782: For C∞ Manifold, Embedded Submanifold with Boundary, and C∞ Vectors Field over Submanifold with Boundary, Differential by Inclusion After Vectors Field Is C∞ over Submanifold with Boundary

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description/proof of that for $C^{\mathrm{\infty }}$ manifold, embedded submanifold with boundary, and $C^{\mathrm{\infty }}$ vectors field over submanifold with boundary, differential by inclusion after vectors field is $C^{\mathrm{\infty }}$ over submanifold with boundary

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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: Natural Language Description
• 3: Note
• 4: 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 $\mathrm{\infty }$.

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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, any embedded submanifold with boundary of the manifold, and any $C^{\mathrm{\infty }}$ vectors field over the submanifold with boundary, the differential by the inclusion after the vectors field is $C^{\mathrm{\infty }}$ over the submanifold with boundary.

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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:
$M^{\prime }$: $\in \{\text{ the }{d}^{\prime }\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
$M$: $\in \{\text{ the embedded submanifolds with boundary of }{M}^{\prime }\}$
$V$: $:M\to TM$, $\in \mathrm{\Gamma }(TM)$
$\iota$: $:M\to M^{\prime }$, $=\text{ the inclusion }$
$d\iota$: $:TM\to T{M}^{\prime }$, $=\text{ the differential }$
$d\iota \circ V$: $:M\to T{M}^{\prime }$
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Statements:
$d\iota \circ V\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
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2: Natural Language Description

For any $d^{\prime }$-dimensional $C^{\mathrm{\infty }}$ manifold, $M^{\prime }$, any embedded submanifold with boundary of $M^{\prime }$, $M$, any $C^{\mathrm{\infty }}$ vectors field, $V:M\to TM$, the inclusion, $\iota :M\to M^{\prime }$, the differential, $d\iota :TM\to T{M}^{\prime }$, and $d\iota \circ V:M\to T{M}^{\prime }$, $d\iota \circ V$ is $C^{\mathrm{\infty }}$.

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3: Note

Although "differential of $V$" usually makes people imagine a vectors field over $M^{\prime }$ (which is invalid as $\iota$ is not diffeomorphic), we are not talking about that.

Anyway, $d\iota$ is valid, because $\iota$ is a $C^{\mathrm{\infty }}$ embedding, by the definition of embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.

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4: Proof

Whole Strategy: Step 1: around any $m\in M$, take an adopted chart, $(U_m^{\prime }\subseteq M^{\prime },{\varphi }_m^{\prime })$, and the corresponding adopting chart, $(U_m:=U_m^{\prime }\cap M\subseteq M,{\varphi }_m:={\pi }_J\circ {\varphi }_m^{\prime }{|}_{U_m})$; Step 2: see what $V$'s being $C^{\mathrm{\infty }}$ means with respect to $(U_m\subseteq M,{\varphi }_m)$; Step 3: see what $d\iota \circ V$'s being $C^{\mathrm{\infty }}$ means with respect to $(U_m^{\prime }\subseteq M^{\prime },{\varphi }_m^{\prime })$; Step 4: conclude the proposition.

Step 1:

Let $m\in M$ be any.

As $M$ is an embedded submanifold with boundary of $M^{\prime }$, there is an adopted chart, $(U_m^{\prime }\subseteq M^{\prime },{\varphi }_m^{\prime })$, for $M$.

There is the corresponding adopting chart, $(U_m:=U_m^{\prime }\cap M\subseteq M,{\varphi }_m:={\pi }_J\circ {\varphi }_m^{\prime }{|}_{U_m})$, where ${\pi }_J:{\mathbb{R}}^{d^{\prime }}\to {\mathbb{R}}^d$ is the projection taking the $J\subseteq \{1,...,d^{\prime }\}$ components.

Step 2:

Let us denote $\pi :TM\to M$ and ${\pi }^{\prime }:T{M}^{\prime }\to M^{\prime }$.

There are the induced charts using the standard bases for $TM$ and $T{M}^{\prime }$, $({\pi }^{-1}(U_m)\subseteq TM,\widetilde{{\varphi }_m})$ and $({\pi }^{\prime -1}(U_m^{\prime })\subseteq T{M}^{\prime },\widetilde{{\varphi }_m^{\prime }})$.

As $V$ is $C^{\mathrm{\infty }}$ at $m$, $\widetilde{{\varphi }_m}\circ V\circ {{\varphi }_m}^{-1}:{\varphi }_m(U_m)\to \widetilde{{\varphi }_m}({\pi }^{-1}(U_m))$ is $C^{\mathrm{\infty }}$ at ${\varphi }_m(m)$.

Step 3:

Whether $d\iota \circ V$ is $C^{\mathrm{\infty }}$ at $m$ or not is about whether $\widetilde{{\varphi }_m^{\prime }}\circ d\iota \circ V\circ {{\varphi }_m}^{-1}:{\varphi }_m(U_m)\to \widetilde{{\varphi }_m^{\prime }}({\pi }^{\prime -1}(U_m^{\prime }))$ is $C^{\mathrm{\infty }}$ at ${\varphi }_m(m)$ or not.

Step 4:

Let us denote the coordinates function of $\iota$ as $f:={\varphi }_m^{\prime }\circ \iota \circ {{\varphi }_m}^{-1}:{\varphi }_m(U_m)\to {\varphi }_m^{\prime }(U_m^{\prime })$.

In order for the simplicity of notations, we can choose ${\varphi }_m^{\prime }$ such that $f:(x^1,...,x^d)\mapsto (x^1,...,x^d,0,...,0)$: we can just reorder the components of ${\varphi }_m^{\prime }$ and translate ${\varphi }_m^{\prime }$. Then, $\partial f^k/\partial x^j={\delta }_j^k$ for $j,k\in \{1,...,d\}$ and $\partial f^k/\partial x^j=0$ otherwise.

Let us denote the $j$-th component of $V$ with respect to $({\pi }^{-1}(U_m)\subseteq TM,\widetilde{{\varphi }_m})$ as $V^j$. Let us denote the $j$-th component of $d\iota \circ V$ with respect to $({\pi }^{\prime -1}(U_m^{\prime })\subseteq T{M}^{\prime },\widetilde{{\varphi }_m^{\prime }})$ as $(d\iota \circ V{)}^j$.

$\widetilde{{\varphi }_m}\circ V\circ {{\varphi }_m}^{-1}:(x^1,...,x^d)\mapsto (x^1,...,x^d,V^1,...,V^d)$, which is $C^{\mathrm{\infty }}$ at ${\varphi }_m(m)$.

As is known well, $(d\iota \circ V{)}^k=\partial f^k/\partial x^j{V}^j$ with the Einstein convention, which is $=V^k$ for each $k\in \{1,...,d\}$ and $=0$ otherwise. That means that $\widetilde{{\varphi }_m^{\prime }}\circ d\iota \circ V\circ {{\varphi }_m}^{-1}:(x^1,...,x^d)\mapsto (x^1,...,x^d,0,...,0,V^1,...,V^d,0,...,0)$, which is $C^{\mathrm{\infty }}$ at ${\varphi }_m(m)$, obviously.

So, $d\iota \circ V$ is $C^{\mathrm{\infty }}$ at $m$.

As $m\in M$ is arbitrary, $d\iota \circ V$ is $C^{\mathrm{\infty }}$ all over $M$.

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References

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