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I need some hints to prove the following lemma (John Lee's $\textit{Introduction to Smooth Manifolds 2nd Ed}$ p.201) :

EXTENSION LEMMA FOR VECTOR FIELDS ON SUBMANIFOLDS: Suppose $M$ is a smooth manifold and $S\subseteq M$ is an embedded submanifold. Given a smooth vector field $X$ on $S$ show that there is a smooth vector field $Y$ on a neighborhood of $S$ in $M$ such that $Y=X$ on $S$. Show that every such vector field extends to all of $M$ if and only if $S$ is properly embedded.

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Take a submanifold atlas on $S$. In each coordinate chart, extend $X$ to $X_\alpha$ in $TU_\alpha$ in the canonical way. Then take a partition of unity subordinate to the $U_\alpha$ and define a vector field on $U = \cup U_\alpha$. Check that at each $p\in S$, this vector field agrees with $X$.

(Slick alternative method: Pick a Riemannian metric on $M$ and use parallel transport along geodesics perpendicular to $S$.)

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    http://math.stackexchange.com/questions/119415/extending-a-vector-field-defined-on-a-closed-submanifold2014-04-27
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    Is there any textbook that has this Lemma in it?2014-06-08
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    @Wauzl If the lemma itself is not explicitly written in it, surely you can prove it with tools from a differential topology textbook. One standard reference is Guillemin-Pollack.2014-06-08
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    @Wauzl The lemma is an exercise in Lee's Smooth Manifolds.2015-06-16
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    Could you please detail what the canonical way of extending $X$ to $X_\alpha$ is?2017-05-31