Is the commutator of vector fields an important notion?

[Zhang Bei]

Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!

 

[fresh_42]

Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!

Here is an easy example in section 6.2: https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

As a geometric intuition, you can imagine to start at a certain point $p$ and follow a flow along vector field $X$ for a small distance, from there along vector field $Y$ for a while and reach point $s$. If you now start again at $p$ but follow first along $Y$ and then $X$, you will usually end up at a different point $t \neq s$. That difference is measured by $[X,Y]=X \circ Y - Y \circ X$. If $[X,Y]=0$ then $t=s$.