Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I really need your help in understanding the following statement in the proof of the extension lemma in Lee's book: Let $A \subseteq M^n $ be a closed submanifold of dimension $k$ , and let $F:A \to \mathbb{R} $ be a smooth function. We want to extend this function to the entire $M$ . THe problem is that I can't understand how to do it locally- Let $ p \in A$ and let $ W_p $ be a neighborhood of $ p$ in $ M$ . I only know how to extend $F$ to a slice chart of $ p$ using projection . How can I do it for $ W_p$ ? The other problem is that the statement should also be true when $A$ is a closed subset of $M$, not necessarily a closed submanifold :(

i.e. how can I extend the function $F$ to a smooth function on $ W_p$ ?

Hope you'll be able to help me !

Thanks !

share|cite|improve this question
I assume, you are asking for smooth extension, I am assuming that $M$ is locally compact and hence $M$ admits Smooth partition of unity. – zapkm Feb 27 '12 at 10:16
When $A$ is just an arbitrary closed subset of $M$, the statement follows from a partition of unity argument and Whitney's extension theorem, which is a nontrivial result in harmonic analysis of Euclidean spaces. The submanifold case is much, much easier. – Willie Wong Feb 27 '12 at 10:27
I'm not that sure about the Whitney's extension theorem, but thanks, I think I understand it now. – joshua Feb 27 '12 at 10:43

I am not sure, if i understood you correctly: Please have following answer:

If $A \subset M^n$ manifold of $k$ dimension.For every point $p\in M$ there is a neighborhood $U_p$ of $p$ in $M$, if points of neighborhood of $U_p$ can be written as $(x_1,x_2,...x_n)$ then in sufficiently small neighborhood, we can think $$U_p\to A \text{,by}$$ $$p:(x_1,x_2,...x_n)\to (x_1,x_2,..x_k)$$ as submersion. Now consider $f \circ p: U\to \mathbb R$. This is a smooth map which is extension of $f$. Now If $M$ has smooth partition of unity $\phi$ subordinate to (locally compact) family $U_p$ , then we have extension of $f$ as $f.\phi$.

share|cite|improve this answer
That's excatly what I was missing. Thanks ! – joshua Feb 27 '12 at 10:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.