867: Restricted C∞ Vectors Bundle

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definition of restricted $C^{\mathrm{\infty }}$ vectors bundle

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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

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle.
• The reader knows a definition of immersed submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary of the manifold with boundary.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.
• The reader admits 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.
• The reader admits the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
• The reader admits the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.

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Target Context

• The reader will have a definition of restricted $C^{\mathrm{\infty }}$ vectors bundle.

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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:
$M^{\prime }$: $\in \{\text{ the }{d}^{\prime }\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$E^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$(E^{\prime },M^{\prime },{\pi }^{\prime })$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$M$: $\in \{\text{ the }d\text{ -dimensional immersed submanifolds with boundary of }M\}$
$\iota$: $:M\to M^{\prime }$, $=\text{ the inclusion }$
$E$: $={\pi }^{\prime -1}(M)\subseteq E^{\prime }$ with the topology and the atlas specified below
$\pi$: $=\pi {|}_E:E\to M$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ locally trivial surjections of rank }k\}$
$\ast (E,M,\pi )$:
//

Conditions:
For each $m\in M$, take a trivializing open subset around $m\in M^{\prime }$, $U_m^{\prime }\subseteq M^{\prime }$.
Take a chart around $m\in M$, $(U_m\subseteq M,{\varphi }_m)$, such that $\iota (U_m)\subseteq U_m^{\prime }$ and $U_m$ is an embedded submanifold with boundary of $M^{\prime }$: as $\iota$ is continuous, an open neighborhood of $m$ on $M$ can be taken to be mapped into $U_m^{\prime }$ under the inclusion and a chart domain can be taken inside the open neighborhood, while the chart domain can be an embedded submanifold with boundary of $M^{\prime }$ because any immersed submanifold with boundary is locally an embedded submanifold with boundary.
$\{U_m|m\in M\}$ is an open cover of $M$, and take any countable subcover, $\{U_{\beta }|\beta \in B\}$, which is possible by the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
Denote the corresponding trivializing open subsets as $\{U_{\beta }^{\prime }|\beta \in B\}$ and take trivializations, $\{{\mathrm{\Phi }}_{\beta }^{\prime }:{\pi }^{\prime -1}(U_{\beta }^{\prime })\to U_{\beta }^{\prime }\times {\mathbb{R}}^k|\beta \in B\}$.
Let ${\mathrm{\Phi }}_{\beta }:{\pi }^{-1}(U_{\beta })\to U_{\beta }\times {\mathbb{R}}^k:=({{\iota }_{\beta }^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{\beta }^{\prime }\circ {\tau }_{\beta }$, where ${\tau }_{\beta }:{\pi }^{-1}(U_{\beta })\to {\pi }^{\prime -1}(U_{\beta }^{\prime })$ is the inclusion, ${\iota }_{\beta }:U_{\beta }\to M^{\prime }$ is the inclusion, and ${\iota }_{\beta }^{\prime }:U_{\beta }\to {\iota }_{\beta }(U_{\beta })\subseteq M^{\prime }$ is the codomain restriction of ${\iota }_{\beta }$.
Let $\lambda :{\mathbb{R}}^{d+k}\to {\mathbb{R}}^{d+k},(r^1,...,r^d,r^{d+1},...,r^{d+k})\mapsto (r^{d+1},...,r^{d+k},r^1,...,r^d)$.
Let $\widetilde{{\varphi }_{\beta }}:{\pi }^{-1}(U_{\beta })\to {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$, $=\lambda \circ ({\varphi }_{\beta },id)\circ {\mathrm{\Phi }}_{\beta }$.
Make the to-be-atlas for $E$, $\{({\pi }^{-1}(U_{\beta }),\widetilde{{\varphi }_{\beta }})|\beta \in B\}$, determine the topology and the atlas of $E$, by the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
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2: Note

Let us see that $\{({\pi }^{-1}(U_{\beta }),\widetilde{{\varphi }_{\beta }})|\beta \in B\}$ is indeed a to-be-atlas mentioned in the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.

$\{{\pi }^{-1}(U_{\beta })|\beta \in B\}$ is indeed a countable cover of $E$.

$\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }))\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$ is open, because $\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }))=\lambda \circ ({\varphi }_{\beta },id)\circ {\mathrm{\Phi }}_{\beta }({\pi }^{-1}(U_{\beta }))=\lambda \circ ({\varphi }_{\beta },id)\circ ({{\iota }_{\beta }^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{\beta }^{\prime }\circ {\tau }_{\beta }({\pi }^{-1}(U_{\beta }))=\lambda \circ ({\varphi }_{\beta },id)(U_{\beta }\times {\mathbb{R}}^k)=\lambda ({\varphi }_{\beta }(U_{\beta })\times {\mathbb{R}}^k)={\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$, which is open on ${\mathbb{R}}^{d+k}$ or ${\mathbb{H}}^{d+k}$.

