823: For C∞ Manifold with Boundary and Embedded Submanifold with Boundary, Inverse of Codomain Restricted Inclusion Is C∞

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description/proof of that for $C^{\mathrm{\infty }}$ manifold with boundary and embedded submanifold with boundary, inverse of codomain restricted inclusion is $C^{\mathrm{\infty }}$

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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 knows a definition of double of $C^{\mathrm{\infty }}$ manifold with boundary.
• 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 $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary.

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Target Context

• The reader will have a description and a proof of 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 $C^{\mathrm{\infty }}$.

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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:
$M^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M$: $\in \{\text{ the embedded submanifolds with boundary of }{M}^{\prime }\}$
$\iota$: $:M\to M^{\prime }$, $=\text{ the inclusion }$
${\iota }^{\prime }$: $:M\to \iota (M)\subseteq M^{\prime }$, $=\text{ the codomain restriction of }\iota$
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Statements:
${{\iota }^{\prime }}^{-1}:\iota (M)\subseteq M^{\prime }\to M\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
//

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2: Note

This logic cannot be applied for immersed submanifold with boundary, which is not guaranteed to have any adopted chart or adopting chart.

$\iota (M)$ equals $M$ sets-wise, but is just a subset of $M^{\prime }$, so, $\iota (M)$ and $M$ need to be distinguished.

The $C^{\mathrm{\infty }}$-ness of ${\iota }^{\prime -1}$ is according to the definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.

An immediate corollary is that when $f:N\to M^{\prime }$ such that $f(N)\subseteq \iota (M)$ is $C^{\mathrm{\infty }}$, $f^{\prime }:N\to M$, with the codomain of $f$ replaced, is $C^{\mathrm{\infty }}$, because $f^{\prime }={{\iota }^{\prime }}^{-1}\circ f$ while $f:N\to \iota (M)\subseteq M^{\prime }$ and ${{\iota }^{\prime }}^{-1}:\iota (M)\subseteq M^{\prime }\to M$ are $C^{\mathrm{\infty }}$, 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: $f:N\to \iota (M)\subseteq M^{\prime }$ 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.

As a caveat, when one tries to prove the $C^{\mathrm{\infty }}$-ness of $f:N_1\to M\to N_2$ by proving the $C^{\mathrm{\infty }}$-nesses of $N_1\to M$ and $M\to N_2$, the $C^{\mathrm{\infty }}$-ness of $N_1\to M^{\prime }$ guarantees the $C^{\mathrm{\infty }}$-ness of $N_1\to M$ by $N_1\to M^{\prime }\to M$ if $M$ is really an embedded submanifold with boundary of $M^{\prime }$ by this proposition, but if $M$ is an immersed submanifold with boundary of $M^{\prime }$, the $C^{\mathrm{\infty }}$-ness of $N_1\to M^{\prime }$ does not guarantee the $C^{\mathrm{\infty }}$-ness of $N_1\to M$, at least by this proposition.

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3: Proof

Whole Strategy: do it in the 2 parts: the 1st part: suppose that $M^{\prime }$ has the empty boundary; the 2nd part: suppose that $M^{\prime }$ has a nonempty boundary; Step 1: suppose that $M^{\prime }$ has the empty boundary; Step 2: for each $m^{\prime }\in \iota (M)$, take an adopted chart, $(U_{m^{\prime }}^{\prime }\subseteq M^{\prime },{\varphi }_{m^{\prime }}^{\prime })$, and the corresponding adopting chart, $(U_{m^{\prime }}\subseteq M,{\varphi }_{m^{\prime }})$, and see that ${{\iota }^{\prime }}^{-1}(U_{m^{\prime }}^{\prime }\cap \iota (M))\subseteq U_{m^{\prime }}$; Step 3: see that ${\varphi }_{m^{\prime }}\circ {{\iota }^{\prime }}^{-1}\circ {{\varphi }_{m^{\prime }}^{\prime }}^{-1}{|}_{{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))}:{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))\to {\varphi }_{m^{\prime }}(U_{m^{\prime }})$ is $C^{\mathrm{\infty }}$ at ${\varphi }_{m^{\prime }}^{\prime }(m^{\prime })$; Step 4: suppose that $m^{\prime }$ has a nonempty boundary; Step 5: take the double of $M^{\prime }$, $D(M^{\prime })$, which has the empty boundary, and apply the 1st part conclusion to conclude the proposition.

Step 1:

For the 1st part, let us suppose that $M^{\prime }$ has the empty boundary.

The reason is that we use the slice charts criterion, which requires $M^{\prime }$ to be without boundary.

Step 2:

For each $m^{\prime }\in \iota (M)$, let us take an adopted chart, $(U_{m^{\prime }}^{\prime }\subseteq M^{\prime },{\varphi }_{m^{\prime }}^{\prime })$, and the corresponding adopting chart, $(U_{m^{\prime }}\subseteq M,{\varphi }_{m^{\prime }})$, where $U_{m^{\prime }}=U_{m^{\prime }}^{\prime }\cap \iota (M)$ and ${\varphi }_{m^{\prime }}={\pi }_J\circ {\varphi }_{m^{\prime }}^{\prime }{|}_{U_{m^{\prime }}}$, where ${\pi }_J:{\mathbb{R}}^{d^{\prime }}\to {\mathbb{R}}^d$ is the projection.

