected, and in particular to prove that those of noncompact type are exactly the bounded symmetric domains in Cn with Riemannian structure given by the Bergman metric. Section 1 gives basic definitions and properties of symmetric spaces with some examples at the end. Section 2 contains a summary of the basics of Lie algebras, and Section 3 provides an overview of the decomposition of symmetric spaces into compact, noncompact, and Euclidean type. We achieve our main goal in Section 4. It should be mentioned that a complete classification of symmetric spaces is possible using the classification of semisimple Lie algebras. While the details of the classification will not be carried out here, Section 5 outlines the basic strategy.