412: For C∞ Vectors Bundle, Chart Open Subset on Base Space Is Not Necessarily Trivializing Open Subset (Probably)

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle, chart open subset on base space is not necessarily trivializing open subset (probably)

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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: Natural Language Description
• 3: Note
• 4: Proof (Not Complete)

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Starting Context

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

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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, a chart open subset on the base space is not necessarily any trivializing open subset (probably).

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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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles }\}$
$U$: $\in \{\text{ the chart domains of }M\}$
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Statements:
Not necessarily $U\in \{\text{ the trivializing open subsets of }M\}$ (probably)
//

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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ vectors bundle, $(E,M,\pi )$, and any chart domain, $U\subseteq M$, $U$ is not necessarily any trivializing open subset (probably).

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3: Note

The author is honestly not sure whether $U$ needs to be a trivializing open subset of $M$ or not.

The reason why we need this proposition even in this unsure state is that at least we need a reminder for not assuming what we have not proved.

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4: Proof (Not Complete)

I know that I need to show a counter example, but I have not managed to.

At least, the definition of $C^{\mathrm{\infty }}$ vectors bundle does not directly require that every chart open subset on the base space is a trivializing open subset, and it seems unlikely that that every chart open subset on the base space is implied to be a trivializing open subset, at 1st glance.

For any tangent vectors bundle case, every chart open subset on the base space is a trivializing open subset (the atlas of the tangent vectors bundle is defined to make it so), but that is not so-immediately generalized to any general $C^{\mathrm{\infty }}$ vectors bundle case.

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References

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