$\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(U_{{\beta }^{\prime }}))=\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }\cap U_{{\beta }^{\prime }}))={\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }})$, which is open on $\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }))={\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$, because $U_{\beta }\cap U_{{\beta }^{\prime }}$ is open on $U_{\beta }$, so, ${\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }})$ is open on ${\varphi }_{\beta }(U_{\beta })$.

$\widetilde{{\varphi }_{\beta }}$ is obviously injective.

$\widetilde{{\varphi }_{{\beta }^{\prime }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(U_{{\beta }^{\prime }}))}=\lambda \circ ({\varphi }_{{\beta }^{\prime }},id)\circ {\mathrm{\Phi }}_{{\beta }^{\prime }}\circ {{\mathrm{\Phi }}_{\beta }}^{-1}\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))}$, because $\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(U_{{\beta }^{\prime }}))=\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }\cap U_{{\beta }^{\prime }}))=\lambda \circ ({\varphi }_{\beta },id)\circ {\mathrm{\Phi }}_{\beta }({\pi }^{-1}(U_{\beta }\cap U_{{\beta }^{\prime }}))=\lambda \circ ({\varphi }_{\beta },id)((U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k)=\lambda \circ ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}),{\mathbb{R}}^k)={\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))$.

$=\lambda \circ ({\varphi }_{{\beta }^{\prime }},id)\circ ({{\iota }_{{\beta }^{\prime }}^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{{\beta }^{\prime }}^{\prime }\circ {\tau }_{{\beta }^{\prime }}\circ (({{\iota }_{\beta }^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{\beta }^{\prime }\circ {\tau }_{\beta }{)}^{-1}\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))}=\lambda \circ ({\varphi }_{{\beta }^{\prime }},id)\circ ({{\iota }_{{\beta }^{\prime }}^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{{\beta }^{\prime }}^{\prime }\circ {\tau }_{{\beta }^{\prime }}\circ {{\tau }_{\beta }}^{-1}\circ {{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}\circ ({{\iota }_{\beta }^{\prime }}^{-1},id{)}^{-1}\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))}=\lambda \circ ({\varphi }_{{\beta }^{\prime }},id)\circ ({{\iota }_{{\beta }^{\prime }}^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{{\beta }^{\prime }}^{\prime }\circ {\tau }_{{\beta }^{\prime }}\circ {{\tau }_{\beta }^{\prime }}^{-1}\circ {{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}\circ ({\iota }_{\beta }^{\prime },id)\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))}$, where ${\tau }_{\beta }^{\prime }:{\pi }^{-1}(U_{\beta })\to {\tau }_{\beta }({\pi }^{-1}(U_{\beta }))\subseteq {\pi }^{\prime -1}(U_{\beta }^{\prime })$ is the codomain restriction of ${\tau }_{\beta }$.

$=\lambda \circ ({\varphi }_{{\beta }^{\prime }},id)\circ ({{\iota }_{{\beta }^{\prime }}^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{{\beta }^{\prime }}^{\prime }\circ {{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}\circ ({\iota }_{\beta }^{\prime },id)\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))}$, which is $:{\mathbb{R}}^k\times ({\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }}))\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}\to {\varphi }_{\beta }(U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k\to (U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq U_{\beta }\times {\mathbb{R}}^k\to (U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq M^{\prime }\times {\mathbb{R}}^k\to {\pi }^{\prime -1}(U_{\beta }\cap U_{{\beta }^{\prime }})\subseteq {\pi }^{\prime -1}(U_{\beta }^{\prime }\cap U_{{\beta }^{\prime }}^{\prime })\to (U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq M^{\prime }\times {\mathbb{R}}^k\to (U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq U_{{\beta }^{\prime }}\times {\mathbb{R}}^k\to {\varphi }_{{\beta }^{\prime }}(U_{\beta }\cap U_{{\beta }^{\prime }})\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k\to {\mathbb{R}}^k\times {\varphi }_{{\beta }^{\prime }}(U_{\beta }\cap U_{{\beta }^{\prime }})\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$.

The point is that ${{\iota }_{{\beta }^{\prime }}^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$. In fact, in order for that, we need $U_{\beta }$ to be embedded in $M^{\prime }$.

