The Exponential Map

Abstract

In this chapter we introduce the exponential map of a Lie group, which is a canonical smooth map from the Lie algebra into the group, mapping lines through the origin in the Lie algebra to one-parameter subgroups. As our first application, we prove the closed subgroup theorem, which says that every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. Next we prove a higher-dimensional generalization of the fundamental theorem on flows: if G is a simply connected Lie group, then any Lie algebra homomorphism from its Lie algebra into the set of complete vector fields on a smooth manifold M generates a smooth action of G on M. Using this theorem, we prove that there is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups. At the end of the chapter, we show that connected normal subgroups of a Lie group correspond to ideals in its Lie algebra.