= Diffeomorphism invariance of a vector field is equivalent to commuting with its local flow
{title2=$\alpha_*X=X\iff\alpha\phi_t=\phi_t\alpha$}
The <chain rule> shows that $\alpha\phi_t\alpha^{-1}$ is the <local flow> of the <pushforward of a vector field> $\alpha_*X$. If this equals $X$, uniqueness of <integral curves of a vector field> gives commutation. Conversely, differentiate the commuting identity at $t=0$ to obtain $d\alpha_pX_p=X_{\alpha(p)}$. All identities hold on their common domains; the <vector field> need not be complete.
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