${{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}$ and ${\mathrm{\Phi }}_{{\beta }^{\prime }}^{\prime }$ are $C^{\mathrm{\infty }}$, because they are some trivializations for the established $C^{\mathrm{\infty }}$ vectors bundle, $(E^{\prime },M^{\prime },{\pi }^{\prime })$.

Each constituent of $\widetilde{{\varphi }_{{\beta }^{\prime }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(U_{{\beta }^{\prime }}))}$ is $C^{\mathrm{\infty }}$, by the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point and the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

So, $\widetilde{{\varphi }_{{\beta }^{\prime }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(U_{{\beta }^{\prime }}))}$ is a legitimate chain of $C^{\mathrm{\infty }}$ maps, and is $C^{\mathrm{\infty }}$, 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.

Also $\widetilde{{\varphi }_{\beta }}\circ {\widetilde{{\varphi }_{{\beta }^{\prime }}}}^{-1}{|}_{\widetilde{{\varphi }_{{\beta }^{\prime }}}({\pi }^{-1}(U_{{\beta }^{\prime }})\cap {\pi }^{-1}(U_{\beta }))}$ is $C^{\mathrm{\infty }}$, by the symmetry.

Let us see that $E$ is Hausdorff: see Note for the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas. Let $e,e^{\prime }\in E$ be any such that $e\ne e^{\prime }$. When $\pi (e)=\pi (e^{\prime })$, there is a ${\pi }^{-1}(U_{\beta })$ such that $e,e^{\prime }\in {\pi }^{-1}(U_{\beta })$. ${\pi }^{-1}(U_{\beta })$ is homeomorphic to ${\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$ and $\widetilde{{\varphi }_{\beta }}(e)\ne \widetilde{{\varphi }_{\beta }}(e^{\prime })$, and there are an open neighborhood of $\widetilde{{\varphi }_{\beta }}(e)$, $U_{\widetilde{{\varphi }_{\beta }}(e)}\subseteq {\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$, and an open neighborhood of $\widetilde{{\varphi }_{\beta }}(e^{\prime })$, $U_{\widetilde{{\varphi }_{\beta }}(e^{\prime })}\subseteq {\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$, such that $U_{\widetilde{{\varphi }_{\beta }}(e)}\cap U_{\widetilde{{\varphi }_{\beta }}(e^{\prime })}=\mathrm{\varnothing }$, because ${\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$ is Hausdorff, because ${\mathbb{R}}^{d+k}$ or ${\mathbb{H}}^{d+k}$ is Hausdorff. Then, ${\widetilde{{\varphi }_{\beta }}}^{-1}(U_{\widetilde{{\varphi }_{\beta }}(e)})$ and ${\widetilde{{\varphi }_{\beta }}}^{-1}(U_{\widetilde{{\varphi }_{\beta }}(e^{\prime })})$ are some open neighborhoods of $e$ and $e^{\prime }$ respectively, and ${\widetilde{{\varphi }_{\beta }}}^{-1}(U_{\widetilde{{\varphi }_{\beta }}(e)})\cap {\widetilde{{\varphi }_{\beta }}}^{-1}(U_{\widetilde{{\varphi }_{\beta }}(e^{\prime })}=\mathrm{\varnothing }$. When $\pi (e)\ne \pi (e^{\prime })$, there are an open neighborhood of $\pi (e)$, $U_{\pi (e)}\subseteq M$, and an open neighborhood of $\pi (e^{\prime })$, $U_{\pi (e^{\prime })}\subseteq M$, such that $U_{\pi (e)}\cap U_{\pi (e^{\prime })}=\mathrm{\varnothing }$, because $M$ is Hausdorff. $U_{\pi (e)}$ and $U_{\pi (e^{\prime })}$ can be taken to be contained in $U_{\beta }$ and $U_{{\beta }^{\prime }}$ respectively, where it may be or not be that $U_{\beta }=U_{{\beta }^{\prime }}$. ${\pi }^{-1}(U_{\pi (e)})\subseteq {\pi }^{-1}(U_{\beta })$ is open because $\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\pi (e)}))={\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\pi (e)})\subseteq {\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta })$, which is open. Likewise, ${\pi }^{-1}(U_{\pi (e^{\prime })})\subseteq {\pi }^{-1}(U_{\beta }^{\prime })$ is open. ${\pi }^{-1}(U_{\pi (e)})\cap {\pi }^{-1}(U_{\pi (e^{\prime })})=\mathrm{\varnothing }$.

So, $\{({\pi }^{-1}(U_{\beta }),\widetilde{{\varphi }_{\beta }})|\beta \in B\}$ is indeed a to-be-atlas mentioned in the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.

