824: For Map Between Embedded Submanifolds with Boundary of C∞ Manifolds with Boundary, Ck-ness Does Not Change When Domain or Codomain Is Regarded to Be Subset

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

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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
• 3: Proof

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Starting Context

• The reader knows a definition of embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• 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 admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$.
• 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 for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range 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 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.

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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:
$M_1^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_1$: $\in \{\text{ the embedded submanifolds with boundary of }{M}_1^{\prime }\}$
${\iota }_1$: $:M_1\to M_1^{\prime }$, $=\text{ the inclusion }$
$M_2$: $\in \{\text{ the embedded submanifolds with boundary of }{M}_2^{\prime }\}$
${\iota }_2$: $:M_2\to M_2^{\prime }$, $=\text{ the inclusion }$
$f_1$: $:M_1\to M_2$
$f_2$: $:M_1\to {\iota }_2(M_2)\subseteq M_2^{\prime }$, $=f_1$ with the codomain replaced
$f_3$: $:{\iota }_1(M_1)\subseteq M_1^{\prime }\to M_2$, $=f_1$ with the domain replaced
$f_4$: $:{\iota }_1(M_1)\subseteq M_1^{\prime }\to {\iota }_2(M_2)\subseteq M_2^{\prime }$, $=f_1$ with the domain and the codomain replaced
//

Statements:
$f_1\in \{\text{ the }{C}^k\text{ maps }\}$
$⟺$
$f_2\in \{\text{ the }{C}^k\text{ maps }\}$
$⟺$
$f_3\in \{\text{ the }{C}^k\text{ maps }\}$
$⟺$
$f_4\in \{\text{ the }{C}^k\text{ maps }\}$
//

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2: Note

This proposition is not so trivial: for example, $C^k$-ness of $f_1$ is judged with respect to the atlas of $M_1$ and the atlas of $M_2$ while $C^k$-ness of $f_4$ is judged with respect to the atlas of $M_1^{\prime }$ and the atlas of $M_2^{\prime }$.

This proposition requires $M_1$ and $M_2$ to be embedded, not more generally immersed.

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3: Proof

Whole Strategy: let ${\iota }_j^{\prime }:M_j\to {\iota }_j(M_j)\subseteq M_j^{\prime }$ be the codomain restriction of ${\iota }_j$, and for from $f_j$ to $f_k$, express $f_k$ with $f_j$ and ${\iota }_1^{\prime },{\iota }_2^{\prime }$, and use the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$; Step 1: express $f_2,f_3,f_4$ with $f_1$ and ${\iota }_1^{\prime },{\iota }_2^{\prime }$, express $f_1,f_3,f_4$ with $f_2$ and ${\iota }_1^{\prime },{\iota }_2^{\prime }$, express $f_1,f_2,f_4$ with $f_3$ and ${\iota }_1^{\prime },{\iota }_2^{\prime }$, and express $f_1,f_2,f_3$ with $f_4$ and ${\iota }_1^{\prime },{\iota }_2^{\prime }$; Step 2: see that ${{\iota }_1^{\prime }}^{-1}$ and ${{\iota }_2^{\prime }}^{-1}$ are $C^{\mathrm{\infty }}$; Step 3: conclude the proposition.

Step 1:

Let ${\iota }_1^{\prime }:M_1\to {\iota }_1(M_1)\subseteq M_1^{\prime }$ be the codomain restriction of ${\iota }_1$.

${\iota }_1^{\prime }$ is $C^{\mathrm{\infty }}$, by the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

Let ${\iota }_2^{\prime }:M_2\to {\iota }_2(M_2)\subseteq M_2^{\prime }$ be the codomain restriction of ${\iota }_2$.

${\iota }_2^{\prime }$ is $C^{\mathrm{\infty }}$, by the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

Let us express $f_2,f_3,f_4$ with $f_1$ and ${\iota }_1^{\prime }$ and ${\iota }_2^{\prime }$.

$f_2={\iota }_2^{\prime }\circ f_1$.

$f_3=f_1\circ {{\iota }_1^{\prime }}^{-1}$.

$f_4={\iota }_2^{\prime }\circ f_1\circ {{\iota }_1^{\prime }}^{-1}$.

Let us express $f_1,f_3,f_4$ with $f_2$ and ${\iota }_1^{\prime }$ and ${\iota }_2^{\prime }$.

$f_1={{\iota }_2^{\prime }}^{-1}\circ f_2$.

$f_3={{\iota }_2^{\prime }}^{-1}\circ f_2\circ {{\iota }_1^{\prime }}^{-1}$.

$f_4=f_2\circ {{\iota }_1^{\prime }}^{-1}$.

Let us express $f_1,f_2,f_4$ with $f_3$ and ${\iota }_1^{\prime }$ and ${\iota }_2^{\prime }$.

$f_1=f_3\circ {\iota }_1^{\prime }$.

$f_2={\iota }_2^{\prime }\circ f_3\circ {\iota }_1^{\prime }$.

$f_4={\iota }_2^{\prime }\circ f_3$.

Let us express $f_1,f_2,f_3$ with $f_4$ and ${\iota }_1^{\prime }$ and ${\iota }_2^{\prime }$.

$f_1={{\iota }_2^{\prime }}^{-1}{f}_4\circ {\iota }_1^{\prime }$.

$f_2=f_4\circ {\iota }_1^{\prime }$.

$f_3={{\iota }_2^{\prime }}^{-1}\circ f_4$.

Step 2:

${{\iota }_1^{\prime }}^{-1}$ and ${{\iota }_2^{\prime }}^{-1}$ are $C^{\mathrm{\infty }}$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$.

Step 3:

Let us suppose $f_1$ is $C^k$.

$f_2,f_3,f_4$ are $C^k$, 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.

Let us suppose $f_2$ is $C^k$.

$f_1,f_3,f_4$ are $C^k$, likewise.

Let us suppose $f_3$ is $C^k$.

$f_1,f_2,f_4$ are $C^k$, likewise.

Let us suppose $f_4$ is $C^k$.

$f_1,f_2,f_3$ are $C^k$, likewise.

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References

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