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Math Help - Lie bracket question

  1. #1
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    Lie bracket question

    Let f:M \rightarrow M be a smooth diffeomorphism and X, Y be smooth vectorfields on M. Why is it true that
    f_* [X,Y] = [f_*X,f_*Y]? Here f_* denotes the push-forward of f.
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  2. #2
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    Just unwind the definition.
    Any smooth function g, and any point p,
    f_*(X)(g) is a smooth function and its value at f(p) is f_*(X)(g)(f(p)) = X(g \circ f)(p) , that is (f_*(X)(g)) \circ f = X(g \circ f)
    Now f_* [X,Y]_{f(p)}(g)=[X,Y]_p(g \circ f )
    = X_p(Y(g \circ f)) - Y_p(X(g \circ f))
    = X_p( (f_*(Y)(g)) \circ f ) - Y_p( (f_*(X)(g)) \circ f )
    = f_*(X)_{f(p)}(f_*(Y)(g)) - f_*(Y)_{f(p)}(f_*(X)(g))
    = [f_*(X),f_*(Y)]_{f(p)}(g)
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