= Projectable vector field
A <vector field> $X$ is projectable through a <smooth map between manifolds> $f:M\to N$ if there is a smooth vector field $Y$ on $N$ with $df_pX_p=Y_{f(p)}$ for every $p$. Agreement on fibres is necessary. For a surjective <submersion> it is also sufficient: smooth local sections of the submersion express $Y$ locally as $df(X)$ and prove its smoothness. An arbitrary non-surjective map may instead give a field only along its image, with extension to $N$ requiring additional choices.
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