706: For Half Euclidean C∞ Manifold with Boundary, Open Half Ball Is Diffeomorphic to Whole Space

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description/proof of that for half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary, open half ball is diffeomorphic to whole space

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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: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.
• The reader admits the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold, any open ball is diffeomorphic to the whole space.

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Target Context

• The reader will have a description and a proof of the proposition that for any half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary, any open half ball is diffeomorphic to the whole space.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
${\mathbb{H}}^d$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold with boundary }$
$p$: $\in Bou({\mathbb{H}}^d)$, where $Bou({\mathbb{H}}^d)$ denotes the manifold boundary of ${\mathbb{H}}^d$
$H_{p,ϵ}$: $=\text{ the open half ball around }p\text{ on }{\mathbb{H}}^d$
$g$: $:H_{p,ϵ}\to {\mathbb{H}}^d,p^{\prime }\mapsto (p^{\prime }-p)/\sqrt{{ϵ}^2-\Vert p^{\prime }-p{\Vert }^2}$
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Statements:
$H_{p,ϵ}{\cong }_g{\mathbb{H}}^d$, where ${\cong }_g$ denotes being diffeomorphic by $g$
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2: Natural Language Description

For the half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary, ${\mathbb{H}}^d$, any open half ball around any $p\in Bou({\mathbb{H}}^d)$, where $Bou({\mathbb{H}}^d)$ denotes the manifold boundary of ${\mathbb{H}}^d$, $H_{p,ϵ}\subseteq {\mathbb{H}}^d$ is diffeomorphic to ${\mathbb{H}}^d$ by $g:H_{p,ϵ}\to {\mathbb{H}}^d,p^{\prime }\mapsto (p^{\prime }-p)/\sqrt{{ϵ}^2-\Vert p^{\prime }-p{\Vert }^2}$.

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3: Proof

Whole Strategy: Step 1: see that $g$ is a restriction of $f$ cited in the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold, any open ball is diffeomorphic to the whole space; Step 2: take the identity chart on ${\mathbb{H}}^d$, $({\mathbb{H}}^d\subseteq {\mathbb{H}}^d,id)$; Step 3: see that $id\circ g\circ i{d}^{-1}{|}_{id(H_{p,ϵ})}:id(H_{p,ϵ})\to id({\mathbb{H}}^d)$ has the $C^{\mathrm{\infty }}$ extension, $f:B_{p,ϵ}\to {\mathbb{R}}^d$; Step 4: see that $id\circ g^{-1}i{d}^{-1}:id({\mathbb{H}}^d)\to id({\mathbb{H}}^d)$ has the $C^{\mathrm{\infty }}$ extension, $f^{-1}:{\mathbb{R}}^d\to {\mathbb{R}}^d$.

Step 1:

$g$ is in fact a restriction of $f$ cited in the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold, any open ball is diffeomorphic to the whole space.

That implies that $g$ is injective; the surjectiveness is obvious: while the $d$-th component of $p$ is zero, for each $p^{″}\in {\mathbb{H}}^d$, there is a $p^{\prime }\in B_{p,ϵ}$ such that $f(p^{\prime })=p^{″}$, but as the $d$-th component of $p^{″}$ is non-negative, the $d$-th component of $p^{\prime }$ is non-negative, which means that $p^{\prime }\in H_{p,ϵ}$.

So, there is the inverse, $g^{-1}:{\mathbb{H}}^d\to H_{p,ϵ}$.

Step 2:

Let us take the identity chart, $({\mathbb{H}}^d\subseteq {\mathbb{H}}^d,id)$.

Step 3:

$C^{\mathrm{\infty }}$-ness of $g$ is about whether $id\circ g\circ i{d}^{-1}{|}_{id(H_{p,ϵ})}:id(H_{p,ϵ})\to id({\mathbb{H}}^d)$ has a $C^{\mathrm{\infty }}$ extension, according to the definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$. But $f:B_{p,ϵ}\to {\mathbb{R}}^d$ is indeed such an extension.

So, $g$ is $C^{\mathrm{\infty }}$.

Step 4:

$C^{\mathrm{\infty }}$-ness of $g^{-1}$ is about whether $id\circ g^{-1}\circ i{d}^{-1}:id({\mathbb{H}}^d)\to id({\mathbb{H}}^d)$ has a $C^{\mathrm{\infty }}$ extension, according to the definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$. But $f^{-1}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is indeed such an extension: $f^{-1}$ is really $f^{-1}:{\mathbb{R}}^d\to B_{p,ϵ}$, but the codomain can be expanded without any harm.

So, $g^{-1}$ is $C^{\mathrm{\infty }}$.

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References

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