874: For C∞ Vectors Bundle and Section from Subset of Base Space Ck at Point Where 0<k, There Is Ck Extension on Open-Neighborhood-of-Point Domain

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle and section from subset of base space $C^k$ at point where $0<k$, there is $C^k$ extension on open-neighborhood-of-point domain

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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: Note
• 3: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle.
• The reader knows a definition of section of continuous surjection.
• The reader knows a definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, there is a chart trivializing open cover.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle and any section from any subset of the base space $C^k$ at any point where $0<k$, there is a $C^k$ extension on an open-neighborhood-of-point domain.

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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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$S$: $\subseteq M$
$p$: $\in S$
$s$: $:S\to {\pi }^{-1}(S)\subseteq E$, $\in \{\text{ the sections of }\pi {|}_{{\pi }^{-1}(S)}\}$, $C^l$ at $p$, where $0<l$
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Statements:
$\mathrm{\exists }{V}_p\subseteq M\in \{\text{ the open neighborhoods of }p\},\mathrm{\exists }{s}^{\prime }:V_p\to {\pi }^{-1}(V_p)\in \{\text{ the sections of }\pi {|}_{{\pi }^{-1}(V_p)}\}(s{|}_{S\cap V_p}=s^{\prime }{|}_{S\cap V_p}\wedge s^{\prime }\in \{\text{ the }{C}^l\text{ maps }\})$
//

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2: Note

Compare with the proposition that for any map from any subset of any $C^{\mathrm{\infty }}$ manifold with boundary into any subset of any $C^{\mathrm{\infty }}$ manifold $C^k$ at any point, there is a $C^k$ extension on an open-neighborhood-of-the-point domain, which requires the codomain to be a subset of a $C^{\mathrm{\infty }}$ manifold without boundary. An issue because of which that proposition cannot be directly applied is that $E$ may be with a nonempty boundary (when $M$ is with a nonempty boundary), and another issue is that the extension is required to be a section, which that proposition does not guarantee. But the basic idea of this proposition is the same with that of that proposition: the rough reason for this proposition is that while $E$ is locally $U_p\times {\mathbb{R}}^k$, the boundary can exist only in the $U_p$ part, but the extension needs to be the identity map with respect to the $U_p$ part (because it is a section), and the concern is really about only the ${\mathbb{R}}^k$ part, which has no boundary.

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3: Proof

Whole Strategy: Step 1: take a chart trivializing subset of $M$ around $p$, with the chart, $(U_p\subseteq M,{\varphi }_p)$, and the induced chart, $({\pi }^{-1}(U_p)\subseteq E,\widetilde{{\varphi }_p})$; Step 2: for the components function, $f:=\widetilde{{\varphi }_p}\circ s\circ {{\varphi }_p}^{-1}{|}_{{\varphi }_p(U_p\cap S)}:{\varphi }_p(U_p\cap S)\to \widetilde{{\varphi }_p}({\pi }^{-1}(U_p))$, take an open neighborhood of ${\varphi }_p(p)$, $U_{{\varphi }_p(p)}\subseteq {\mathbb{R}}^d$, and an extension of $f$, $f^{\prime }:U_{{\varphi }_p(p)}\to {\mathbb{R}}^{d+k}$; Step 3: tweak $f^{\prime }$ to have $f^{″}:U_{{\varphi }_p(p)}\to {\mathbb{R}}^k\times {\varphi }_p(U_p)$; Step 4: take $V_p:={{\varphi }_p}^{-1}({\varphi }_p(U_p)\cap U_{{\varphi }_p(p)})$ and $s^{\prime }:={\widetilde{{\varphi }_p}}^{-1}\circ f^{″}\circ {\varphi }_p{|}_{V_p}\to {\pi }^{-1}(V_p)$; Step 5: see that $s^{\prime }$ satisfies the requirements.

Step 1:

Let us take a chart trivializing subset of $M$ around $p$, with the chart, $(U_p\subseteq M,{\varphi }_p)$, which is possible by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, there is a chart trivializing open cover.

Let us take the induced chart, $({\pi }^{-1}(U_p)\subseteq E,\widetilde{{\varphi }_p})$, which is possible by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.

$s(U_p)\subseteq {\pi }^{-1}(U_p)$, because $s$ is a section.

Step 2:

Let the components function be $f:=\widetilde{{\varphi }_p}\circ s\circ {{\varphi }_p}^{-1}{|}_{{\varphi }_p(U_p\cap S)}:{\varphi }_p(U_p\cap S)\to \widetilde{{\varphi }_p}({\pi }^{-1}(U_p))$.

There is an open neighborhood of ${\varphi }_p(p)$, $U_{{\varphi }_p(p)}\subseteq {\mathbb{R}}^d$, and a $C^l$ extension of $f$, $f^{\prime }:U_{{\varphi }_p(p)}\to {\mathbb{R}}^{d+k}$, because that is what the definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$ requires.

