definition of chart ball around point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of chart on \(C^\infty\) manifold with boundary.
Target Context
- The reader will have a definition of chart ball 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( p\): \(\in M\)
\(*B_p\): \(\in \{\text{ the open neighborhoods of } p \text{ on } M\}\)
//
Conditions:
\(\exists (B_p \subseteq M, \phi_p) \in \{\text{ the charts on } M\} \)
\(\land\)
\(\phi_p (B_p) \in \{\text{ the open balls around } \phi_p (p) \text{ on } \mathbb{R}^d\}\)
//
2: Natural Language Description
For any \(d\)-dimensional \(C^\infty\) manifold with boundary, \(M\), and any point, \(p \in M\), any open neighborhood of \(p\), \(B_p \subseteq M\), such that there is a chart, \((B_p \subseteq M, \phi_p)\), and \(\phi_p (B_p) \subseteq \mathbb{R}^d\) is an open ball around \(\phi_p (p)\)
3: Note
There is no chart ball around \(p\) when \(p\) is a boundary point.
There is always a chart ball around \(p\) when \(p\) is an interior point, by the proposition that for any \(C^\infty\) manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball.
This concept is different from 'open ball around point on metric space': on any metric space, \(M\), around each \(p \in M\), \(B_{p, \epsilon} := \{p' \in M \vert dist (p', p) \lt \epsilon\}\) is always an open ball, which is not necessarily homeomorphic to any open ball on \(\mathbb{R}^d\).
It can be called also "chart open ball around \(p\) on \(M\)", but of course, each chart domain is known to be an open subset.
