Lie Groups

Abstract

This chapter presents a brief introduction to matrix Lie groups and their Lie algebras and their actions on manifolds. We review left-invariant 1-forms and the Maurer–Cartan form of a Lie group, and the adjoint representation of the Lie group on its Lie algebra. The treatment of principal bundles is self-contained. We derive basic properties of transitive actions. We define the notion of a slice for nontransitive actions. In many instances this is just a submanifold cutting each orbit uniquely and transitively such that the isotropy subgroup at each point of the submanifold is the same. The general idea of a slice is used, however, for the ubiquitous action by conjugation of the orthogonal group on symmetric matrices. We review the Frobenius theory of smooth distributions. The chapter concludes with statements and proofs of the Cartan–Darboux uniqueness and existence theorems. Our proof of the global existence theorem is simpler than those proofs we have seen in the literature.