827: C∞ Trivializing Open Subset and C∞ Local Trivialization

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definition of $C^{\mathrm{\infty }}$ trivializing open subset and $C^{\mathrm{\infty }}$ local trivialization

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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

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Starting Context

• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.
• The reader knows a definition of trivializing open subset and local trivialization.
• The reader knows a definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.
• The reader knows a definition of Euclidean $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of finite-product $C^{\mathrm{\infty }}$ manifold with boundary.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ trivializing open subset and $C^{\mathrm{\infty }}$ local trivialization.

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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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$E$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$k$: $\in \mathbb{N}\setminus \{0\}$
$\pi$: $:E\to M$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$\ast U$: $\in \{\text{ the trivializing open subsets of }M\}$
$\ast \mathrm{\Phi }$: $:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$, $\in \{\text{ the local trivializations }\}$
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Conditions:
$\mathrm{\Phi }\in \{\text{ the diffeomorphisms }\}$
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${\mathbb{R}}^k$ and $U\times {\mathbb{R}}^k$ are the Euclidean $C^{\mathrm{\infty }}$ manifold and the product $C^{\mathrm{\infty }}$ manifold with boundary.

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References

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