Let $\displaystyle f:M \rightarrow M$ be a smooth diffeomorphism and $\displaystyle X, Y$ be smooth vectorfields on M. Why is it true that
$\displaystyle f_* [X,Y] = [f_*X,f_*Y]$? Here $\displaystyle f_*$ denotes the push-forward of f.
Let $\displaystyle f:M \rightarrow M$ be a smooth diffeomorphism and $\displaystyle X, Y$ be smooth vectorfields on M. Why is it true that
$\displaystyle f_* [X,Y] = [f_*X,f_*Y]$? Here $\displaystyle f_*$ denotes the push-forward of f.
Just unwind the definition.
Any smooth function g, and any point p,
$\displaystyle f_*(X)(g)$ is a smooth function and its value at f(p) is $\displaystyle f_*(X)(g)(f(p)) = X(g \circ f)(p) $, that is $\displaystyle (f_*(X)(g)) \circ f = X(g \circ f)$
Now $\displaystyle f_* [X,Y]_{f(p)}(g)=[X,Y]_p(g \circ f )$
$\displaystyle = X_p(Y(g \circ f)) - Y_p(X(g \circ f))$
$\displaystyle = X_p( (f_*(Y)(g)) \circ f ) - Y_p( (f_*(X)(g)) \circ f )$
$\displaystyle = f_*(X)_{f(p)}(f_*(Y)(g)) - f_*(Y)_{f(p)}(f_*(X)(g))$
$\displaystyle = [f_*(X),f_*(Y)]_{f(p)}(g)$