469: Map Between Arbitrary Subsets of C∞ Manifolds with Boundary Ck at Point, Where k Excludes 0 and Includes ∞

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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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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 $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of 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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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, $S_1\subseteq M_1,S_2\subseteq M_2$, any point, $p\in S_1$, and any natural number (excluding 0) or $\mathrm{\infty }$ $k$, any map, $f:S_1\to S_2$, such that there are a chart, $(U_p^{\prime }\subseteq M_1,{\varphi }_p^{\prime })$, around $p$ and a chart, $(U_{f(p)}\subseteq M_2,{\varphi }_{f(p)})$, around $f(p)$ such that $f(U_p^{\prime }\cap S_1)\subseteq U_{f(p)}$ and ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)\to {\varphi }_{f(p)}(U_{f(p)})$ is $C^k$ at ${\varphi }_p^{\prime }(p)$ by the definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

When this definition is satisfied, there is the open subset, $U_{{\varphi }_p^{\prime }(p)}$, cited in the definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$, on the whole of which $f^{\prime }$ is $C^k$ and $f^{\prime }{|}_{U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}={\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}$.

As $U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime })$ is an open subset of ${\varphi }_p^{\prime }(U_p^{\prime })$, ${{\varphi }_p^{\prime }}^{-1}(U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime }))$ is open on $U_p^{\prime }$ and on $M_1$, and so, ${{\varphi }_p^{\prime }}^{-1}(U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime }))$ can be taken to be $U_p^{\prime }$ instead satisfying the conditions, and then, ${\varphi }_p^{\prime }(U_p^{\prime })\subseteq U_{{\varphi }_p^{\prime }(p)}$.

When ${\varphi }_p^{\prime }(U_p^{\prime })$ is open on ${\mathbb{R}}^{d_1}$, $U_{{\varphi }_p^{\prime }(p)}$ can be taken to be contained in ${\varphi }_p^{\prime }(U_p^{\prime })$, and then, $U_{{\varphi }_p^{\prime }(p)}\subseteq {\varphi }_p^{\prime }(U_p^{\prime })$. If furthermore ${{\varphi }_p^{\prime }}^{-1}(U_{{\varphi }_p^{\prime }(p)}\cap {\varphi }_p^{\prime }(U_p^{\prime }))$ is taken to be $U_p^{\prime }$ instead as in the previous paragraph, ${\varphi }_p^{\prime }(U_p^{\prime })=U_{{\varphi }_p^{\prime }(p)}$.

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2: Note

$k=0$ is excluded because that case has been already defined as map continuous at point.

But when $f$ is $C^k$ at $p$ where $1\le k$, $f$ is $C^0$ at $p$: for any open neighborhood, $U_{f(p)}^{\prime }\subseteq S_2$, of $f(p)$, we can think of only the case, $U_{f(p)}^{\prime }\subseteq U_{f(p)}$, because otherwise, we can think of $U_{f(p)}^{\prime }\cap U_{f(p)}$, which is open on $S_2$, instead; $U_{f(p)}^{\prime }=U_{f(p)}^{″}\cap S_2$ where $U_{f(p)}^{″}\subseteq M_2$ is open and we can take $U_{f(p)}^{″}\subseteq U_{f(p)}$; ${\varphi }_{f(p)}(U_{f(p)}^{″})\subseteq {\varphi }_{f(p)}(U_{f(p)})$ is open on ${\mathbb{H}}^{d_2}$ or ${\mathbb{R}}^{d_2}$ and ${\varphi }_{f(p)}(U_{f(p)}^{″})=U_{{\varphi }_{f(p)}(f(p))}\cap {\mathbb{H}}^{d_2}\text{ or }{U}_{{\varphi }_{f(p)}(f(p))}\cap {\mathbb{R}}^{d_2}$ where $U_{{\varphi }_{f(p)}(f(p))}\subseteq {\mathbb{R}}^{d_2}$ is open; the extension of ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}$, $f^{\prime }:U_{{\varphi }_p(p)}^{\prime }\to {\mathbb{R}}^{d_2}$, is $C^k$ and continuous at ${\varphi }_p(p)$; there is an open neighborhood, $U_{{\varphi }_p(p)}^{″}\subseteq U_{{\varphi }_p(p)}^{\prime }$, of ${\varphi }_p(p)$ such that $f^{\prime }(U_{{\varphi }_p(p)}^{″})\subseteq U_{{\varphi }_{f(p)}(f(p))}$; $U_{{\varphi }_p(p)}^{″}\cap {\varphi }_p^{\prime }(U_p^{\prime })$ is open on ${\varphi }_p^{\prime }(U_p^{\prime })$ and ${{\varphi }_p^{\prime }}^{-1}(U_{{\varphi }_p(p)}^{″}\cap {\varphi }_p^{\prime }(U_p^{\prime }))\subseteq U_p^{\prime }$ is open on $U_p^{\prime }$ and on $M_1$; $U_p^{″}:={{\varphi }_p^{\prime }}^{-1}(U_{{\varphi }_p(p)}^{″}\cap {\varphi }_p^{\prime }(U_p^{\prime }))\cap S_1$ is open on $S_1$; $f(U_p^{″})={{\varphi }_{f(p)}}^{-1}\circ {\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}\circ {\varphi }_p^{\prime }(U_p^{″})={{\varphi }_{f(p)}}^{-1}\circ f^{\prime }\circ {\varphi }_p^{\prime }(U_p^{″})$, but ${\varphi }_p^{\prime }(U_p^{″})\subseteq U_{{\varphi }_p(p)}^{″}$, so, $f^{\prime }\circ {\varphi }_p^{\prime }(U_p^{″})\subseteq U_{{\varphi }_{f(p)}(f(p))}$ and it is also contained in ${\mathbb{H}}^{n^{\prime }}$ or ${\mathbb{R}}^{n^{\prime }}$ because the coordinates function maps into there, so, it is contained in ${\varphi }_{f(p)}(U_{f(p)}^{″})$; so, $f(U_p^{″})\subseteq U_{f(p)}^{″}\cap S_2=U_{f(p)}^{\prime }$.

When $M_1$ and $M_2$ are any Euclidean $C^{\mathrm{\infty }}$ manifolds, this definition coincides with the definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$, because in that case, if $f$ satisfies the latter definition, we can take $(U_p^{\prime }=M_1\subseteq M_1,{\varphi }_p^{\prime }=id)$ and $(U_{f(p)}=M_2\subseteq M_2,{\varphi }_{f(p)}=id)$ for the former definition, and ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}=f$, which is $C^k$ at ${\varphi }_p^{\prime }(p)=p$ by the latter definition; if $f$ satisfies the former definition, while ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}$ is $C^k$ at ${\varphi }_p^{\prime }(p)$, $id\circ {{\varphi }_{f(p)}}^{-1}\circ {\varphi }_{f(p)}\circ f\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap S_1)}\circ {\varphi }_p^{\prime }\circ {id}^{-1}{|}_{U_p^{\prime }\cap S_1}=f{|}_{U_p^{\prime }\cap S_1}$ is $C^k$ at $p$, because the transition maps are diffeomorphisms, by the proposition that for any map between arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, its restriction on any subset that contains the point is $C^k$ at the point and the proposition that the composition of any maps between arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, is $C^k$ at the point, and so, $f$ is $C^k$ at $p$, by the proposition that any map between arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary is $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, if its restriction on any open subspace domain that contains the point is $C^k$ at the point.

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References

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