Let $M$ and $N$ be manifolds and let $f:M\rightarrow N$ be a smooth map. A vector field $X$ on $M$ and a vector field $Y$ on $N$ are said to be $f$-related if
$$ df_p(X_p)=Y_{f(p)},\forall p\in M. $$This means that for every function
$$ h:N\rightarrow R $$ $$ X_p[h\circ f]=Y_{f(p)} h. $$Suppose that $X$ is $f$-related to $Z$ and $Y$ is $f$-related to $W$. Show that $[X,Y]$ is $f$-related to $[Z,W]$.
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Author of the notes: Antonio J. Pan-Collantes
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