840: Local C∞ Frame on C∞ Vectors Bundle

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definition of local $C^{\mathrm{\infty }}$ frame on $C^{\mathrm{\infty }}$ 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 section of continuous map.

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Target Context

• The reader will have a definition of local $C^{\mathrm{\infty }}$ frame on $C^{\mathrm{\infty }}$ 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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$U$: $\in \{\text{ the open subsets of }M\}$
$\ast \{s_1,...,s_k\}$: $s_j:U\to {\pi }^{-1}(U)\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ sections of }\pi {|}_{{\pi }^{-1}(U)}\}$
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Conditions:
$\mathrm{\forall }m\in U(\{s_1(m),...,s_k(m)\}\in \{\text{ the bases of }{\pi }^{-1}(m)\})$
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2: Note

Whether the domain or the codomain of $s_j:U\to {\pi }^{-1}(U)$ is regarded to be the subset of the ambient $M$ or $E$ or is regarded to be the embedded submanifold with boundary of $M$ or the embedded submanifold with boundary of $E$ as the restricted $C^{\mathrm{\infty }}$ vectors bundle does not really matter, by the proposition that for any map between any embedded submanifolds with boundary of any $C^{\mathrm{\infty }}$ manifolds with boundary, $C^{\mathrm{\infty }}$-ness does not change when the domain or the codomain is regarded to be the subset.

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References

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