1053: C∞ Map Induced from C∞ Map from Finite-Product C∞ Manifold with Boundary by Fixing Some Domain Components Based on Point

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definition of $C^{\mathrm{\infty }}$ map induced from $C^{\mathrm{\infty }}$ map from finite-product $C^{\mathrm{\infty }}$ manifold with boundary by fixing some domain components based on point

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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

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Starting Context

• The reader knows a definition of finite-product $C^{\mathrm{\infty }}$ 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 definition of $C^{\mathrm{\infty }}$ map induced from $C^{\mathrm{\infty }}$ map from finite-product $C^{\mathrm{\infty }}$ manifold with boundary by fixing some domain components based on point.

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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:
$\{M_1,...,M_{n-1}\}$: $\subseteq \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
$M_n$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_1\times ...\times M_n$: $=\text{ the finite-product }{C}^{\mathrm{\infty }}\text{ manifold with boundary }$
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f$: $:M_1\times ...\times M_n\to M$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$m_0$: $\in M_1\times ...\times M_n$
$S$: $=\{j_1,...,j_s\}\subseteq \{1,...,n\}$
$\ast f_{S,m_0}$: $:M_1\times ...\times M_n\to M,m=(m^1,...m^n)\mapsto f(m_0^1,...,m^{j_1},...,m^{j_s},...,m_0^n)$, which means that $\{m_0^1,...,m_0^n\}$ are used except $\{m^{j_1},...,m^{j_s}\}$
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Conditions:
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When $S=\{j\}$, $f_{S,m_0}$ is denoted also as $f_{j,m_0}$, which is a typical case.

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2: Note

Let us see that $f_{S,m_0}$ is indeed $C^{\mathrm{\infty }}$.

Let $m=(m^1,...,m^n)\in M_1\times ...\times M_n$ be any.

$\tilde{m}:=(m_0^1,...,m^{j_1},...,m^{j_s},...,m_0^n)=({\tilde{m}}^1,...,{\tilde{m}}^n)$.

As $f$ is $C^{\mathrm{\infty }}$ at $\tilde{m}$, there are a chart around $\tilde{m}$, $(U_{\tilde{m}}\subseteq M_1\times ...\times M_n,{\varphi }_{\tilde{m}})$, and a chart around $f(\tilde{m})$, $(U_{f(\tilde{m})}\subseteq M,{\varphi }_{f(\tilde{m})})$, such that $f(U_{\tilde{m}})\subseteq U_{f(\tilde{m})}$ and ${\varphi }_{f(\tilde{m})}\circ f\circ {{\varphi }_{\tilde{m}}}^{-1}:{\varphi }_{\tilde{m}}(U_{\tilde{m}})\to {\varphi }_{f(\tilde{m})}(U_{f(\tilde{m})})$ is $C^{\mathrm{\infty }}$ at ${\varphi }_{\tilde{m}}(\tilde{m})$.

By the definition of finite-product $C^{\mathrm{\infty }}$ manifold with boundary, $(U_{\tilde{m}}\subseteq M_1\times ...\times M_n,{\varphi }_{\tilde{m}})$ can be chosen as $U_{\tilde{m}}=U_{1,{\tilde{m}}^1}\times ...\times U_{n,{\tilde{m}}^n}$ and ${\varphi }_{\tilde{m}}={\varphi }_{1,{\tilde{m}}^1}\times ...\times {\varphi }_{n,{\tilde{m}}^n}$, where $(U_{j,{\tilde{m}}^j}\subseteq M_j,{\varphi }_{j,{\tilde{m}}^j})$ is a chart for $M_j$. $U_{\tilde{m}}=U_{1,{m_0}^1}\times ...\times U_{j_1,m^{j_1}}\times ...\times U_{j_s,m^{j_s}}\times ...\times U_{n,{m_0}^n}$ and ${\varphi }_{\tilde{m}}={\varphi }_{1,{m_0}^1}\times ...\times {\varphi }_{j_1,m^{j_1}}\times ...\times {\varphi }_{j_s,m^{j_s}}\times ...\times {\varphi }_{n,{m_0}^n}$.

Obviously, ${{\varphi }_{\tilde{m}}}^{-1}={{\varphi }_{1,{m_0}^1}}^{-1}\times ...\times {{\varphi }_{j_1,m^{j_1}}}^{-1}\times ...\times {{\varphi }_{j_s,m^{j_s}}}^{-1}\times ...\times {{\varphi }_{n,{m_0}^n}}^{-1}$.

