1046: For C∞ Manifold, Regular Domain, Inclusion, and 2 C∞ Vectors Fields, Lie Bracket of Vectors Fields Is Pulled-Back Lie Bracket of Pushed-Forward-and-Extended Vectors Fields

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description/proof of that for $C^{\mathrm{\infty }}$ manifold, regular domain, inclusion, and 2 $C^{\mathrm{\infty }}$ vectors fields, Lie bracket of vectors fields is pulled-back Lie bracket of pushed-forward-and-extended vectors fields

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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 1
• 3: Proof
• 4: Note 2

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Starting Context

• The reader knows a definition of regular domain of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors field on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of Lie bracket of $C^{\mathrm{\infty }}$ vectors fields on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at point.
• The reader admits 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.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold, its any regular domain, any $C^{\mathrm{\infty }}$ manifold with boundary, and any $C^{\mathrm{\infty }}$ map from the regular domain into the $C^{\mathrm{\infty }}$ manifold with boundary, the corresponding map with the domain regarded as the subset of the manifold is $C^{\mathrm{\infty }}$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of the manifold with boundary is properly embedded if and only if it is closed.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.
• The reader admits the proposition that for any map from any subset of any $C^{\mathrm{\infty }}$ manifold with boundary into any subset of any $C^{\mathrm{\infty }}$ manifold $C^k$ at any point, there is a $C^k$ extension on an open-neighborhood-of-the-point domain.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ map between any $C^{\mathrm{\infty }}$ manifolds with boundary, any the-map-related 2 $C^{\mathrm{\infty }}$ vectors fields on the domain and any 2 $C^{\mathrm{\infty }}$ vectors fields on the codomain, the Lie bracket of the vectors fields on the domain is the-map-related to the Lie bracket of the vectors fields on the codomain.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any regular domain, the differential of the inclusion at each point on the regular domain is a 'vectors spaces - linear morphisms' isomorphism.

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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 regular domain of the $C^{\mathrm{\infty }}$ manifold, the inclusion of the regular domain into the $C^{\mathrm{\infty }}$ manifold, and any 2 $C^{\mathrm{\infty }}$ vectors fields over the regular domain, the Lie bracket of the vectors fields is the pulled-back Lie bracket of any pushed-forward-and-extended vectors fields.

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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:
$M^{\prime }$: $\in \{\text{ the }{d}^{\prime }\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
$M$: $\in \{\text{ the regular domains of }{M}^{\prime }\}$
$\iota$: $:M\to M^{\prime }$, $=\text{ the inclusion }$
$V$: $:M\to TM$, $\in \mathrm{\Gamma }(TM)$
$W$: $:M\to TM$, $\in \mathrm{\Gamma }(TM)$
$d\iota$: $:TM\to T{M}^{\prime }$, $=\text{ the differential }$
//

Statements:
$[V,W]=\lambda ([\tau V,\tau W])$, where $\tau$ pushes-forward each vector on $TM$ into $T{M}^{\prime }$ by $d\iota$ and extends it smoothly over $M^{\prime }$ and $\lambda$ pulls-back each vector pushed-forward by $d\iota$
//

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2: Note 1

In order for $\tau$ to make sense, $d\iota \circ V:M\to T{M}^{\prime }$ and $d\iota \circ W:M\to T{M}^{\prime }$ have to be proved to be $C^{\mathrm{\infty }}$ and $d\iota \circ V$ and $d\iota \circ W$ have to be proved to be able to be extended smoothly, which will be done in Proof.

In order for $\lambda$ to make sense, $[\tau V,\tau W]$ at each point on $\iota (M)$ has to be proved to be the pushed-forward of a vector on $TM$, which will be done in Proof.

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3: Proof

Whole Strategy: Step 1: see that $d\iota \circ V:M\to T{M}^{\prime }$ and $d\iota \circ W:M\to T{M}^{\prime }$ are some $C^{\mathrm{\infty }}$ maps; Step 2: see that $d\iota \circ V:M\to T{M}^{\prime }$ and $d\iota \circ W:M\to T{M}^{\prime }$ can be extended to some $C^{\mathrm{\infty }}$ $\tau V\in \mathrm{\Gamma }(T{M}^{\prime })$ and $\tau W\in \mathrm{\Gamma }(T{M}^{\prime })$; Step 3: see that $[V,W]$ is $\iota$-related to $[\tau V,\tau W]$, which means that $[V,W]$ can be pushed-forward-and-extended to $\tau ([V,W])=[\tau V,\tau W]$; Step 4: do $\lambda \circ \tau ([V,W])=\lambda ([\tau V,\tau W])$.

