I am confused with the definition and notation in differential geometry. Take the solution to the problem reply here for example.
X,Y are Vectorfields on M and $\psi: M \rightarrow N$, $g\in C^{\infty}(N)$ Can someone explain to me why: $$ \psi_*[X,Y](g) = [X,Y](g\circ \psi) $$ Our definition in class was the following: $$ \psi_*X = T\psi \circ X \circ \psi^{-1} $$ Shouldn't the same thing happen here: $[X,Y]$ is also a vector field on M. Thus $\psi_*[X,Y]$ should be a vector field on N. Hence, I would think that: $$ \psi_*[X,Y] = T\psi \circ [X,Y] \circ \psi^{-1} $$ How do these definitions reconcile?
Also am I correct that $X(f) = Tf(X)$ where Tf is the tangent map?
I would very much appreciate every help!
Thanks in advance!
Max
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EDIT: Is that correct? We have in general: $$ \psi_*X(f) = Tf(\psi_*X) = Tf(T\psi \circ X \circ \psi^{-1}) = Tf \circ T\psi \circ X \circ \psi^{-1} = T(f\circ \psi)(X) \circ \psi^{-1} = X(f\circ \psi) \circ \psi^{-1} \quad (*)$$ We can thus write: $$ \psi_*[X,Y](f) \stackrel{*}{=} [X,Y](f\circ \psi) \circ \psi^{-1} = XY(f\circ \psi) \circ \psi^{-1} - YX(f\circ \psi) \circ \psi^{-1} $$ $$ = X(Y(f\circ \psi)) \circ \psi^{-1}-Y(X(f\circ \psi)) \circ \psi^{-1} $$ $$ = X(Y(f\circ \psi) \circ \psi^{-1} \circ \psi) \circ \psi^{-1}-Y(X(f\circ \psi) \circ \psi^{-1} \circ \psi) \circ \psi^{-1} $$ $$ \stackrel{*}{=} X(\psi_*Y(f) \circ \psi) \circ \psi^{-1}-Y(\psi_*X(f) \circ \psi) \circ \psi^{-1} $$ $$ \stackrel{*}{=} \psi_*X(\psi_*Y(f)) - \psi_*Y(\psi_*X(f) ) $$ $$ = [\psi_*X,\psi_*Y](f) $$
For that I assumed, that: $$ X(f) = Tf(X) $$ Is that correct?