486: Diffeomorphism Between Arbitrary Subsets of C∞ Manifolds with Boundary

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A definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary

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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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of bijection.
• 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 $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.

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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: Definition

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 $C^{\mathrm{\infty }}$ bijection, $f:S_1\to S_2$, such that also the inverse, $f^{-1}:S_2\to S_1$ is $C^{\mathrm{\infty }}$, where $C^{\mathrm{\infty }}$-ness is by the definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

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2: Note

It is important to be aware that this definition is not by the existence of a diffeomorphic extension: there are a $C^{\mathrm{\infty }}$ extension for a direction and a $C^{\mathrm{\infty }}$ extension for the other direction, but the $C^{\mathrm{\infty }}$ extensions are not dictated to be the inverses to each other or each $C^{\mathrm{\infty }}$ extension is not dictated to be even injective. If you want to claim that there is a diffeomorphic extension, you will have to prove it.

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References

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