702: r-Open-Ball Chart Around Point on C∞ Manifold with Boundary

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definition of $r$-open-ball chart around point on $C^{\mathrm{\infty }}$ manifold 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: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of chart on $C^{\mathrm{\infty }}$ manifold with boundary.

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Target Context

• The reader will have a definition of $r$-open-ball chart around point on $C^{\mathrm{\infty }}$ manifold 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: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$m$: $\in M$
$r$: $\in \{r^{\prime }\in \mathbb{R}|0<r^{\prime }\}$
$\ast (B_{m,r}\subseteq M,{\varphi }_m)$: $\in \{\text{ the charts around }m\text{ on }M\}$
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Conditions:
${\varphi }_m(B_{m,r})=B_{{\varphi }_m(m),r}\subseteq {\mathbb{R}}^d$ where $B_{{\varphi }_m(m),r}$ is the open ball centered at ${\varphi }_m(m)$ with the radius $r$
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2: Note

There is no $r$-open-ball chart around $m$ when $m$ is a boundary point.

There is always an $r$-open-ball chart around $m$ for any positive $r$ when $m$ is an interior point, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has an $r$-open-ball chart and each boundary point has an $r$-open-half-ball chart for any positive $r$.

This concept is different from 'open ball around point on metric space': on any metric space, $M$, around each $m\in M$, $B_{m,r}:=\{m^{\prime }\in M|dist(m^{\prime },m)<r\}$ is always an open ball, which is not necessarily homeomorphic to any open ball on ${\mathbb{R}}^d$: even when $m$ is a boundary point, $B_{m,r}$ is well-defined.

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References

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