Integral Curves and Flows

Abstract

In this chapter we return to the study of vector fields. The primary geometric objects associated with smooth vector fields are their “integral curves,” which are smooth curves whose tangent vector at each point is equal to the value of the vector field there. The collection of all integral curves of a given vector field on a manifold determines a family of diffeomorphisms of (open subsets of) the manifold, called a “flow.” Any smooth R-action is a flow, for example; but we will see that there are flows that are not R-actions because the diffeomorphisms may not be defined on the whole manifold for all t ∈ R.