$\pi$ is $C^{\mathrm{\infty }}$: for each $e\in E$, there is a $\beta \in B$ such that $e\in {\pi }^{-1}(U_{\beta })$; take the charts, $({\pi }^{-1}(U_{\beta })\subseteq E,\widetilde{{\varphi }_{\beta }})$ and $(U_{\beta }\subseteq M,{\varphi }_{\beta })$; ${\varphi }_{\beta }\circ \pi \circ {\widetilde{{\varphi }_{\beta }}}^{-1}$ is $:(v,{\varphi }_{\beta }(p)\mapsto {\varphi }_{\beta }(p))$, which is obviously $C^{\mathrm{\infty }}$.

Let us see that ${\mathrm{\Phi }}_{\beta }$ is a trivialization of rank $k$.

As $\widetilde{{\varphi }_{\beta }}=\lambda \circ ({\varphi }_{\beta },id)\circ {\mathrm{\Phi }}_{\beta }$, ${\mathrm{\Phi }}_{\beta }=({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}\circ \widetilde{{\varphi }_{\beta }}=({{\varphi }_{\beta }}^{-1},id)\circ {\lambda }^{-1}\circ \widetilde{{\varphi }_{\beta }}$, which is diffeomorphic, because ${{\varphi }_{\beta }}^{-1}$ and $\widetilde{{\varphi }_{\beta }}$ are diffeomorphic.

For each $m\in U_{\beta }$, ${\mathrm{\Phi }}_{\beta }{|}_{{\pi }^{-1}(m)}$ is 'vectors spaces - linear morphisms' isomorphic, because it equals ${\mathrm{\Phi }}_{\beta }^{\prime }{|}_{{\pi }^{-1}(m)}$.

So, $(E,M,\pi )$ is indeed a $C^{\mathrm{\infty }}$ vectors bundle.

Let us see that the topology and the atlas are uniquely defined: the procedure above ostensibly depends on the choices of $\{(U_{\beta },{\varphi }_{\beta })|\beta \in B\}$ and $\{{\mathrm{\Phi }}_{\beta }^{\prime }|\beta \in B\}$.

Let another choices be $\{(\overline{U_{\gamma }},\overline{{\varphi }_{\gamma })}|\gamma \in \bar{B}\}$ and $\{\overline{{\mathrm{\Phi }}_{\gamma }^{\prime }}|\gamma \in \bar{B}\}$.

Then, we have $\overline{{\mathrm{\Phi }}_{\gamma }}=({{\iota }_{\gamma }^{\prime }}^{-1},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }^{\prime }}\circ {\tau }_{\gamma }$ and $\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}:{\pi }^{-1}(\overline{U_{\gamma }})\to {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}=\lambda \circ (\overline{{\varphi }_{\gamma }},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }}$.

We are going to apply the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.

$\{{\pi }^{-1}(U_{\beta })|\beta \in B)\}$ is a chart domains open cover for the former topology-atlas pair and $\{{\pi }^{-1}(\overline{U_{\gamma }})|\gamma \in \bar{B})\}$ is a chart domains open cover for the latter topology-atlas pair. $\{{\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }})|(\beta ,\gamma )\in B\times \bar{B})\}$ is a countable common chart domains open cover, because ${\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }})={\pi }^{-1}(U_{\beta }\cap \overline{U_{\gamma }})={{\mathrm{\Phi }}_{\beta }}^{-1}((U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k)$, which is open on ${\pi }^{-1}(U_{\beta })$, and likewise, it is open on ${\pi }^{-1}(\overline{U_{\gamma }})$.

$\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }}))}=\lambda \circ (\overline{{\varphi }_{\gamma }},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }}\circ {{\mathrm{\Phi }}_{\beta }}^{-1}\circ ({\varphi }_{\beta },id{)}^{-1}\circ {\lambda }^{-1}{|}_{{\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta }\cap \overline{U_{\gamma }})}$, because $\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }}))=\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta }\cap \overline{U_{\gamma }}))={\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta }\cap \overline{U_{\gamma }})$.

