821: For C∞ Embedding Between C∞ Manifolds with Boundary, Restriction of Embedding on Embedded Submanifold with Boundary Domain Is C∞ Embedding

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description/proof of that for $C^{\mathrm{\infty }}$ embedding between $C^{\mathrm{\infty }}$ manifolds with boundary, restriction of embedding on embedded submanifold with boundary domain is $C^{\mathrm{\infty }}$ embedding

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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
• 3: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ embedding.
• The reader knows a definition of embedded submanifold with boundary 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 any restriction of any continuous map on the domain and the codomain is continuous.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ embedding between any $C^{\mathrm{\infty }}$ manifolds with boundary, the restriction of the embedding on any embedded submanifold with boundary domain is a $C^{\mathrm{\infty }}$ embedding.

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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:
$M_1^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f$: $:M_1^{\prime }\to M_2$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
$M_1$: $\in \{\text{ the embedded submanifolds with boundary of }{M}_1^{\prime }\}$
$f^{\prime }$: $:M_1\to M_2,m\mapsto f(m)$
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Statements:
$f^{\prime }\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
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2: Proof

Whole Strategy: Step 1: see that $f^{\prime }$ is injective; Step 2: see that $f^{\prime }$ is $C^{\mathrm{\infty }}$; Step 3: see that $f^{\prime }$ is a $C^{\mathrm{\infty }}$ immersion; Step 4: see that the codomain restriction of $f^{\prime }$, $f^{″}:M_1\to f^{\prime }(M_1)$, is homeomorphic; Step 5: conclude the proposition.

Step 1:

$f^{\prime }$ is injective as $f$ is so.

Step 2:

Let $\iota :M_1\to M_1^{\prime }$ be the inclusion.

$\iota$ is a $C^{\mathrm{\infty }}$ embedding, because $M_1$ is an embedded submanifold with boundary of $M_1^{\prime }$.

$f^{\prime }=f\circ \iota$ 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

The differential is, $d{f}^{\prime }=df\circ d\iota$. $d\iota$ is injective, because $\iota$ is a $C^{\mathrm{\infty }}$ embedding. $df$ is injective, because $f$ is a $C^{\mathrm{\infty }}$ embedding. So, $d{f}^{\prime }$ is injective.

So, $f^{\prime }$ is a $C^{\mathrm{\infty }}$ immersion.

Step 4:

Let $f^{″}:M_1\to f^{\prime }(M_1)$ be the codomain restriction of $f^{\prime }$.

Let us see that $f^{″}$ is homeomorphic.

Let ${\iota }^{\prime }:M_1\to \iota (M_1)$ be the codomain restriction of $\iota$, which is homeomorphic because $\iota$ is a $C^{\mathrm{\infty }}$ embedding.

Let $f^{‴}:\iota (M_1)\to f(\iota (M_1))$ be the domain and the codomain restriction of $f$, which is homeomorphic because $f$ is a $C^{\mathrm{\infty }}$ embedding, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

$f^{″}=f^{‴}\circ {\iota }^{\prime }$, which is homeomorphic as a composition of homeomorphisms.

Step 5:

So, $f^{\prime }$ is an injective $C^{\mathrm{\infty }}$ embedding whose codomain restriction is homeomorphic, which means that $f^{\prime }$ is a $C^{\mathrm{\infty }}$ embedding.

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3: Note

As an immediate corollary, when $M_1$ is an open submanifold with boundary, $f^{\prime }$ is a $C^{\mathrm{\infty }}$ embedding: any open submanifold with boundary is an embedded submanifold with boundary.

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References

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