Vector fields are familiar objects of study in multivariable calculus. In this chapter we show how to define vector fields on smooth manifolds, as certain kinds of maps from the manifold to its tangent bundle. Then we introduce the Lie bracket operation, which is a way of combining two smooth vector fields to obtain another. The most important application of Lie brackets is to Lie groups: the set of all smooth vector fields on a Lie group that are invariant under left multiplication is closed under Lie brackets, and thus forms an algebraic object naturally associated with the group, called the Lie algebra of the Lie group. We show how Lie group homomorphisms induce homomorphisms of their Lie algebras, from which it follows that isomorphic Lie groups have isomorphic Lie algebras.
KeywordsVector Field Smooth Manifold Real Vector Space Smooth Vector Field Vector Space Isomorphism