Let us take any chart around $m$, $(U_m\subseteq M_1\times ...\times M_n,{\varphi }_m)$, such that $U_m:=U_{1,m^1}\times ...\times U_{j_1,m^{j_1}}\times ...\times U_{j_s,m^{j_s}}\times ...U_{n,m^n}$ and ${\varphi }_m={\varphi }_{1,m^1}\times ...\times {\varphi }_{j_1,m^{j_1}}\times ...\times {\varphi }_{j_s,m^{j_s}}\times ...\times {\varphi }_{n,m^n}$, where $(U_{j,m^j}\subseteq M_j,{\varphi }_{j,m^j})$ is a chart for $M_j$, while $U_{j_1,m^{j_1}},...,U_{j_s,m^{j_s}}$ and ${\varphi }_{j_1,m^{j_1}},...,{\varphi }_{j_s,m^{j_s}}$ are the ones introduced above.

$f_{S,m_0}(U_m)\subseteq U_{f(\tilde{m})}$, because for each $m^{\prime }=(m^{\prime 1},...,m^{\prime n})\in U_m$, $f_{S,m_0}(m^{\prime })=f((m_0^1,...,m^{\prime j_1},...,m^{\prime j_s},...,m_0^n))$, but $(m_0^1,...,m^{\prime j_1},...,m^{\prime j_s},...,m_0^n)\in U_{1,{m_0}^1}\times ...\times U_{j_1,m^{j_1}}\times ...\times U_{j_s,m^{j_s}}\times ...\times U_{n,{m_0}^n}=U_{\tilde{m}}$ while $f(U_{\tilde{m}})\subseteq U_{f(\tilde{m})}$.

Let us think of ${\varphi }_{f(\tilde{m})}\circ f_{S,m_0}\circ {{\varphi }_m}^{-1}:{\varphi }_m(U_m)\to {\varphi }_{f(\tilde{m})}(U_{f(\tilde{m})})$.

It is ${\varphi }_{f(\tilde{m})}\circ f_{S,m_0}\circ {{\varphi }_{1,m^1}}^{-1}\times ...\times {{\varphi }_{j_1,m^{j_1}}}^{-1}\times ...\times {{\varphi }_{j_s,m^{j_s}}}^{-1}\times ...\times {{\varphi }_{n,m^n}}^{-1}:(x^{\prime 1},...,x^{\prime n})\mapsto {\varphi }_{f(\tilde{m})}\circ f_{S,m_0}((m^{\prime 1},...,m^{\prime n}))={\varphi }_{f(\tilde{m})}\circ f((m_0^1,...,m^{\prime j_1},...,m^{\prime j_s},...,m_0^n))$.

That is in fact same with ${\varphi }_{f(\tilde{m})}\circ f\circ {{\varphi }_{\tilde{m}}}^{-1}$ where $({\varphi }_{{\tilde{m}}^1}(m_0^1),...,\widehat{x^{\prime j_1}},...,\widehat{x^{\prime j_s}},...,{\varphi }_{{\tilde{m}}^n}(m_0^n))$ is fixed.

As ${\varphi }_{f(\tilde{m})}\circ f\circ {{\varphi }_{\tilde{m}}}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_{\tilde{m}}(\tilde{m})$, it is $C^{\mathrm{\infty }}$ with respect to $(x^{\prime j_1},...,x^{\prime j_s})$, so, ${\varphi }_{f(\tilde{m})}\circ f_{S,m_0}\circ {{\varphi }_m}^{-1}$ is $C^{\mathrm{\infty }}$ with respect to $(x^{\prime j_1},...,x^{\prime j_s})$.

${\varphi }_{f(\tilde{m})}\circ f_{S,m_0}\circ {{\varphi }_m}^{-1}$ is constant with respect to $(x^{\prime 1},...,\widehat{x^{\prime j_1}},...,\widehat{x^{\prime j_s}},...,x^{\prime n})$, so, is $C^{\mathrm{\infty }}$ with respect to $(x^{\prime 1},...,\widehat{x^{\prime j_1}},...,\widehat{x^{\prime j_s}},...,x^{\prime n})$.

So, ${\varphi }_{f(\tilde{m})}\circ f_{S,m_0}\circ {{\varphi }_m}^{-1}$ is $C^{\mathrm{\infty }}$ with respect $x^{\prime 1},...,x^{\prime n}$.

So, $f_{S,m_0}$ is $C^{\mathrm{\infty }}$ at $m$.

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References

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