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.
As a differential equation
The flow is the unique curve in solving the differential equation
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.