492: Map Between Arbitrary Subsets of C∞ Manifolds with Boundary Locally Diffeomorphic at Point Is C∞ at Point

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A description/proof of that map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at point is $C^{\mathrm{\infty }}$ at point

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Topics

About: $C^{\mathrm{\infty }}$ manifold with boundary

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at point.

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Target Context

• The reader will have a description and a proof of the proposition that any map between arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at any point is $C^{\mathrm{\infty }}$ 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, $S_1\subseteq M_1,S_2\subseteq M_2$, and any point, $p\in S_1$, any map, $f:S_1\to S_2$, that is locally diffeomorphic at $p$ is $C^{\mathrm{\infty }}$ at $p$.

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2: Proof

There are an open neighborhood, $U_p\subseteq M_1$, of $p$ and an open neighborhood, $U_{f(p)}\subseteq M_2$, of $f(p)\in S_2$, such that $f{|}_{U_p\cap S_1}:U_p\cap S_1\to U_{f(p)}\cap S_2$ is a diffeomorphism.

As $f{|}_{U_p\cap S_1}$ is $C^{\mathrm{\infty }}$, there are a chart, $(U_p^{\prime }\subseteq M_1,{\varphi }_p^{\prime })$, and a chart, $(U_{f(p)}^{\prime }\subseteq M_2,{\varphi }_{f(p)}^{\prime })$, such that $f{|}_{U_p\cap S_1}(U_p^{\prime }\cap U_p\cap S_1)\subseteq U_{f(p)}^{\prime }$ and ${\varphi }_{f(p)}^{\prime }\circ f{|}_{U_p\cap S_1}\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap U_p\cap S_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap U_p\cap S_1)\to {\varphi }_{f(p)}^{\prime }(U_{f(p)}^{\prime })$ is $C^{\mathrm{\infty }}$ at ${\varphi }_p^{\prime }(p)$.

Also $(U_p^{\prime }\cap U_p\subseteq M_1,{\varphi }_p^{\prime }{|}_{U_p^{\prime }\cap U_p})$ is a chart, and $f(U_p^{\prime }\cap U_p\cap S_1)=f{|}_{U_p\cap S_1}(U_p^{\prime }\cap U_p\cap S_1)\subseteq U_{f(p)}^{\prime }$ and ${\varphi }_{f(p)}^{\prime }\circ f\circ {{\varphi }_p^{\prime }{|}_{U_p^{\prime }\cap U_p}}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap U_p\cap S_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap U_p\cap S_1)\to {\varphi }_{f(p)}^{\prime }(U_{f(p)}^{\prime })={\varphi }_{f(p)}^{\prime }\circ f{|}_{U_p\cap S_1}\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap U_p\cap S_1)}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_p^{\prime }(p)$.

So, $f$ is $C^{\mathrm{\infty }}$ at $p$.

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3: Note

Typically, $S_1=M_1$ and $S_2=M_2$, and any map between any $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at $p$ is $C^{\mathrm{\infty }}$ at $p$.

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References

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