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Let $M$ be a manifold and $X \in \Gamma(TM)$ a smooth vector field. Suppose that $X_p = 0$ for some $p \in M$.

Let $Y \in T_ p M$ and suppose that $Y'$ is a vector field defined on an open neighborhood $U$ of $p$ such that $Y' p = Y$ . Show that $[X_{|U} , Y' ]_p \in T_pM$ depends only on $Y$ , i.e. show that it is independent of the choice of $U$ and $Y'$ .

From the definition of $[X_{|U} , Y' ]_p = X_p (Y'f)-Y'_p(X_{|U} f)=0-Y'_p(X_{|U} f)$

Why is $-Y'_p(X_{|U} f)$ independent of U?

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You have $Y'(f)=df.Y', X(Y'f)=d^2f.(X,Y)+df.dY'.X$ this implies that $[X,Y'](f)(p)=df.(dY'.X-dX.Y')=df.(dX.Y')(p)$ since $X_p=0$.

$df.(dX.Y')(p)$ depends only on $Y'(p)$ i.e $Y'(p)=Y"(p), (Y'-Y")(p)=0$ and $df.(dX.Y')(p)=df(dX.Y")(p)$.

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