I know that not necessarily you can extend a smooth vector field defined over a subset of a manifold to ALL of the maniffold, but, can you extend it at least to an open set? (Of course I'm talking about smooth extensions)
[Math] Can you extend vector fields on a manifold
differential-geometry
Best Answer
You're probably already aware of the result that not every smooth vector field defined on an open subset admits a smooth extension to a larger open set. For example, the vector field $V(x) = \frac{1}{x(x-1)}\ \mathbf{e_1}$ on $(0,1)\subset\mathbb{R}$ admits no smooth extension to a larger open set.
On the other hand, one usually defines a smooth vector field on a non-open subset $A$ as one that admits a smooth extension in a neighborhood of every point of $A$. (More precisely, a smooth vector field $V$ on a set $A$ is one that for every $p \in A$, there is an open set $U$ containing $p$ and a smooth extension $\widetilde{V}$ defined on $U$ whose restriction to $U \cap A$ agrees with $V$.)
However, it is a fact that every smooth vector field $V$ defined on an embedded submanifold $S \subset M$ that is closed as a subset of $M$ admits a smooth extension to all of $M$. I don't remember how the proof goes offhand, but it probably involves a partition of unity argument.