1166: For C∞ Vectors Bundle, Union of k-Dimensional Vectors Subspaces of Fibers That Allows Local C∞ Frames Is C∞ Vectors Subbundle with Subspace Topology and Adopting Atlas

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle, union of $k$-dimensional vectors subspaces of fibers that allows local $C^{\mathrm{\infty }}$ frames is $C^{\mathrm{\infty }}$ vectors subbundle with subspace topology and adopting atlas

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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: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors subbundle of rank $k$ of $C^{\mathrm{\infty }}$ vectors bundle of rank $k^{\prime }$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle and any set of local $C^{\mathrm{\infty }}$ sections over any open domain that is linearly independent, of each point of the domain, there is a possibly smaller open neighborhood over which there is a local $C^{\mathrm{\infty }}$ frame that contains the restricted sections set.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ frame exists over and only over any trivializing open subset.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
• 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.
• The reader admits the proposition that any subset of any $C^{\mathrm{\infty }}$ manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.

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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, the union of any $k$-dimensional vectors subspaces of the fibers that allows any local $C^{\mathrm{\infty }}$ frames is a $C^{\mathrm{\infty }}$ vectors subbundle with the subspace topology and the adopting atlas.

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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:
$(E^{\prime },M,{\pi }^{\prime })$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }{k}^{\prime }\}$
$k$: $\in \mathbb{N}\setminus \{0\}$ such that $k\le k^{\prime }$
$\{E_m\in \{\text{ the }k\text{ -dimensional vectors subspaces of }{\pi }^{\prime -1}(m)\}|m\in M\}$
$E$: $={\cup }_{m\in M}{E}_m\subseteq E^{\prime }$ with the subspace topology
$\pi$: $={\pi }^{\prime }{|}_E:E\to M$
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Statements:
$\mathrm{\forall }m\in M(\mathrm{\exists }{U}_m\subseteq M\in \{\text{ the open neighborhoods of }m\},\mathrm{\exists }{s}_1,...,s_k\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ sections of }{\pi }^{\prime }{|}_{{\pi }^{\prime -1}(U_m)}\}(\mathrm{\forall }{m}^{\prime }\in U_m(\{s_1(m^{\prime }),...,s_k(m^{\prime })\}\in \{\text{ the bases for }{E}_{m^{\prime }}\})))$
$⟹$
(
$E$ with the adopting atlas $\in \{\text{ the embedded submanifolds with boundary of }{E}^{\prime }\}$
$\wedge$
$(E,M,\pi )\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors subbundles of }(E^{\prime },M,{\pi }^{\prime })\}$
)
//

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2: Proof

Whole Strategy: Step 1: take a trivializing chart around $m$, $(V_m\subseteq M,{\varphi }_m)$, such that $V_m\subseteq U_m$ and a local $C^{\mathrm{\infty }}$ frame over $V_m$ on $E^{\prime }$, $\{s_1,...,s_k,s_{k+1},...,s_{k^{\prime }}\}$; Step 2: take the corresponding chart for $E^{\prime }$, $({\pi }^{\prime -1}(V_m)\subseteq E^{\prime },\widetilde{{\varphi }_m})$, and see that the chart is an adopted chart for $E$, so, give $E$ the adopting atlas making $E$ an embedded submanifold with boundary of $E^{\prime }$; Step 3: see that $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors bundle; Step 4: see that $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors subbundle of $(E^{\prime },M,{\pi }^{\prime })$.

Step 1:

There are an open neighborhood of $m$, $V_m^{\prime }\subseteq M$, such that $V_m^{\prime }\subseteq U_m$ and a local $C^{\mathrm{\infty }}$ frame over $V_m^{\prime }$ on $E^{\prime }$, $\{s_1,...,s_k,s_{k+1},...,s_{k^{\prime }}\}$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle and any set of local $C^{\mathrm{\infty }}$ sections over any open domain that is linearly independent, of each point of the domain, there is a possibly smaller open neighborhood over which there is a local $C^{\mathrm{\infty }}$ frame that contains the restricted sections set.

$V_m^{\prime }$ is a trivializing open subset for $E^{\prime }$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ frame exists over and only over any trivializing open subset, with the trivialization, ${\mathrm{\Phi }}_m^{\prime }:{\pi }^{\prime -1}(V_m^{\prime })\to V_m^{\prime }\times {\mathbb{R}}^{k^{\prime }},v=s^j{s}_j\mapsto ({\pi }^{\prime }(v),s^1,...,s^k,s^{k+1}...,s^{k^{\prime }})$.

There is a trivializing chart around $m$, $(V_m\subseteq M,{\varphi }_m)$, such that $V_m\subseteq V_m^{\prime }$, with the trivialization, ${\mathrm{\Phi }}_m{|}_{{\pi }^{\prime -1}(V_m)}:{\pi }^{\prime -1}(V_m)\to V_m\times {\mathbb{R}}^{k^{\prime }},v=s^j{s}_j\mapsto ({\pi }^{\prime }(v),s^1,...,s^k,s^{k+1}...,s^{k^{\prime }})$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.

