Vanishing Lie derivative of vector fields means $L_X Y=0$

Question

This is some restricted case of my previous post Meaning of vanishing Lie bracket

I want to know meaning of vanishing Lie derivatives of vector fields \begin{align} L_X Y=[X,Y]= 0 \end{align} In this case i am not concerning for any $Y$ condition.

Simply i can guess, $X$ and $Y$ are coordinate basis or $Y$ is proportional to $X$, is there any other description?

Answer 1

On basically every differential geometry book you will find a formula expressing the Lie bracket in terms of a relation between flows of the two vector fields. Try to figure out what $[X,Y]=0$ tells you about the flow of $X$ and the one of $Y$.

There is no general condition. In the specific case of invariant vector fields on a Lie group you can think of the two vector fields as two elements of the Lie algebra. If $G=GL(n;\mathbb R)$ this means choosing a pair of $n\times n$ commuting matrices. They maybe very far from being one multiple of the other...

Answer 2

A way to understand it geometrically is this : it is equivalent to first go through X and then Y, than going first through Y and then X. Kind of you don't care in which order you apply them.