I am a high schooler self-studying differential geometry. I am stuck trying to show the following:
Let $\phi:M\rightarrow N$ be a diffeomorphism, and $X$ and $Y$ are vector fields on $M$. Show that $\phi_*[X,Y]=[\phi_*X,\phi_*Y]$.
I want to still solve the problem by myself, and I have played around with different definitions of the push forward and commutator, but I have been unable to show the property holds. Could I have a sketch of the proof / some pointers on good directions to head in order to show it? The main thing I’ve tried doing is rewriting $(\phi_*)X_{\phi(p)}=(D_p\phi)X_p$, but I am unsure how to proceed from here.
We first prove that $$X(f\circ\phi)=((\phi_*X)f)\circ\phi.$$ On one hand, $$(\phi_*X)_{\phi(p)}f=(D_p\phi)X_pf=X_p(f\circ\phi)=X(f\circ\phi)(p).$$ On the other hand, $$(\phi_*X)_{\phi(p)}f=((\phi_*X)f)\circ\phi(p).$$ Thus $$X(f\circ\phi)(p)=((\phi_*X)f)\circ\phi(p).$$ Now we prove the desired result $$\phi_*[X,Y]=[\phi_*X,\phi_*Y].$$ Note that $$XY(f\circ\phi)=X(((\phi_*Y)f)\circ\phi)=((\phi_*X\phi_*Y)f)\circ\phi.$$ Similarly, $$YX(f\circ\phi)=((\phi_*Y\phi_*X)f)\circ\phi.$$ Therefore \begin{equation*} \begin{aligned}\phi_*[X,Y]f&=[X,Y](f\circ\phi)\\ &=XY(f\circ\phi)-YX(f\circ\phi)\\ &=((\phi_*X\phi_*Y)f)\circ\phi-((\phi_*Y\phi_*X)f)\circ\phi\\ &=([\phi_*X,\phi_*Y]f)\circ\phi. \end{aligned} \end{equation*}