1133: For Diffeomorphism Between C∞ Manifolds with Boundary and C∞ Vectors Field over Domain, There Is Unique C∞ Vectors Field over Codomain Map-Related with Vectors Field over Domain

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description/proof of that for diffeomorphism between $C^{\mathrm{\infty }}$ manifolds with boundary and $C^{\mathrm{\infty }}$ vectors field over domain, there is unique $C^{\mathrm{\infty }}$ vectors field over codomain map-related with vectors field over domain

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Topics

About: vectors space

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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 diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.
• The reader knows a definition of map-related vectors fields pair for $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ immersion between any $C^{\mathrm{\infty }}$ manifolds with boundary, its global differential is a $C^{\mathrm{\infty }}$ immersion.
• 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 diffeomorphism between any $C^{\mathrm{\infty }}$ manifolds with boundary and any $C^{\mathrm{\infty }}$ vectors field over the domain, there is the unique $C^{\mathrm{\infty }}$ vectors field over the codomain map-related with the vectors field over the domain.

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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:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f$: $:M_1\to M_2$, $\in \{\text{ the diffeomorphisms }\}$
$V_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors fields over }{M}_1\}$
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Statements:
$!\mathrm{\exists }{V}_2\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors fields over }{M}_2\}((V_1,V_2)\in \{\text{ the }f\text{ -related vectors fields pairs }\})$
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2: Proof

Whole Strategy: Step 1: define $V_2$ as $:M_2\to T{M}_2,m\mapsto d{f}_{f^{-1}(m)}{V}_1(f^{-1}(m))$; Step 2: see that $V_2$ is $C^{\mathrm{\infty }}$; Step 3: see that $V_2$ is unique.

Step 1:

Let us define $V_2:M_2\to T{M}_2,m\mapsto d{f}_{f^{-1}(m)}{V}_1(f^{-1}(m))$.

Step 2:

$V_2=df\circ V_1\circ f^{-1}$, where $df:T{M}_1\to T{M}_2$ is the global differential of $f$.

$f^{-1}$ is $C^{\mathrm{\infty }}$, $V_1$ is $C^{\mathrm{\infty }}$, and $df$ is $C^{\mathrm{\infty }}$ by the proposition that for any $C^{\mathrm{\infty }}$ immersion between any $C^{\mathrm{\infty }}$ manifolds with boundary, its global differential is a $C^{\mathrm{\infty }}$ immersion: any diffeomorphism is a $C^{\mathrm{\infty }}$ immersion.

So, $V_2$ 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.

Step 3:

$V_2$ is unique, because for each $m\in M_2$, $f(f^{-1}(m))=m$, so, there is no other option but $V_2(m)=d{f}_{f^{-1}(m)}{V}_1(f^{-1}(m))$.

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References

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