description/proof of that for \(C^\infty\) manifold with boundary, interior point has \(r'\)-\(r\)-open-balls charts pair and boundary point has \(r'\)-\(r\)-open-half-balls charts pair
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(r'\)-\(r\)-open-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(r'\)-\(r\)-open-half-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) 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 \(r\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\).
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(m\): \(\in M\)
\(r'\): \(\in \{r'' \in \mathbb{R} \vert 0 \lt r''\}\)
\(r\): \(\in \{r'' \in \mathbb{R} \vert 0 \lt r''\}\) such that \(r \lt r'\)
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Statements:
(
\(m \in \{\text{ the interior points of } M\}\)
\(\implies\)
\(\exists ((B_{m, r'} \subseteq M, \phi_m), (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})) \in \{\text{ the } r' \text{ - } r \text{ -open-balls charts pairs around } m \text{ on } M\}\)
)
\(\land\)
(
\(m \in \{\text{ the boundary points of } M\}\)
\(\implies\)
\(\exists ((H_{m, r'} \subseteq M, \phi_m), (H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}})) \in \{\text{ the } r' \text{ - } r \text{ -open-half-balls charts pairs around } m \text{ on } M\}\)
)
//
2: Proof
Whole Strategy: Step 1: suppose that \(m\) is any interior point, and take any \(r'\)-open-ball chart around \(m\), \((B_{m, r'} \subseteq M, \phi_m)\); Step 2: take \(B_{\phi_m (m), r} \subset B_{\phi_m (m), r'}\) and define \(B_{m, r} := \phi_m^{-1} (B_{\phi_m (m), r})\); Step 3: see that \((B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\) is an \(r\)-open-ball chart around \(m\); Step 4: suppose that \(m\) is any boundary point, and take any \(r'\)-open-half-ball chart around \(m\), \((H_{m, r'} \subseteq M, \phi_m)\); Step 5: take \(H_{\phi_m (m), r} \subset H_{\phi_m (m), r'}\) and define \(H_{m, r} := \phi_m^{-1} (H_{\phi_m (m), r})\); Step 6: see that \((H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}})\) is an \(r\)-open-half-ball chart around \(m\).
Step 1:
Let us suppose that \(m\) is any interior point.
Let us take any \(r'\)-open-ball chart around \(m\), \((B_{m, r'} \subseteq M, \phi_m)\), which is possible, by the proposition that for any \(C^\infty\) 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 \(r\).
Step 2:
Let us take \(B_{\phi_m (m), r} \subset B_{\phi_m (m), r'}\).
Let us define \(B_{m, r} := \phi_m^{-1} (B_{\phi_m (m), r}) \subset B_{m, r'}\).
\(B_{m, r}\) is an open neighborhood of \(m\) on \(B_{m, r}\) and on \(M\).
Step 3:
\((B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\) is a chart, because \(\phi_m \vert_{B_{m, r}}: B_{m, r} \to B_{\phi_m (m), r}\) is a homeomorphism and \((B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\) is \(C^\infty\) compatible with larger \((B_{m, r'} \subseteq M, \phi_m)\).
So, \((B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\) is an \(r\)-open-ball chart around \(m\).
So, \(((B_{m, r'} \subseteq M, \phi_m), (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}}))\) is an \(r'\)-\(r\)-open-balls charts pair around \(m\).
Step 4:
Let us suppose that \(m\) is any boundary point.
Let us take any \(r'\)-open-half-ball chart around \(m\), \((H_{m, r'} \subseteq M, \phi_m)\), which is possible, by the proposition that for any \(C^\infty\) 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 \(r\).
Step 5:
Let us take \(H_{\phi_m (m), r} \subset H_{\phi_m (m), r'}\).
Let us define \(H_{m, r} := \phi_m^{-1} (H_{\phi_m (m), r}) \subset H_{m, r'}\).
\(H_{m, r}\) is an open neighborhood of \(m\) on \(H_{m, r}\) and on \(M\).
Step 6:
\((H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}})\) is a chart, because \(\phi_m \vert_{H_{m, r}}: H_{m, r} \to H_{\phi_m (m), r}\) is a homeomorphism and \((H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}})\) is \(C^\infty\) compatible with larger \((H_{m, r'} \subseteq M, \phi_m)\).
So, \((H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}})\) is an \(r\)-open-half-ball chart around \(m\).
So, \(((H_{m, r'} \subseteq M, \phi_m), (H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}}))\) is an \(r'\)-\(r\)-open-half-balls charts pair around \(m\).