${{\iota }^{\prime }}^{-1}(U_{m^{\prime }}^{\prime }\cap \iota (M))\subseteq U_{m^{\prime }}$, obviously.

Step 3:

Whether ${{\iota }^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$ at $m^{\prime }$ is about whether ${\varphi }_{m^{\prime }}\circ {{\iota }^{\prime }}^{-1}\circ {{\varphi }_{m^{\prime }}^{\prime }}^{-1}{|}_{{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))}:{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))\to {\varphi }_{m^{\prime }}(U_{m^{\prime }})$ is $C^{\mathrm{\infty }}$ at ${\varphi }_{m^{\prime }}^{\prime }(m^{\prime })$, by the definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.

${\varphi }_{m^{\prime }}\circ {{\iota }^{\prime }}^{-1}\circ {{\varphi }_{m^{\prime }}^{\prime }}^{-1}{|}_{{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))}$ is $(x^1,...,x^d,0,...,0)\mapsto (x^1,...,x^d)$, by the natures of adopted chart and adopting chart.

It can be extended to the $C^{\mathrm{\infty }}$ map, $f^{\prime }:{\mathbb{R}}^{d^{\prime }}\to {\mathbb{R}}^d,(x^1,...,x^d,0,...,0)\mapsto (x^1,...,x^d)$.

So, ${\varphi }_{m^{\prime }}\circ {{\iota }^{\prime }}^{-1}\circ {{\varphi }_{m^{\prime }}^{\prime }}^{-1}{|}_{{\varphi }_{m^{\prime }}^{\prime }(U_{m^{\prime }}^{\prime }\cap \iota (M))}$ is indeed $C^{\mathrm{\infty }}$ at ${\varphi }_{m^{\prime }}^{\prime }$, and so, ${{\iota }^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$ at $m^{\prime }$.

As $m^{\prime }\in \iota (M)$ is arbitrary, ${{\iota }^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$ all over $\iota (M)$.

Step 4:

For the 2nd part, let us suppose that $M^{\prime }$ has a nonempty boundary.

Step 5:

Let us take the double of $M^{\prime }$, $D(M^{\prime })$.

$D(M^{\prime })$ is a $C^{\mathrm{\infty }}$ manifold without boundary that has a regular domain, $\widetilde{M^{\prime }}$, that is diffeomorphic to $M^{\prime }$.

Let a diffeomorphism be $g:M^{\prime }\to \widetilde{M^{\prime }}$.

Let the restriction of $g$ be $g^{\prime }:M\to g(M):=\tilde{M}\subseteq \widetilde{M^{\prime }}$, where $\tilde{M}$ is the embedded submanifold with boundary of $\widetilde{M^{\prime }}$ that ($\tilde{M}$) is diffeomorphic to $M$: it is "restriction" map-between-sets-wise, but the domain and the codomain are replaced by the $C^{\mathrm{\infty }}$ manifolds with boundary, $M$ and $\tilde{M}$. $\tilde{M}$ can be indeed construed so, because while $\widetilde{M^{\prime }}$ is diffeomorphic to $M^{\prime }$, $\tilde{M}$ is to $\widetilde{M^{\prime }}$ what $M$ is to $M^{\prime }$.

$\tilde{M}$ is an embedded submanifold with boundary of $D(M^{\prime })$, because $\widetilde{M^{\prime }}$ is an embedded submanifold with boundary (in fact, a regular domain) of $D(M^{\prime })$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary.

Let $\tilde{\iota }:\tilde{M}\to D(M^{\prime })$ be the inclusion.

Let ${\tilde{\iota }}^{\prime }:\tilde{M}\to \tilde{\iota }(\tilde{M})\subseteq D(M^{\prime })$ be the codomain restriction of $\tilde{\iota }$.

By the 1st part conclusion, ${{\tilde{\iota }}^{\prime }}^{-1}:\tilde{\iota }(\tilde{M})\subseteq D(M^{\prime })\to \tilde{M}$ is $C^{\mathrm{\infty }}$.

Let $\tilde{\tau }:\widetilde{M^{\prime }}\to D(M^{\prime })$ be the inclusion, which is $C^{\mathrm{\infty }}$, because $\widetilde{M^{\prime }}$ is an embedded submanifold with boundary of $D(M^{\prime })$.

Let ${\tilde{\tau }}^{\prime }:\widetilde{M^{\prime }}\to \tilde{\tau }(\widetilde{M^{\prime }})\subseteq D(M^{\prime })$ be the codomain restriction of $\tilde{\tau }$, also which is $C^{\mathrm{\infty }}$.

${{\iota }^{\prime }}^{-1}:\iota (M)\subseteq M^{\prime }\to M=g^{\prime -1}\circ {{\tilde{\iota }}^{\prime }}^{-1}\circ {\tilde{\tau }}^{\prime }\circ g{|}_{\iota (M)}:\iota (M)\subseteq M^{\prime }\to g(\iota (M))=\tilde{\iota }(\tilde{M})\subseteq \widetilde{M^{\prime }}\to \tilde{\tau }(\tilde{\iota }(\tilde{M}))=\tilde{\iota }(\tilde{M})\subseteq D(M^{\prime })\to \tilde{M}\to M,m^{\prime }\mapsto g(m^{\prime })\mapsto \tilde{\tau }(g(m^{\prime }))=g(m^{\prime })\mapsto g(m^{\prime })\to m^{\prime }$.

As it is a composition of $C^{\mathrm{\infty }}$ maps, it 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.

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References

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