Compact Groups, Klein Forms of Symmetric Spaces

Abstract

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.