Step 2:

Let us take the chart induced by the trivialization, $({\pi }^{\prime -1}(V_m)\subseteq E^{\prime },\widetilde{{\varphi }_m})$, where $\widetilde{{\varphi }_m}:{\pi }^{\prime -1}(V_m)\to {\mathbb{R}}^{d+k^{\prime }}\text{ or }{\mathbb{H}}^{d+k^{\prime }},v\mapsto ({\pi }_2({\mathrm{\Phi }}_m(v)),{\varphi }_m({\pi }^{\prime }(v)))$, 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.

Let us see that the chart is an adopted chart for $E$ around any $e\in E_m$.

It is about that there are a $J\subseteq \{1,...,k^{\prime }+d\}$ and a $u\in {\pi }^{\prime -1}(V_m)$ such that ${\pi }^{\prime -1}(V_m)\cap E=S_{J,u}({\pi }^{\prime -1}(V_m))$.

Let $J=\{1,...,k,k^{\prime }+1,...,k^{\prime }+d\}$ and $u=e$: $u\in {\pi }^{\prime -1}(V_m)$, because $e\in E_m\subseteq {\pi }^{\prime -1}(V_m)$.

$\widetilde{{\varphi }_m}({\pi }^{\prime -1}(V_m)\cap E)=\{(v^1,...,v^k,0,...,0,{\varphi }_m(m^{\prime }))|v^1,...,v^k\in \mathbb{R},m^{\prime }\in V_m\}$.

$\widetilde{{\varphi }_m}({\pi }^{\prime -1}(V_m))=\{(v^1,...,v^k,v^{k+1},...,v^{k^{\prime }},{\varphi }_m(m^{\prime }))|v^1,...,v^{k^{\prime }}\in \mathbb{R},m^{\prime }\in V_m\}$.

$\widetilde{{\varphi }_m}(u)=(\widetilde{{\varphi }_m}(u{)}^1,...,\widetilde{{\varphi }_m}(u{)}^k,0,...,0,{\varphi }_m(m))$.

$S_{J,\widetilde{{\varphi }_m}(u)}({\mathbb{R}}^{k^{\prime }+d})=\{(v^1,...,v^k,0,...,0,x^1,...,x^d)\in {\mathbb{R}}^{k^{\prime }+d}|v^1,...,v^k\in \mathbb{R},x^1,...,x^d\in \mathbb{R}\}$.

So, $\widetilde{{\varphi }_m}({\pi }^{\prime -1}(V_m))\cap S_{J,\widetilde{{\varphi }_m}(u)}({\mathbb{R}}^{k^{\prime }+d})=\{(v^1,...,v^k,0,...,0,{\varphi }_m(m^{\prime }))|v^1,...,v^k\in \mathbb{R},m^{\prime }\in V_m\}$.

So, $\widetilde{{\varphi }_m}({\pi }^{\prime -1}(V_m)\cap E)=\widetilde{{\varphi }_m}({\pi }^{\prime -1}(V_m))\cap S_{J,\widetilde{{\varphi }_m}(u)}({\mathbb{R}}^{k^{\prime }+d})$.

So, ${\pi }^{\prime -1}(V_m)\cap E=S_{J,u}({\pi }^{\prime -1}(V_m))$.

So, $E$ satisfies the local-slice condition, so, $E$ with the subspace topology and the adopting atlas, $\{({\pi }^{\prime -1}(V_m)\cap E={\pi }^{-1}(V_m)\subseteq E,\widetilde{{\varphi }_m}={\pi }_J\circ {\widetilde{{\varphi }_m}}^{\prime }{|}_{{\pi }^{-1}(V_m)})|m\in M\}$, is an embedded submanifold with boundary of $E^{\prime }$, by the proposition that any subset of any $C^{\mathrm{\infty }}$ manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.

Step 3:

Let us see that $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors bundle.

$\pi$ is $C^{\mathrm{\infty }}$, because for each $e\in E$, denoting $m=\pi (e)$, there are a chart, $(U_m\subseteq M,{\varphi }_m)$, and the corresponding chart, $({\pi }^{-1}(V_m)\subseteq E,\widetilde{{\varphi }_m})$, and the components function is $:(v^1,...,v^k,{\varphi }_m(m^{\prime }))\mapsto {\varphi }_m(m^{\prime })$, which is $C^{\mathrm{\infty }}$.

For each $m\in M$, the chart, $(V_m\subseteq M,{\varphi }_m)$, taken in Step 1 is a $C^{\mathrm{\infty }}$ trivializing open neighborhood of $m$, because $:{\pi }^{-1}(V_m)\to V_m\times {\mathbb{R}}^k,v\mapsto (\pi (v),{\pi }_2(\widetilde{{\varphi }_m}(v)))$ is a $C^{\mathrm{\infty }}$ trivialization: it is a restriction of the $C^{\mathrm{\infty }}$ trivialization, ${\mathrm{\Phi }}_m:{\pi }^{\prime -1}(V_m)\to V_m\times {\mathbb{R}}^{k^{\prime }}$.

Step 4:

So, $E$ is an embedded submanifold with boundary of $E^{\prime }$, $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors bundle, and for each $m\in M$, ${\pi }^{-1}(m)=E_m$ is a $k$-dimensional vectors subspace of ${\pi }^{\prime -1}(m)$, so, $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors subbundle of $(E^{\prime },M,{\pi }^{\prime })$.

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References

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