$=\lambda \circ (\overline{{\varphi }_{\gamma }},id)\circ ({{\iota }_{\gamma }^{\prime }}^{-1},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }^{\prime }}\circ {\tau }_{\gamma }\circ (({{\iota }_{\beta }^{\prime }}^{-1},id)\circ {\mathrm{\Phi }}_{\beta }^{\prime }\circ {\tau }_{\beta }{)}^{-1}\circ ({{\varphi }_{\beta }}^{-1},id)\circ {\lambda }^{-1}=\lambda \circ (\overline{{\varphi }_{\gamma }},id)\circ ({{\iota }_{\gamma }^{\prime }}^{-1},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }^{\prime }}\circ {\tau }_{\gamma }\circ {{\tau }_{\beta }}^{-1}\circ {{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}\circ ({{\iota }_{\beta }^{\prime }}^{-1},id{)}^{-1}\circ ({{\varphi }_{\beta }}^{-1},id)\circ {\lambda }^{-1}=\lambda \circ (\overline{{\varphi }_{\gamma }},id)\circ ({{\iota }_{\gamma }^{\prime }}^{-1},id)\circ \overline{{\mathrm{\Phi }}_{\gamma }^{\prime }}\circ {{\mathrm{\Phi }}_{\beta }^{\prime }}^{-1}\circ ({{\iota }_{\beta }^{\prime }}^{-1},id{)}^{-1}\circ ({{\varphi }_{\beta }}^{-1},id)\circ {\lambda }^{-1}$.

That is $:{\mathbb{R}}^k\times {\varphi }_{\beta }(U_{\beta }\cap \overline{U_{\gamma }})\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}\to {\varphi }_{\beta }(U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k\to (U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq U_{\beta }\times {\mathbb{R}}^k\to (U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq M^{\prime }\times {\mathbb{R}}^k\to {\pi }^{-1}(U_{\beta }\cap \overline{U_{\gamma }})\subseteq {\pi }^{-1}(U_{\beta }^{\prime }\cap \overline{U_{\gamma }^{\prime }})\to (U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq M^{\prime }\times {\mathbb{R}}^k\to (U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq \overline{U_{\gamma }}\times {\mathbb{R}}^k\to \overline{{\varphi }_{\gamma }}(U_{\beta }\cap \overline{U_{\gamma }})\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k\to {\mathbb{R}}^k\times \overline{{\varphi }_{\gamma }}(U_{\beta }\cap \overline{U_{\gamma }})\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$, which is a legitimate chain of $C^{\mathrm{\infty }}$ maps.

So, $\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }}))}$ is $C^{\mathrm{\infty }}$, 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.

Also, $\widetilde{{\varphi }_{\beta }}\circ {\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}}^{-1}{|}_{\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}({\pi }^{-1}(\overline{U_{\gamma }})\cap {\pi }^{-1}(U_{\beta }))}$ is $C^{\mathrm{\infty }}$, by the symmetry.

So, $\widetilde{\stackrel{―}{{\varphi }_{\gamma }}}\circ {\widetilde{{\varphi }_{\beta }}}^{-1}{|}_{\widetilde{{\varphi }_{\beta }}({\pi }^{-1}(U_{\beta })\cap {\pi }^{-1}(\overline{U_{\gamma }}))}$ is diffeomorphic.

So, by the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same, the 2 topology-atlas pairs are the same, which means that the topology and the atlas are uniquely determined by the specification.

Let us see that $E$ is an immersed submanifold with boundary of $E^{\prime }$.

Let $\tau :E\to E^{\prime }$ be the inclusion.

Although $U_{\beta }^{\prime }$ was chosen to be just a trivializing open subset, $U_{\beta }^{\prime }$ can be a chart trivializing open subset, which we do now.

For each $e\in E$, let us choose the charts, $({\pi }^{-1}(U_{\beta })\subseteq E,\widetilde{{\varphi }_{\beta }})$ and $({\pi }^{\prime -1}(U_{\beta }^{\prime })\subseteq E^{\prime },\widetilde{{\varphi }_{\beta }^{\prime }})$, such that $e\in {\pi }^{-1}(U_{\beta })$, which inevitably implies that $\tau (e)\in {\pi }^{\prime -1}(U_{\beta }^{\prime })$ and $\tau ({\pi }^{-1}(U_{\beta }))\subseteq {\pi }^{\prime -1}(U_{\beta }^{\prime })$.

$\widetilde{{\varphi }_{\beta }^{\prime }}\circ \tau \circ {\widetilde{{\varphi }_{\beta }}}^{-1}$ is $:(v,{\varphi }_{\beta }(p))\mapsto (v,{\varphi }_{\beta }^{\prime }(p))=(id,{\varphi }_{\beta }^{\prime }\circ {{\varphi }_{\beta }}^{-1})$, which implies that $\tau$ is an injective $C^{\mathrm{\infty }}$ immersion, because ${\varphi }_{\beta }^{\prime }\circ {{\varphi }_{\beta }}^{-1}$ satisfies the characteristics that makes $\iota :M\to M^{\prime }$ be an injective $C^{\mathrm{\infty }}$ immersion.

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References

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