888: For 2 C∞ Vectors Bundles over Same C∞ Manifold with Boundary, Bijective C∞ Vectors Bundle Homomorphism Is 'C∞ Vectors Bundles - C∞ Vectors Bundle Homomorphisms' Isomorphism

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description/proof of that for 2 $C^{\mathrm{\infty }}$ vectors bundles over same $C^{\mathrm{\infty }}$ manifold with boundary, bijective $C^{\mathrm{\infty }}$ vectors bundle homomorphism is '$C^{\mathrm{\infty }}$ vectors bundles - $C^{\mathrm{\infty }}$ vectors bundle homomorphisms' isomorphism

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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 knows a definition of bijection.
• The reader knows a definition of %structure kind name% homomorphism.
• The reader knows a definition of %category name% isomorphism.
• 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 open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• 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.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any 2 real finite-dimensional vectors spaces turned $C^{\mathrm{\infty }}$ manifolds, any $C^{\mathrm{\infty }}$ bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism.
• The reader admits the proposition that for any map between any embedded submanifolds with boundary of any $C^{\mathrm{\infty }}$ manifolds with boundary, $C^k$-ness does not change when the domain or the codomain is regarded to be the subset.
• 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.
• The reader admits the proposition that any bijective linear map is a 'vectors spaces - linear morphisms' isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that for any 2 $C^{\mathrm{\infty }}$ vectors bundles over any same $C^{\mathrm{\infty }}$ manifold with boundary, any bijective $C^{\mathrm{\infty }}$ vectors bundle homomorphism is a '$C^{\mathrm{\infty }}$ vectors bundles - $C^{\mathrm{\infty }}$ vectors bundle homomorphisms' isomorphism.

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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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles }\}$
$(E^{\prime },M,{\pi }^{\prime })$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles }\}$
$f$: $:E\to E^{\prime }$, $\in \{\text{ the bijections }\}$, such that $(id:M\to M,f)\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundle homomorphisms }\}$
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Statements:
$f\in \{\text{ the ' }{C}^{\mathrm{\infty }}\text{ vectors bundles - }{C}^{\mathrm{\infty }}\text{ vectors bundle homomorphisms' isomorphisms }\}$
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2: Proof

Whole Strategy: see that $(i{d}^{-1},f^{-1})$ is a $C^{\mathrm{\infty }}$ vectors bundle homomorphism; Step 1: see that $\pi \circ f^{-1}=i{d}^{-1}\circ {\pi }^{\prime }$; Step 2: see that $f^{-1}$ is $C^{\mathrm{\infty }}$; Step 3: for each $m\in M$, see that $f^{-1}{|}_{{\pi }^{\prime -1}(m)}:{\pi }^{\prime -1}(m)\to {\pi }^{-1}(m)$ is linear; Step 4: conclude the proposition.

Step 1:

As $f$ is bijective, there is $f^{-1}$.

Let us see that $\pi \circ f^{-1}=i{d}^{-1}\circ {\pi }^{\prime }$.

${\pi }^{\prime }\circ f=id\circ \pi$, because $(id:M\to M,f)$ is a $C^{\mathrm{\infty }}$ vectors bundle homomorphism.

$i{d}^{-1}\circ {\pi }^{\prime }=i{d}^{-1}\circ {\pi }^{\prime }\circ f\circ f^{-1}=i{d}^{-1}\circ id\circ \pi \circ f^{-1}=\pi \circ f^{-1}$.

Step 2:

Step 2 Strategy: Step 2-1: around each $m\in M$, take a trivializing open subset, $U_m\subseteq M$, a trivialization, ${\mathrm{\Phi }}_m:{\pi }^{-1}(U_m)\to U_m\times {\mathbb{R}}^k$, and a trivialization, ${\mathrm{\Phi }}_m^{\prime }:{\pi }^{\prime -1}(U_m)\to U_m\times {\mathbb{R}}^k$; Step 2-2: think of ${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$, and see that it is a $C^{\mathrm{\infty }}$ bijection that is 1st-factor-preserving and 1st-factor-fixed linear; Step 2-3: conclude that ${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is diffeomorphic; Step 2-4: conclude that $f$ is diffeomorphic; Step 3: conclude the proposition.

Step 2-1:

Around each $m\in M$, let us take a trivializing open subset, $U_m\subseteq M$, a trivialization, ${\mathrm{\Phi }}_m:{\pi }^{-1}(U_m)\to U_m\times {\mathbb{R}}^k$, and a trivialization, ${\mathrm{\Phi }}_m^{\prime }:{\pi }^{\prime -1}(U_m)\to U_m\times {\mathbb{R}}^k$, which is possible because while there are a trivializing open subset, $V_m$, for $E$ and a trivializing open subset, $V_m^{\prime }$, for $E^{\prime }$, $U_m:=V_m\cap V_m^{\prime }$ will do, by the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

Step 2-2:

Let us think of ${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}:U_m\times {\mathbb{R}}^k\to U_m\times {\mathbb{R}}^k$.

