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

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A definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary 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: Definition
• 2: Note

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Starting Context

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

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Target Context

• The reader will have a definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary locally diffeomorphic at 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: Definition

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, $f:S_1\to S_2$, such that 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

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2: Note

Typically, $S_1=M_2$ and $S_2=M_2$, and then, $f$ is a map between some $C^{\mathrm{\infty }}$ manifolds with boundary local diffeomorphic at $p$.

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References

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