476: For Map Between Arbitrary Subsets of C∞ Manifolds with Boundary Ck at Point, Restriction on Domain That Contains Point Is Ck at Point

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A description/proof of that for map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, restriction on domain that contains point is $C^k$ at 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: Description
• 2: Proof

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Starting Context

• The reader knows a definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.
• The reader admits the proposition that for any map between any arbitrary subsets of any Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.

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Target Context

• The reader will have a description and a proof of 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 on any domain that contains the point is $C^k$ at the 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: Description

For any $C^{\mathrm{\infty }}$ manifolds with (possibly empty) boundary, $M_1,M_2$, any subsets, $S_1,S_1^{\prime }\subseteq M_1,S_2\subseteq M_2$, such that $S_1^{\prime }\subseteq S_1$, any point, $p\in S_1^{\prime }$, any natural number (including 0) or $\mathrm{\infty }$ $k$, and any map, $f:S_1\to S_2$, such that $f$ is $C^k$ at $p$, $f{|}_{S_1^{\prime }}:S_1^{\prime }\to S_2$ is $C^k$ at $p$.

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2: Proof

Let us suppose that $k=0$.

For any open neighborhood, $U_{f(p)}\subseteq S_2$, of $f(p)$, there is an open neighborhood, $U_p\subseteq S_1$, of $p$ such that $f(U_p)\subseteq U_{f(p)}$. $U_p\cap S_1^{\prime }\subseteq S_1^{\prime }$ is an open neighborhood of $p$ on $S_1^{\prime }$, and $f{|}_{S_1^{\prime }}(U_p\cap S_1^{\prime })\subseteq U_{f(p)}$.

Let us suppose that $1\le k$ including $\mathrm{\infty }$.

There are a chart, $(U_p^{\prime }\subseteq M_1,{\varphi }_p^{\prime })$, around $p$ and a chart, $(U_{f(p)}\subseteq M_2,{\varphi }_{f(p)})$, around $f(p)$ such that $f(U_p^{\prime }\cap S_1)\subseteq U_{f(p)}$ and ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}$ is $C^k$ at ${\varphi }_p^{\prime }(p)$.

$f{|}_{S_1^{\prime }}(U_p^{\prime }\cap S_1^{\prime })\subseteq U_{f(p)}$ and ${\varphi }_{f(p)}\circ f{|}_{S_1^{\prime }}\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1^{\prime })}={\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1^{\prime })}$ is $C^k$ at ${\varphi }_p^{\prime }(p)$, by the proposition that for any map between any arbitrary subsets of any Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.

So, the same charts pair can be used for $f{|}_{S_1^{\prime }}$.

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References

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