This may be a rather trivial question, but I just learned the very basic definitions.
I am currently studying differential geometry using Jeffrey Lee's book. Lee defines the Lie bracket of two vector fields as follows:
Let $X, Y$ be vector fields on a smooth manifold $M$. The Lie bracket of $X$ and $Y$ is defined as the unique vector field whose Lie derivative is $L_X L_Y – L_Y L_X $, where $L_X $ and $ L_Y$ are the Lie derivatives of $X $ and $Y$, respectively.
My questions is : Is the restriction (to an open subset of $M$) of the Lie bracket same as the Lie bracket of the restrictions? Succinctly, does the formula $[X \upharpoonright _U , Y \upharpoonright _U] = [X,Y]\upharpoonright_U$ hold? (Here, the subscripts denote restriction of function domains to $U \subset M$.)
Best Answer
Yes, it is equal to the restriction since the Lie braket is defined locally, you can also define $[L_X,L_Y]=-dX.Y+dY.X$.