We are going to see that it is diffeomorphic, using the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any 2 real finite-dimensional vectors spaces turned $C^{\mathrm{\infty }}$ manifolds, any $C^{\mathrm{\infty }}$ bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism. Step 2-2 is about seeing that ${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ satisfies the requirements for the proposition. "1st-factor-preserving" and "1st-factor-fixed-linear" hereafter means what are meant in the proposition.

$U_m$ is a $C^{\mathrm{\infty }}$ manifold with boundary, by Note for the definition of open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.

Note the proposition that for any map between any embedded submanifolds with boundary of any $C^{\mathrm{\infty }}$ manifolds with boundary, $C^k$-ness does not change when the domain or the codomain is regarded to be the subset.

${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is bijective, because ${{\mathrm{\Phi }}_m}^{-1}:U_m\times {\mathbb{R}}^k\to {\pi }^{-1}(U_m)$ is bijective, $f{|}_{{\pi }^{-1}(U_m)}:{\pi }^{-1}(U_m)\to {\pi }^{-1}(U_m)$ is bijective: $f$ is fiber-preserving and is 'vectors spaces - linear morphisms' isomorphic (so, bijective) on each fiber, and ${\mathrm{\Phi }}_m^{\prime }:{\pi }^{-1}(U_m)\to U_m\times {\mathbb{R}}^k$ is bijective.

${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is $C^{\mathrm{\infty }}$, 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.

${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is 1st-factor-preserving, because ${{\mathrm{\Phi }}_m}^{-1}$ maps $\{p\}\times {\mathbb{R}}^k$ into ${\pi }^{-1}(p)$, $f$ maps ${\pi }^{-1}(p)$ into ${\pi }^{\prime -1}(p)$, and ${\mathrm{\Phi }}_m^{\prime }$ maps ${\pi }^{\prime -1}(p)$ into $\{p\}\times {\mathbb{R}}^k$.

${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is 1st-factor-fixed linear, because ${{\mathrm{\Phi }}_m}^{-1}{|}_{\{p\}\times {\mathbb{R}}^k}$ is linear into ${\pi }^{-1}(p)$, $f{|}_{{\pi }^{-1}(p)}$ is linear into ${\pi }^{\prime -1}(p)$, and ${\mathrm{\Phi }}_m^{\prime }{|}_{{\pi }^{\prime -1}(p)}$ is linear into $\{p\}\times {\mathbb{R}}^k$.

Step 2-3:

So, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any 2 real finite-dimensional vectors spaces turned $C^{\mathrm{\infty }}$ manifolds, any $C^{\mathrm{\infty }}$ bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism, ${\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}$ is diffeomorphic.

So, $({\mathrm{\Phi }}_m^{\prime }\circ f\circ {{\mathrm{\Phi }}_m}^{-1}{)}^{-1}={\mathrm{\Phi }}_m\circ f^{-1}\circ {{\mathrm{\Phi }}_m^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$.

Step 2-4:

Then, $f^{-1}={{\mathrm{\Phi }}_m}^{-1}\circ {\mathrm{\Phi }}_m\circ f^{-1}\circ {{\mathrm{\Phi }}_m^{\prime }}^{-1}\circ {\mathrm{\Phi }}_m^{\prime }$ is $C^{\mathrm{\infty }}$, 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.

So, $f$ is a diffeomorphism.

Step 3:

For each $m\in M$, let us see that $f^{-1}{|}_{{\pi }^{\prime -1}(m)}:{\pi }^{\prime -1}(m)\to {\pi }^{-1}(m)$ is linear.

As $f$ is fiber-preserving and bijective, $f^{-1}{|}_{{\pi }^{\prime -1}(m)}$ is the inverse of $f{|}_{{\pi }^{-1}(m)}:{\pi }^{-1}(m)\to {\pi }^{\prime -1}(m)$, which is linear by the definition of $C^{\mathrm{\infty }}$ vectors bundle homomorphism. By the proposition that any bijective linear map is a 'vectors spaces - linear morphisms' isomorphism, $f{|}_{{\pi }^{-1}(m)}$ is a 'vectors spaces - linear morphisms' isomorphism, and so, $f^{-1}{|}_{{\pi }^{\prime -1}(m)}$ is linear.

Step 4:

So, $(i{d}^{-1},f^{-1})$ is a $C^{\mathrm{\infty }}$ vectors bundle homomorphism.

So, $(id,f)$ is a '$C^{\mathrm{\infty }}$ vectors bundles - $C^{\mathrm{\infty }}$ vectors bundle homomorphisms' isomorphism.

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References

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