Lie Groups with Bi-invariant Metrics

Abstract

This chapter deals with Lie groups with special types of Riemannian metrics: bi-invariant metrics. Every compact Lie group admits one such metric (see Proposition 2.24), which plays a very important role in the study of its geometry. In what follows, we use tools from Riemannian geometry to give concise proofs of several classical results on compact Lie groups. We begin by reviewing some auxiliary facts of Riemannian geometry. Basic results on bi-invariant metrics and Killing forms are then discussed, e.g., we prove that the exponential map of a compact Lie group is surjective (Theorem 2.27), and we show that a semisimple Lie group is compact if and only if its Killing form is negative-definite (Theorem 2.35). We also prove that a simply-connected Lie group admits a bi-invariant metric if and only if it is a product of a compact Lie group with a vector space (Theorem 2.45). Finally, we prove that if the Lie algebra of a compact Lie group G is simple, then the bi-invariant metric on G is unique up to rescaling (Proposition 2.48).