# Proving The Extension Lemma For Vector Fields On Submanifolds

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.

## 1 Answer

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 $S$.)

• math.stackexchange.com/questions/119415/… – rmznyzgyr Apr 27 '14 at 19:08
• Is there any textbook that has this Lemma in it? – Wauzl Jun 8 '14 at 17:06
• @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. – Neal Jun 8 '14 at 22:09
• @Wauzl The lemma is an exercise in Lee's Smooth Manifolds. – Alex G. Jun 16 '15 at 20:44
• Could you please detail what the canonical way of extending $X$ to $X_\alpha$ is? – A. Salguero-Alarcón May 31 '17 at 17:02