Semisimple Groups and Riemannian Symmetric Spaces

The prerequisites for this book are roughly the following:

In §§1 to 4 we review some concepts and facts of the theory of differentiable manifolds, in particular Lie derivatives, covariant differentiations, linear connections, and Cartan equations. No proofs are given so that this is not meant as a self-contained introduction to the subject, for which the readers may for instance consult [6,7,10].

Let M be a connected Riemannian manifold. The distance function d(x, y) (x, y ∈ M), which is by definition the inf limit of the lengths of the rectifiable arcs joining x to y, defines a metric compatible with the topology of M. By the Hopf-Rinow theorem, the following conditions are equivalent:

The first three paragraphs are essentially group theoretical, and depend on Chapters I, II rather than on III, IV. Differential geometry occurs only when proving that a connected Lie group with compact Lie algebra is covered by its one-parameter subgroups (1.1). § 1 gives some classical properties of compact Lie groups, including H. Weyl’s theorem that a group with compact semi-simple Lie algebra is compact. § 2 introduces the diagram, in the global sense, of a compact orthogonal involutive Lie algebra, and discusses some of its properties. In § 3 it is proved that the fixed point set of an automorphism of a compact, connected, simply connected Lie group is connected. Although of some independent interest, this result is, in the context of this chapter, only a preliminary to § 4, where it allows one to show that a compact, simply connected, Riemannian symmetric space may be realized as the space of transvections in the universal covering of its group of isometries. § 4 is devoted to the Klein forms of a simply connected, Riemannian symmetric space M. If M has negative curvature, there is only M itself; if M has positive curvature, there may be several Klein forms, and their determination generalizes the characterisation of the groups with a given compact semi-simple Lie algebra.

This Chapter is devoted to the complex analogues of Riemannian symmetric spaces: Hermitian manifolds in which each point is an isolated fixed point of some involutive automorphism. They are Riemannian symmetric with respect to the underlying Riemannian structure.

This chapter is chiefly devoted to a fundamental theorem which asserts, roughly, that if G is a Lie group with finitely many connected components, then G has maximal compact subgroups, is homeomorphic to the product of any one of them by a euclidean space, and any two maximal compact subgroups of G are conjugate by an inner automorphism. The precise statement, and two corollaries, are given in §1, the proof in §2. The latter makes use of a theorem of Iwasawa on extensions of vector groups by compact groups, and of several results obtained in the previous chapters, notably the decomposition Ad g = _ωK × P, and the conjugacy of maximal compact subgroups in Aut \(\mathfrak{g}\), where \(\mathfrak{g}\) is a semisimple Lie algebra. In order to make this chapter more self-contained, we give in §3 alternative proofs of those statements on semi-simple Lie groups, which do not make use of Riemannian geometry, and reproduce a proof of the Iwasawa theorem in §4.