Brackets and Flows
Now, what does the Lie bracket of two vector fields really measure? We’ve gone through all this time defining and manipulating a bunch of algebraic expressions, but this is supposed to be geometry! What does the bracket actually mean? It turns out that the bracket of two vector fields measures the extent to which their flows fail to commute.
We won’t work this all out today, but we’ll start with an important first step: the bracket of two vector fields vanishes if and only if their flows commute. That is, if and are vector fields with flows and , respectively, then if and only if for all and .
Conversely, let’s assume that . For any we can define the curve in the tangent space by . Since the Lie derivative vanishes, we know that , and I say that for all , or (equivalently) that .
Fixing any we can set . Then we calculate
Now this means that is -invariant for all , meaning that and commute for all and , as asserted.
As a special case, if is a coordinate patch then we have the coordinate vector fields . The fact that partial derivatives commute means that the brackets disappear:
This corresponds to the fact that adding to the th coordinate and to the th coordinate can be done in either order. That is, their flows commute.