409: For C∞ Vectors Bundle, Trivialization of Chart Trivializing Open Subset Induces Canonical Chart Map

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle, trivialization of chart trivializing open subset induces canonical chart map

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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 1
• 3: Proof
• 4: Note 2

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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 $C^{\mathrm{\infty }}$ manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean $C^{\mathrm{\infty }}$ manifold or closed upper half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary is a chart map if and only if it is a diffeomorphism.
• The reader admits the proposition that the $d$-dimensional Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces whose (the product's) dimension equals $d$.
• The reader admits the proposition that the $d$-dimensional closed upper half Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces and a closed upper half Euclidean space whose (the product's) dimension equals $d$.
• The reader admits the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at 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, the trivialization of any chart trivializing open subset induces the canonical chart map.

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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:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$(U\subseteq M,\varphi )$: $\in \{\text{ the charts of }M\}$, such that $U\in \{\text{ the trivializing open subsets }\}$
$\mathrm{\Phi }$: $:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$, $\in \{\text{ the trivializations }\}$
${\pi }_2$: $:(U\times {\mathbb{R}}^k)\to {\mathbb{R}}^k$, $=\text{ the projection }$
$\tilde{\varphi }$: $:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k\to {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k},v\mapsto ({\pi }_2(\mathrm{\Phi }(v)),\varphi (\pi (v)))$
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Statements:
$({\pi }^{-1}(U)\subseteq E,\tilde{\varphi })\in \{\text{ the charts of }E\}$
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2: Note 1

When the boundary of $M$ is empty, $\tilde{\varphi }$ is usually chosen to be such that $v\mapsto (\varphi (\pi (v)),{\pi }_2(\mathrm{\Phi }(v)))$ because $(\varphi (\pi (v)),{\pi }_2(\mathrm{\Phi }(v)))\in {\mathbb{R}}^{d+k}$ anyway, but when the boundary of $M$ is not empty, $(\varphi (\pi (v)),{\pi }_2(\mathrm{\Phi }(v)))\notin {\mathbb{H}}^{d+k}$ in general, so, we have chosen $\tilde{\varphi }$ as above, which is fine also for when the boundary of $M$ is empty.

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3: Proof

Whole Strategy: apply the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean $C^{\mathrm{\infty }}$ manifold or closed upper half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary is a chart map if and only if it is a diffeomorphism; Step 1: see that $\tilde{\varphi }({\pi }^{-1}(U))\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$ is open; Step 2: see that $\tilde{\varphi }$ is a diffeomorphism; Step 3: conclude the proposition.

Step 1:

Let us see that $\tilde{\varphi }({\pi }^{-1}(U))\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$ is open.

$\tilde{\varphi }({\pi }^{-1}(U))={\mathbb{R}}^k\times \varphi (U)$.

${\mathbb{R}}^{d+k}\cong {\mathbb{R}}^k\times {\mathbb{R}}^d$ or ${\mathbb{H}}^{d+k}\cong {\mathbb{R}}^k\times {\mathbb{H}}^d$, where $\cong$ means being homeomorphic, by the proposition that the $d$-dimensional Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces whose (the product's) dimension equals $d$ or the proposition that the $d$-dimensional closed upper half Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces and a closed upper half Euclidean space whose (the product's) dimension equals $d$.

${\mathbb{R}}^k\times \varphi (U)\subseteq {\mathbb{R}}^k\times {\mathbb{R}}^d\text{ or }{\mathbb{R}}^k\times {\mathbb{H}}^d$ is an open subset, by the definition of product topology, because $\varphi (U)\subseteq {\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$ is an open subset.

So, $\tilde{\varphi }({\pi }^{-1}(U))\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$ is open.

Step 2:

Let us see that $\tilde{\varphi }$ is diffeomorphic.

$\tilde{\varphi }=\lambda \circ (\varphi ,id)\circ \mathrm{\Phi }$, where $\lambda :{\mathbb{R}}^{d+k}\to {\mathbb{R}}^{d+k},(r^1,...,r^d,r^{d+1},...,r^{d+k})\mapsto (r^{d+1},...,r^{d+k},r^1,...,r^d)$.

$\mathrm{\Phi }:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$ is diffeomorphic; $(\varphi ,id):U\times {\mathbb{R}}^k\to \varphi (U)\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k$ is obviously diffeomorphic; and $\lambda {|}_{\varphi (U)\times {\mathbb{R}}^k}:\varphi (U)\times {\mathbb{R}}^k\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^k\text{ or }{\mathbb{H}}^d\times {\mathbb{R}}^k\to {\mathbb{R}}^k\times \varphi (U)\subseteq {\mathbb{R}}^{d+k}\text{ or }{\mathbb{H}}^{d+k}$ is obviously diffeomorphic.

So, $\tilde{\varphi }$ is diffeomorphic, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

As $\tilde{\varphi }$ is a diffeomorphism, $({\pi }^{-1}(U),\tilde{\varphi })$ is a chart of $E$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean $C^{\mathrm{\infty }}$ manifold or closed upper half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary is a chart map if and only if it is a diffeomorphism.

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4: Note 2

$U$ has to be a chart trivializing open subset, not just a trivializing open subset, because otherwise, $\varphi$ would not exist.

By the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, there is a chart trivializing open cover, the chart trivializing open cover can be used to have a chart over any point, $p\in M$.

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References

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