Global flow
A (global) flow is a continuous left -action on a smooth manifold .
Motivation
While this definition is technically correct, it doesn’t leave any intuition. Given a vector field , a (global) flow is a function that “follows” integral curves from points. We want these integral curves to be started from , so Then for a fixed letting values of range through would give the function which is a curve on the M. We want this curve to be an integral curve of .
Local flow
The only problem, is generally an integral curve of with initial condition is not defined for all of . Therefore, we pick a special (open) subset such that the integral curve we care about is defined on each set This is called a flow domain.
Then a local flow is a continuous map that follows the same general structure as the global flow.
Maximal flows
A maximal flow is a flow where each is a maximal integral curve. In other words, the flow domain cannot be made any bigger for any point .
Properties
- Thus, is a group homomorphsim.
- Flows commute if and only if the infinitesimal vector field commutes.