Step 1:

Let us see that $d\iota \circ V:M\to T{M}^{\prime }$ is $C^{\mathrm{\infty }}$.

In fact, that has been proved by 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.

$d\iota \circ W:M\to T{M}^{\prime }$ is $C^{\mathrm{\infty }}$, likewise.

Step 2:

While $d\iota \circ V:M\to T{M}^{\prime }$ is $C^{\mathrm{\infty }}$ with the domain regarded as the regular domain, it is $C^{\mathrm{\infty }}$ also with the domain regarded as the subset of $M^{\prime }$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold, its any regular domain, any $C^{\mathrm{\infty }}$ manifold with boundary, and any $C^{\mathrm{\infty }}$ map from the regular domain into the $C^{\mathrm{\infty }}$ manifold with boundary, the corresponding map with the domain regarded as the subset of the manifold is $C^{\mathrm{\infty }}$.

$\iota (M)\subseteq M^{\prime }$ is closed on $M^{\prime }$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of the manifold with boundary is properly embedded if and only if it is closed.

$d\iota \circ V$ can be extended to a $\tau V\in \mathrm{\Gamma }(T{M}^{\prime })$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, any $C^{\mathrm{\infty }}$ section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset, while $d\iota \circ V$ is indeed a $C^{\mathrm{\infty }}$ vectors field along $\iota (M)$, by the proposition that for any map from any subset of any $C^{\mathrm{\infty }}$ manifold with boundary into any subset of any $C^{\mathrm{\infty }}$ manifold $C^k$ at any point, there is a $C^k$ extension on an open-neighborhood-of-the-point domain: the definition of $C^{\mathrm{\infty }}$ vectors field along subset is exactly that there is a $C^{\mathrm{\infty }}$ extension.

$d\iota \circ W$ can be extended to a $\tau V\in \mathrm{\Gamma }(T{M}^{\prime })$, likewise.

Step 3:

$V$ is $\iota$-related to $\tau V$, because $d\iota V_p=(\tau V{)}_{\iota (p)}$.

$W$ is $\iota$-related to $\tau W$, likewise.

So, $[V,W]$ is $\iota$-related to $[\tau V,\tau W]$, by the proposition that for any $C^{\mathrm{\infty }}$ map between any $C^{\mathrm{\infty }}$ manifolds with boundary, any the-map-related 2 $C^{\mathrm{\infty }}$ vectors fields on the domain and any 2 $C^{\mathrm{\infty }}$ vectors fields on the codomain, the Lie bracket of the vectors fields on the domain is the-map-related to the Lie bracket of the vectors fields on the codomain.

That means that $[V,W]$ can be pushed-forward and extended to be $[\tau V,\tau W]$, which is denoted as $\tau ([V,W])=[\tau V,\tau W]$.

Step 4:

$\lambda \circ \tau ([V,W])$ is valid because $d{\iota }_p$ is bijective at each $p\in M$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any regular domain, the differential of the inclusion at each point on the regular domain is a 'vectors spaces - linear morphisms' isomorphism and $\tau ([V,W])$ at each $\iota (p)$ is the pushed-forward of the vector in $T_{p}M$, $[V,W{]}_p$.

Let us do $\lambda \circ \tau ([V,W])=\lambda ([\tau V,\tau W])$, but $\lambda \circ \tau =id$, the identity map, so, $[V,W]=\lambda \circ \tau ([V,W])=\lambda ([\tau V,\tau W])$.

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4: Note 2

An immediate corollary of this proposition is that $\lambda ([\tau V,\tau W])$ does not depend on $M^{\prime }$ or the extension (because it equals $[V,W]$, which is ignorant of $M^{\prime }$ or the extension), which is in fact the immediate purpose of this proposition: the Lie derivative of $W$ by $V$ at any boundary point, $p\in \partial M$, is defined to be the pulled-back Lie derivative of $\tau W$ by $\tau V$ at $\iota (p)$, which equals $\lambda ([\tau V,\tau W{]}_{\iota (p)})$, which needs to be proved to be independent of $M^{\prime }$ or the extension.

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References

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