497: For Map Between Arbitrary Subsets of C∞ Manifolds with Boundary Locally Diffeomorphic at Point, Restriction on Open Subset of Domain That Contains Point Is Locally Diffeomorphic at Point

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A description/proof of that for map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at point, restriction on open subset of domain that contains point is locally diffeomorphic 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

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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.
• The reader admits 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.
• The reader admits 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 or expansion on any codomain that contains the range 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 arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at any point, the restriction on any open subset of the domain that contains the point is locally diffeomorphic 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 $S_1\subseteq S_1^{\prime }$ is open on $S_1^{\prime }$, any point, $p\in S_1$, and any map, $f^{\prime }:S_1^{\prime }\to S_2$, such that $f^{\prime }$ is locally diffeomorphic at $p$, the domain restriction, $f:S_1\to S_2$ is locally diffeomorphic at $p$.

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2: Proof

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

$f{|}_{U_p^{\prime }\cap S_1}=f^{\prime }{|}_{U_p^{\prime }\cap S_1^{\prime }}{|}_{U_p^{\prime }\cap S_1}:U_p^{\prime }\cap S_1\to f{|}_{U_p^{\prime }\cap S_1}(U_p^{\prime }\cap S_1)$ is a diffeomorphism, by 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 and 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 or expansion on any codomain that contains the range is $C^k$ at the point.

But the issue is that the range has not been proved to be an open subset of $S_2$ (the invariance of domain theorem does not apply here, because the diffeomorphism is not from any open subset of any $C^{\mathrm{\infty }}$ manifold without boundary).

$U_p^{\prime }\cap S_1$ is an open subset of $U_p^{\prime }\cap S_1^{\prime }$, because as $S_1$ is open on $S_1^{\prime }$, $S_1=U_p^{″}\cap S_1^{\prime }$, where $U_p^{″}\subset M_1$ is an open subset of $M_1$, so, $U_p^{\prime }\cap S_1=U_p^{\prime }\cap U_p^{″}\cap S_1^{\prime }=U_p^{″}\cap U_p^{\prime }\cap S_1^{\prime }$. So, $f^{\prime }{|}_{U_p^{\prime }\cap S_1^{\prime }}(U_p^{\prime }\cap S_1)$ is open on $U_{f(p)}\cap S_2$, so, $f^{\prime }{|}_{U_p^{\prime }\cap S_1^{\prime }}(U_p^{\prime }\cap S_1)=U_{f(p)}^{\prime }\cap U_{f(p)}\cap S_2$, where $U_{f(p)}^{\prime }$ is an open subset of $M_2$.

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

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References

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