408: For C∞ Vectors Bundle, There Is Chart Trivializing Open Cover

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle, there is chart trivializing open cover

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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 bundle of rank $k$.
• 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 $C^{\mathrm{\infty }}$ manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
• The reader admits the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

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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, there is a chart trivializing open cover.

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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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
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Statements:
$\mathrm{\exists }\text{ a chart trivializing open cover of }M$
//

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2: Proof

Whole Strategy: Step 1: take any trivializing open cover, $\{U_{\beta }|\beta \in B\}$; Step 2: for each $p\in U_{\beta }$, take a chart trivializing open subset, $U_{\beta ,p}$, and see that $\{U_{\beta ,p}|\beta \in B,p\in U_{\beta }\}$ covers $M$; Step 3: see that $\{U_{\beta ,p}|\beta \in B,p\in U_{\beta }\}$ can be refined to be locally finite, if so desired.

Step 1:

There is a trivializing open cover, $\{U_{\beta }|\beta \in B\}$, where $B$ is any possibly uncountable index set, by the definition of $C^{\mathrm{\infty }}$ vectors bundle.

But $U_{\beta }$ is not necessarily a chart open subset, so, we are going to find a trivializing open cover whose each constituent is a chart open subset, which this proposition is about.

Step 2:

For each $p\in U_{\beta }$, there is a chart trivializing open subset, $U_{\beta ,p}$ such that $U_{\beta ,p}\subseteq U_{\beta }$, 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.

Although $\{U_{\beta ,p}|\beta \in B,p\in U_{\beta }\}$ may have some duplications, such duplications are automatically eliminated by the definition of set.

$\{U_{\beta ,p}|\beta \in B,p\in U_{\beta }\}$ obviously covers $M$.

Step 3:

The proposition has been already proved, but if one desires the open cover to be smaller, $\{U_{\beta ,p}|\beta \in B,p\in U_{\beta }\}$ can be refined to be locally finite, because any $C^{\mathrm{\infty }}$ manifold with boundary is paracompact: each element of the refinement is a chart trivializing open subset, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart and the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

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References

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