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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 !

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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

1 Answer 1

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 $f.\phi$.

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That's excatly what I was missing. Thanks ! –  joshua Feb 27 '12 at 10:42

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