Smooth Extension of Locally Defined Function on Manifold

In summary, a bump function is used to extend a locally defined function to a smooth function on the whole manifold.
  • #1
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how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?
 
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  • #2
AFAIK, partitions of unity are used to extend. I have not seen bump functions, since I think these are defined on ℝn, and are real-valued. Could you tell us the method you are referring to?
 
  • #3
Hi,
i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.
i don't see how such an extention is done.
thank's
Hedi
 
  • #4
hedipaldi said:
Hi,
i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.
i don't see how such an extention is done.
thank's
Hedi

For instance, say F is defined on a coordinate chart U diffeomorphic to R^n. Let D(1) be the closed subset of U corresponding to the closed unit disk under the coordinate map, and let B(2) be the open subset of U corresponding to the open disk of radius 2 under the coordinate map. Then you know there is a smooth bump function b that is equal to 1 on all of D(1), and has support in B(2), right? So define F', an "extension" of F, by setting

F'(x):=b(x)F(x) if x is in U
F'(x)=0 otherwise

It is obvious that F' is smooth since it is smooth on U and it is smooth "away from U": the only potentially problematic points are at the boundary of U. But at those points, F' is locally identically 0 by construction so all is well.

Now F' is not an extension of F in the strictest sense since we have modified it outside of D(1). But for many purposes, this is just fine.
 
  • #5
It was very helpfull indeed.Thank"s a lot.
Hedi
 

1. What is a smooth extension of a locally defined function on a manifold?

A smooth extension of a locally defined function on a manifold is a way of extending a function defined on a subset of a manifold to a larger domain while maintaining smoothness. This allows the function to be defined and continuous on the entire manifold, rather than just a portion of it.

2. Why is smooth extension important in manifold theory?

Smooth extension is important in manifold theory because it allows for the study of functions on the entire manifold, rather than just on a subset. This can provide a more complete understanding of the manifold and its properties, and is often necessary in order to apply other mathematical tools and techniques.

3. How is a smooth extension of a locally defined function achieved?

A smooth extension of a locally defined function on a manifold can be achieved by using techniques such as partitions of unity or bump functions. These methods allow for the construction of a smooth function that agrees with the original function on the subset where it is defined, and is also smooth on the rest of the manifold.

4. What conditions must be satisfied for a smooth extension to exist?

In order for a smooth extension to exist, the original function must be continuous and at least C^k (k times continuously differentiable) on the subset of the manifold where it is defined. Additionally, the manifold must be a paracompact space, which ensures that a partition of unity exists.

5. Can any locally defined function on a manifold be smoothly extended?

No, not all locally defined functions on a manifold can be smoothly extended. There are certain functions that are not continuous or differentiable, and therefore cannot be extended in a smooth manner. Additionally, the existence of a smooth extension depends on the specific properties of the manifold and the function in question.

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