definition of \(r\)-open-ball chart around point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of chart on \(C^\infty\) manifold with boundary.
Target Context
- The reader will have a definition of \(r\)-open-ball chart around point on \(C^\infty\) manifold with boundary.
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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( m\): \(\in M\)
\( r\): \(\in \{r' \in \mathbb{R} \vert 0 \lt r'\}\)
\(*(B_{m, r} \subseteq M, \phi_m)\): \(\in \{\text{ the charts around } m \text{ on } M\}\)
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Conditions:
\(\phi_m (B_{m, r}) = B_{\phi_m (m), r} \subseteq \mathbb{R}^d\) where \(B_{\phi_m (m), r}\) is the open ball centered at \(\phi_m (m)\) 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 \(C^\infty\) 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 \(m \in M\), \(B_{m, r} := \{m' \in M \vert dist (m', m) \lt r\}\) 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.