Step 3:

Denote $f^{\prime }:(x^1,...,x^d)\mapsto (f^{\prime 1}(x^1,...,x^d),...,f^{\prime k}(x^1,...,x^d),f^{\prime k+1}(x^1,...,x^d),...,,f^{\prime d+k}(x^1,...,x^d))$. Each $f^{\prime j}(x^1,...,x^d)$ is $C^l$.

Let us define $f^{″}:U_{{\varphi }_p(p)}\to {\mathbb{R}}^k\times U_{{\varphi }_p(p)},(x^1,...,x^d)\mapsto (f^{\prime 1}(x^1,...,x^d),...,f^{\prime k}(x^1,...,x^d),x^1,...,,x^d)$.

$f^{″}$ is obviously $C^l$.

$f{|}_{{\varphi }_p(U_p\cap S)}=f^{\prime }{|}_{{\varphi }_p(U_p\cap S)}=f^{″}{|}_{{\varphi }_p(U_p\cap S)}$, because $f{|}_{{\varphi }_p(U_p\cap S)}(x^1,...,x^d)=f^{\prime }{|}_{{\varphi }_p(U_p\cap S)}(x^1,...,x^d)=(f^{\prime 1}(x^1,...,x^d),...,f^{\prime k}(x^1,...,x^d),f^{\prime k+1}(x^1,...,x^d),...,,f^{\prime d+k}(x^1,...,x^d))=(f^{\prime 1}(x^1,...,x^d),...,f^{\prime k}(x^1,...,x^d),x^1,...,,x^d)=f^{″}{|}_{{\varphi }_p(U_p\cap S)}(x^1,...,x^d)$, because $s$ is fiber-preserving and $\widetilde{{\varphi }_p}$ is induced from ${\varphi }_p$.

So, $f^{″}$ is a $C^l$ extension of $f$.

Step 4:

${\varphi }_p(U_p)\cap U_{{\varphi }_p(p)}\subseteq {\varphi }_p(U_p)$ is an open neighborhood of ${\varphi }_p(p)$ on ${\varphi }_p(U_p)$.

Let us define $V_p:={{\varphi }_p}^{-1}({\varphi }_p(U_p)\cap U_{{\varphi }_p(p)})\subseteq M$ such that $V_p\subseteq U_p$, which is an open neighborhood of $p$.

Let us define $s^{\prime }:={\widetilde{{\varphi }_p}}^{-1}\circ f^{″}\circ {\varphi }_p{|}_{V_p}:V_p\to {\pi }^{-1}(V_p)$, which is possible because $f^{″}{|}_{{\varphi }_p{|}_{V_p}(V_p)}$ is into ${\mathbb{R}}^k\times (U_{{\varphi }_p(p)}\cap {\varphi }_p(U_p))\subseteq {\mathbb{R}}^k\times {\varphi }_p(U_p)=\widetilde{{\varphi }_p}({\pi }^{-1}(U_p))$.

Step 5:

$s^{\prime }$ is indeed a section, obviously.

$s^{\prime }$ is $C^l$, because while ${\varphi }_p{|}_{V_p}$ is $C^{\mathrm{\infty }}$ as $:V_p\to {\varphi }_p(V_p)\subseteq {\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$, $f^{″}$ is $C^l$ as $:U_{{\varphi }_p(p)}\subseteq {\mathbb{R}}^d\to {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$, and ${\widetilde{{\varphi }_p}}^{-1}$ is $C^{\mathrm{\infty }}$ as $:{\mathbb{R}}^k\times {\varphi }_p(U_p)\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}\to {\pi }^{-1}(U_p)\subseteq E$, the only concern for $s^{\prime }$ to be a legitimate chain of $C^l$ maps is that ${\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$ for the codomain of ${\varphi }_p{|}_{V_p}$ is different from ${\mathbb{R}}^d$ for the domain of $f^{″}$, but ${\varphi }_p{|}_{V_p}$ can be regarded to be ${\varphi }_p{|}_{V_p}:V_p\to {\varphi }_p(V_p)\subseteq {\mathbb{R}}^d$, which obviously does not change $C^{\mathrm{\infty }}$-ness, so, $s^{\prime }$ is indeed a legitimate chain of $C^l$ maps, and is $C^l$, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

$s^{\prime }{|}_{V_p\cap S}=s{|}_{V_p\cap S}$, because $s^{\prime }{|}_{V_p\cap S}={\widetilde{{\varphi }_p}}^{-1}\circ f^{″}\circ {\varphi }_p{|}_{V_p}{|}_{V_p\cap S}={\widetilde{{\varphi }_p}}^{-1}\circ f\circ {\varphi }_p{|}_{V_p}{|}_{V_p\cap S}={\widetilde{{\varphi }_p}}^{-1}\circ \widetilde{{\varphi }_p}\circ s\circ {{\varphi }_p}^{-1}{|}_{{\varphi }_p(U_p\cap S)}\circ {\varphi }_p{|}_{V_p}{|}_{V_p\cap S}=s{|}_{V_p}{|}_{V_p\cap S}=s{|}_{V_p\cap S}$.

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References

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