I just finished my own proof of one of the problems in Lee's Smooth Manfolds, 2nd ed., but I wonder if anyone knows a better (less messy) solution. It's problem 8-15, the "Extension Lemma for Vector Fields on Submanifolds." Here's the part of the exercise which I had some trouble with:

Let $S$ be an embedded submanifold with or without boundary of a manifold $M$. Every vector field $X \in \mathfrak{X}(S)$ extends to all of $M$ if and only if $S$ is properly embedded.

I proved that if $S$ is properly embedded, then all vector fields extend globally easily. It's the reverse direction I had trouble with. My idea was that if $S$ is not closed in $M$, then take some point $p \in \overline{S} \setminus S$. I showed that one can find a smooth path $\gamma: (-\epsilon, \epsilon) \to M$ such that $\gamma$ is an embedding, $\gamma(0) = p$, $\gamma(-\epsilon, 0) \subset S$, and in some coordinate neighborhood of $p$, $|\gamma'(t)| = 1$. Then I defined the function $f: \gamma(-\epsilon, 0) \to \mathbb{R}$ as $f(x) = -1/\gamma^{-1}(x)$ (the idea is that this is a smooth function which blows up as $x \to p$). Finally I defined the vector field $f(x) \gamma'(\gamma^{-1}(x))$ on $\gamma[-\epsilon/2, 0)$. This is a smooth vector field on a closed subset of $S$, hence by the extension lemma proved earlier in the text, it extends to a smooth vector field on $S$. Clearly it does not extend to a smooth vector field on all of $M$ though, because in the local coordinate chart near $p$, its norm blows up.

The above argument feels very messy to me. I don't like the process I went through to find a smooth function on $S$ which blows up near $p$; surely there's a simpler way to do this?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.