Vector Flow - Vector Flow in Differential Topology

Vector Flow in Differential Topology

Relevant concepts: (flow, infinitesimal generator, integral curve, complete vector field)

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain. For each p ∈ M the map D_p → M is the unique maximal integral curve of V starting at p.

A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete if it generates a global flow. Every vector field on a compact manifold without boundary is complete.