493: Map Between Arbitrary Subsets of C∞ Manifolds with Boundary Bijective and Locally Diffeomorphic at Each Point Is Diffeomorphism

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

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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 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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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 bijective and locally diffeomorphic at each point is a diffeomorphism.

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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, $M_1,M_2$, and any subsets, $S_1\subseteq M_1,S_2\subseteq M_2$, any map, $f:S_1\to S_2$, that is bijective and locally diffeomorphic at each point is a diffeomorphism.

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2: Proof

As $f$ is bijective, there is the inverse, $f^{-1}:S_2\to S_1$.

$f$ is $C^{\mathrm{\infty }}$, by 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.

For each $f(p)\in S_2$, there are an open neighborhood, $U_p\subseteq M_1$, of $p\in S_1$ and an open neighborhood, $U_{f(p)}\subseteq M_2$, of $f(p)$ such that $f{|}_{U_p\cap S_1}:U_p\cap S_1\to U_{f(p)}\cap S_2$ is a diffeomorphism. That means that there are an open neighborhood, $U_{f(p)}\subseteq M_2$, of $f(p)$ and an open neighborhood, $U_p\subseteq M_1$, of $p=f^{-1}(f(p))\in S_1$ such that $f^{-1}{|}_{U_{f(p)}\cap S_2}:U_{f(p)}\cap S_2\to U_p\cap S_1$ is a diffeomorphism, so, $f^{-1}$ is locally diffeomorphic at each $f(p)$, and is $C^{\mathrm{\infty }}$, by 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.

So, $f$ is a diffeomorphism.

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References

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