A Can Smooth Functions be Extended on Manifolds?

[JYM]

I have been stuck several days with the following problem.
Suppose M and N are smooth manifolds, U an open subset of M , and F: U → N is smooth. Show that there exists a neighborhood V of any p in U, V contained in U, such that F can be extended to a smooth mapping F*: M → N with F(q)=F*(q) for all q in V. [smooth means C-infinity].
I try to use charts to get a smooth map between Euclidean spaces and I know how to extend such maps. but I get a difficulty in transforming to the original problem. If you have some hint please well come. thanks in advance.

 

[fresh_42]

Are you sure this is the correct set-up? I do not understand the role of $U$. You have $p \in V \subseteq U \stackrel{F}{\longrightarrow} N$ no information about $F$ outside of $U$ and want to extend $F|_V$. What do you need $U$ for, and what if $F$ isn't smooth on $M-U\;$? If we don't have to bother points not in $U$, then we probably need to extend $F$ trivially outside of $U$, independent of $F$. But then, what do we need $V$ for? I'm confused.

 

[JYM]

Yes it is correct. It is given that the map F is smooth only on U.

 

[fresh_42]

Yes it is correct. It is given that the map F is smooth only on U.

Do you have a group action on $M$ which allows to transport $U$ to other neighborhoods?
As stated you could choose $V=U$.

 

[JYM]

I new for the group action concept and the problem is before that. what i want to use is charts. i upload my trial.

 

[fresh_42]

O.k. then it doesn't play a role what happens outside of $U$. All we need is any smooth function extending $F$. That's where the bump function comes into play. The w.l.o.g. assumption is only, because the language is easier this way. It works in all other cases the same way, we would only need two additional diffeomorphisms to "translate" the bump function into the general case back and forth. This would complicate the notation, but wouldn't change the statement. The restriction to $V\subseteq U$ is only to gain space for the bump function to change smoothly from $1$ inside $V$ and $0$ outside of $U$.

 

[WWGD]

I know this is generic but usually partitions of unity usually are used in these extensions. But I think you need a closedness condition, e.g., for 1/x , smooth on (0,1), which does not extend to the left.

 

[fresh_42]

I know this is generic but usually partitions of unity usually are used in these extensions. But I think you need a closedness condition, e.g., for 1/x , smooth on (0,1), which does not extend to the left.

That's where $V=(\frac{1}{4},\frac{3}{4})$ comes into play: now it extends to the left, as nobody is interested in $U=(0,1)$ anymore.

 

[WWGD]

That's where $V=(\frac{1}{4},\frac{3}{4})$ comes into play: now it extends to the left, as nobody is interested in $U=(0,1)$ anymore.

But the order seems to be wrong. If V contained in U, how are we making an extension? Isn't it a restriction in this case?

 

[fresh_42]

But the order seems to be wrong. If V contained in U, how are we making an extension? Isn't it a restriction in this case?

That's what confused me, too, at the beginning but the picture cleared it. $F^*$ has little to do with $F$ except on $V$. It is any smooth extension of $F|_V$. The restriction from $U$ to $V$ is only needed to have space for the bump. The entire theorem could as well be stated as: We always have a smooth bump function $F^*$ whenever a function $F$ is smooth somewhere, extending it at (in a neighborhood of) this point.

Something like this:

 

[mathwonk]

this is not an easy problem for a beginner but there is a standard technique for it. Using products of translates and reflections of the function e^(-1/x^2), one constructs a function that equals 1 on an interval around a given point and equals zero off a slightly larger interval, (bump function). then by taking products one does the same for a cube around a given point in n space. Then given that a manifold looks locally like an open cube, one gives the desired construction. a standard reference is perhaps spivak's calculus on manifolds.

see: https://en.wikipedia.org/wiki/Bump_function