707: For C∞ Manifold with Boundary, Interior Point Has Chart Whose Range Is Whole Euclidean Space and Boundary Point Has Chart Whose Range Is Whole Half Euclidean Space

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description/proof of that for $C^{\mathrm{\infty }}$ manifold with boundary, interior point has chart whose range is whole Euclidean space and boundary point has chart whose range is whole half Euclidean 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 admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball.
• The reader admits the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold, any open ball is diffeomorphic to the whole space.
• The reader admits 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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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any interior point has a chart whose range is the whole Euclidean space and any boundary point has a chart whose range is the whole half Euclidean 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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
$p$: $\in M$
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Statements:
(
$p\in \{\text{ the interior points of }M\}$
$⟹$
$\mathrm{\exists }(B_p\subseteq M,{\varphi }_p)\in \{\text{ the charts around }p\text{ on }M\}({\varphi }_p(B_p)={\mathbb{R}}^d)$
)
$\wedge$
(
$p\in \{\text{ the boundary points of }M\}$
$⟹$
$\mathrm{\exists }(H_p\subseteq M,{\varphi }_p)\in \{\text{ the charts around }p\text{ on }M\}({\varphi }_p(H_p)={\mathbb{H}}^d)$
)
//

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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifold with boundary, $M$, and any $p\in M$, if $p$ is any interior point of $M$, there is a chart around $p$, $(B_p\subseteq M,{\varphi }_p)$, such that ${\varphi }_p(B_p)={\mathbb{R}}^d$, and if $p$ is any boundary point of $M$, there is a chart around $p$, $(H_p\subseteq M,{\varphi }_p)$, such that ${\varphi }_p(H_p)={\mathbb{H}}^d$.

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3: Proof

Whole Strategy: Step 1: suppose that $p$ is any interior point and take any chart around $p$, $(B_p\subseteq M,{\varphi }_p)$, with $B_p$ as a chart ball; Step 2: take the diffeomorphism, $f$, from the chart range open ball onto the Euclidean space; Step 3: take the chart as $(B_p\subseteq M,f\circ {\varphi }_p)$; Step 4: suppose that $p$ is any boundary point and take any chart around $p$, $(H_p\subseteq M,{\varphi }_p)$, with $H_p$ as a chart half ball; Step 5: take the diffeomorphism, $g$, from the chart range open half ball onto the half Euclidean space; Step 6: take the chart as $(H_p\subseteq M,g\circ {\varphi }_p)$.

Step 1:

Let us suppose that $p$ is any interior point.

Let us take any chart around $p$, $(B_p\subseteq M,{\varphi }_p)$, with $B_p$ as a chart ball, which is possible, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball. ${\varphi }_p(B_p)\subseteq {\mathbb{R}}^d$ is an open ball around ${\varphi }_p(p)$.

Step 2:

Let us take the diffeomorphism, $f:{\varphi }_p(B_p)\to {\mathbb{R}}^d$, which is possible, by the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold, any open ball is diffeomorphic to the whole space.

Step 3:

Let us take the chart, $(B_p\subseteq M,f\circ {\varphi }_p)$, which is indeed a chart, because $f\circ {\varphi }_p(B_p)={\mathbb{R}}^d$ is an open subset of ${\mathbb{R}}^d$; $f\circ {\varphi }_p:B_p\to {\mathbb{R}}^d$ is homeomorphic; and the charts transition, $f\circ {\varphi }_p\circ {{\varphi }_p}^{-1}:{\varphi }_p(B_p)\to {\mathbb{R}}^d$ is $f$, which is diffeomorphic.

Step 4:

Let us suppose that $p$ is any boundary point.

Let us take any chart around $p$, $(H_p\subseteq M,{\varphi }_p)$, with $H_p$ as a chart half ball, which is possible, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball. ${\varphi }_p(H_p)\subseteq {\mathbb{H}}^d$ is an open half ball around ${\varphi }_p(p)$.

Step 5:

Let us take the diffeomorphism, $g:{\varphi }_p(H_p)\to {\mathbb{H}}^d$, which is possible, by the proposition that for any half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary, any open half ball is diffeomorphic to the whole space.

Step 6:

Let us take the chart, $(H_p\subseteq M,g\circ {\varphi }_p)$, which is indeed a chart, because $g\circ {\varphi }_p(H_p)={\mathbb{H}}^d$ is an open subset of ${\mathbb{H}}^d$; $g\circ {\varphi }_p:H_p\to {\mathbb{H}}^d$ is homeomorphic; and the charts transition, $g\circ {\varphi }_p\circ {{\varphi }_p}^{-1}:{\varphi }_p(H_p)\to {\mathbb{H}}^d$ is $g$, which is diffeomorphic.

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References

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