1163: For C∞ Vectors Bundle and Set of Local C∞ Sections over Open Domain That Is Linearly Independent, of Each Point of Domain, There Is Possibly Smaller Open Neighborhood over Which There Is Local C∞ Frame That Contains Restricted Sections Set

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle and set of local $C^{\mathrm{\infty }}$ sections over open domain that is linearly independent, of each point of domain, there is possibly smaller open neighborhood over which there is local $C^{\mathrm{\infty }}$ frame that contains restricted sections set

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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 local $C^{\mathrm{\infty }}$ frame on $C^{\mathrm{\infty }}$ vectors bundle.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.
• The reader admits the proposition that the set of any $C^{\mathrm{\infty }}$ sections of any $C^{\mathrm{\infty }}$ vectors bundle that (the set) is linearly independent at a point is linearly independent on an open neighborhood of the point.

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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 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.

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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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$U$: $\in \{\text{ the open subsets of }M\}$
$\{s_1,...,s_n\}$: $s_j:U\to E\in \{\text{ the local }{C}^{\mathrm{\infty }}\text{ sections of }\pi \}$ such that $\mathrm{\forall }u\in U(\{s_1(u),...,s_n(u)\}\in \{\text{ the linearly independent subsets of }{\pi }^{-1}(u)\})$
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Statements:
$\mathrm{\forall }u\in U(\mathrm{\exists }{U}_u\subseteq M\in \{\text{ the open neighborhoods of }u\}\text{ such that }{U}_u\subseteq U,\mathrm{\exists }\{s_{n+1},...,s_k\}\text{ where }{s}_{n+j}:M\to E\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ sections of }\pi \}(\{s_1{|}_{U_u},...,s_k{|}_{U_u}\}\in \{\text{ the local }{C}^{\mathrm{\infty }}\text{ frames }\}))$
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2: Proof

Whole Strategy: Step 1: take any basis for ${\pi }^{-1}(u)$, $\{s_1(u),...,s_n(u),v_{n+1},...,v_k\}$; Step 2: for each $j\in \{1,...,k-n\}$, take a $C^{\mathrm{\infty }}$ section of $\pi$, $s_{n+j}:M\to E$, such that $s_{n+j}(u)=v_{n+j}$; Step 3: see that there is an open neighborhood of $u$, $U_u\subseteq M$, such that $U_u\subseteq U$ over which $\{s_1,...,s_n,s_{n+1},...,s_k\}$ is linearly independent.

Step 1:

There is a basis for ${\pi }^{-1}(u)$, $\{s_1(u),...,s_n(u),v_{n+1},...,v_k\}$.

Step 2:

Let $j\in \{1,...,k-n\}$ be any.

$\{u\}\subseteq M$ is a closed subset of $M$.

$s_{n+j}:\{u\}\to E,u\mapsto v_{n+j}$ is a $C^{\mathrm{\infty }}$ section along $\{u\}$: take a trivializing chart around $u$ and the corresponding chart for $E$, then, the components function of $s_{n+j}$ has the obvious $C^{\mathrm{\infty }}$ extension over the trivializing chart, $:x\mapsto (v_{n+j}^1,...,v_{n+j}^k,x)$.

There is a $C^{\mathrm{\infty }}$ section of $\pi$, $s_{n+j}:M\to E$, such that $s_{n+j}(u)=v_{n+j}$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.

Step 3:

Now, we have a set of local $C^{\mathrm{\infty }}$ sections over $U$, $\{s_1,...,s_n,s_{n+1}{|}_U,...,s_k{|}_U\}$, such that $\{s_1(u),...,s_k(u)\}$ is linearly independent.

There is an open neighborhood of $u$, $U_u\subseteq M$, such that $U_u\subseteq U$ over which $\{s_1,...,s_k\}$ is linearly independent, by the proposition that the set of any $C^{\mathrm{\infty }}$ sections of any $C^{\mathrm{\infty }}$ vectors bundle that (the set) is linearly independent at a point is linearly independent on an open neighborhood of the point: although the proposition ostensibly requires the sections to be over the entire $M$, the proposition can be applied to this case with the local sections as is clear from that Proof.

So, $\{s_1{|}_{U_u},...,s_k{|}_{U_u}\}$ is a local $C^{\mathrm{\infty }}$ frame over $U_u$.

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References

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