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Given a smooth map $f: M\rightarrow N$ between smooth manifolds, by definition (wiki reference or this post) we have a pull back tangent bundle $f^{*}TN\subset M\times TN$.

Can we define a Lie bracket $[X,Y]$ for $X,Y\in f^{*}TN$?

(It should because I have seen some property like $f_{*}([X,Y])=[f_{*}X,f_{*}Y]$ where $X,Y\in TM$. (ref: Answers in this post))

But there's a problem, in convention $[X,Y]=XY-YX$, but what does $X(Y)-Y(X)$ mean exactly for $X, Y\in f^{*}TN$? For $p\in M$, given $h\in C^{\infty}{(V)}$, where $f(p)\in V$ open in $N$, it seems we should have $[X,Y]|_{p} h =X_p({Y(h)})-Y_p({X(h)})$. But $X_p\in T_{f(p)}N$, therefore $Y(h)$ should be in $C^{\infty}(V)$, but how could it be defined?

CYC
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  • What does the juxtaposition $XY$ mean in this situation? Sections of $f^*(TN)$ are not derivations. – Kajelad Oct 14 '20 at 09:18
  • I have no idea, so $[f_{}X, f_{}Y]$ doesn’t make sense? – CYC Oct 14 '20 at 09:45
  • What are $X$ and $Y$, and $f_$ in that expression? The differential $f_$ maps $TM$ to $TN$; it doesn't act in an obvious way on $f^*(TN)$. – Kajelad Oct 14 '20 at 09:50
  • I've added some reference, I'm just reading some notes without detail explanation, so sorry I'm not exactly sure. Is there a possible explanation, possibly with abuse of notations, that make $[f_{}X, f_{}Y]$ make sense? – CYC Oct 14 '20 at 10:38

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