The geometry of Lie groups

Abstract

The two main results in this chapter concern a class of Lie groups which contains all compact connected Lie groups, namely the class of complete connected Lie groups which possess invariant Riemannian metrics. For groups in this class it is first shown that geodesies and translates of 1-parameter subgroups are essentially the same (Theorem 4. 3. 3) and then, as a corollary, that the associated exponential map is surjective (Corollary 4. 3. 5). These results are presented solely for their own interest and are not needed to establish the main structure Theorems described in Chapter 6. The first section of this chapter contains a brief discussion of Riemannian manifolds (with some statements being given without proof) while the second section is concerned primarily with the establishment of a necessary and sufficient condition for a Lie group to possess an invariant Riemannian metric. In the third section these preliminaries culminate with the proofs of the main results described above. Riemannian manifolds Throughout this section we will suppose that M is an analytic manifold. By selecting p in M and imposing a norm on T p (M) we can arrive at a notion of distance in an ‘infinitesimal’ neighbourhood of p in the following way. Temporarily ignoring all the usual caveats against talking about infinitesimals, let q be a point in this neighbourhood of p and let ξ be an analytic curve on M with ξ(0) = p and ξ(δt)=q.