71: For C∞ Function on Open Neighborhood, There Exists C∞ Function on C∞ Manifold with Boundary That Equals Function on Possibly Smaller Neighborhood

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description/proof of that for $C^{\mathrm{\infty }}$ function on open neighborhood, there exists $C^{\mathrm{\infty }}$ function on $C^{\mathrm{\infty }}$ manifold with boundary that equals function on possibly smaller neighborhood

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof
• 4: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, Where $k$ Includes $\mathrm{\infty }$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any closed subset, and any open neighborhood of the subset, there is a $C^{\mathrm{\infty }}$ bump function that is supported in the open neighborhood and is $1$ on a neighborhood of the closed subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ function on any point open neighborhood of any $C^{\mathrm{\infty }}$ manifold with boundary, there exists a $C^{\mathrm{\infty }}$ function on the whole manifold with boundary that equals the original function on a possibly smaller neighborhood of the point.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$p_0$: $\in M$
$U_{p_0}$: $\in \{\text{ the open neighborhoods of }{p}_0\text{ on }M\}$
$f$: $:U_{p_0}\to \mathbb{R}$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
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Statements:
$\mathrm{\exists }\tilde{f}:M\to \mathbb{R},\mathrm{\exists }{N}_{p_0}\in \{\text{ the neighborhoods of }{p}_0\text{ on }M\}\text{ such that }{N}_{p_0}\subseteq U_{p_0}(f{|}_{N_{p_0}}=\tilde{f}{|}_{N_{p_0}})$
//

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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifold with boundary, $M$, any point, $p_0\in M$, any open neighborhood, $U_{p_0}\subseteq M$, of $p_0$, and any $C^{\mathrm{\infty }}$ map, $f:U_{p_0}\to \mathbb{R}$, there is a $C^{\mathrm{\infty }}$ function, $\tilde{f}:M\to \mathbb{R}$, that equals the original function, $f$, on a possibly smaller neighborhood, $N_{p_0}\subseteq M$, of the point, $p_0$.

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3: Proof

Whole Strategy: Step 1: take a $C^{\mathrm{\infty }}$ bump function, $b:M\to \mathbb{R}$, that is supported in $U_{p_0}$ and equals $1$ on a possibly smaller neighborhood, $N_{p_0}\subseteq U_{p_0}$; Step 2: define $\tilde{f}(p):=b(p)f(p)\text{ for }p\in U_{p_0};0\text{ for }p\notin U_{p_0}$; Step 3: see that $\tilde{f}$ satisfies the requirements.

Step 1:

There is a $C^{\mathrm{\infty }}$ bump function, $b:M\to \mathbb{R}$, that is supported in $U_{p_0}$ and equals $1$ on a possibly smaller neighborhood, $N_{p_0}\subseteq U_{p_0}$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any closed subset, and any open neighborhood of the subset, there is a $C^{\mathrm{\infty }}$ bump function that is supported in the open neighborhood and is $1$ on a neighborhood of the closed subset: take $\{p_0\}$ as the closed subset and $U_{p_0}$ as the open neighborhood.

Step 2:

Let us define $\tilde{f}(p):=b(p)f(p)\text{ for }p\in U_{p_0};0\text{ for }p\notin U_{p_0}$.

Step 3:

$\tilde{f}$ is $C^{\mathrm{\infty }}$, because for each $p\in U_{p_0}$, it is a product of $C^{\mathrm{\infty }}$ functions, $C^{\mathrm{\infty }}$; for each $p\notin U_{p_0}$, $p\notin suppb$ because $suppb\subseteq U_{p_0}$, and as $suppb$ is closed, $M\setminus suppb$ is open where $p$ belongs, and so, there is a neighborhood of $p$ contained in $M\setminus suppb$ in which $b$ is $0$, and so, $\tilde{f}$ is $0$ there, $C^{\mathrm{\infty }}$; as $\tilde{f}$ is $C^{\mathrm{\infty }}$ on an open neighborhood of each point on $M$, $\tilde{f}$ is $C^{\mathrm{\infty }}$.

On $N_{p_0}$, $\tilde{f}=f$ because $b=1$ there.

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4: Note

An expression like "$f$ can be extended to $\tilde{f}$" tends to be used, but $\tilde{f}$ is not really any extension of $f$, because $\tilde{f}$ is not guaranteed to equal $f$ on the whole $U_{p_0}$.

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References

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