description/proof of that for \(C^\infty\) 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
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: Proof
Starting Context
- The reader admits 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.
- The reader admits the proposition that for any Euclidean \(C^\infty\) manifold, any open ball is diffeomorphic to the whole space.
- The reader admits the proposition that for any half Euclidean \(C^\infty\) manifold with boundary, any open half ball is diffeomorphic to the whole space.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) 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.
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:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds }\}\)
\(p\): \(\in M\)
//
Statements:
(
\(p \in \{\text{ the interior points of } M\}\)
\(\implies\)
\(\exists (B_p \subseteq M, \phi_p) \in \{\text{ the charts around } p \text{ on } M\} (\phi_p (B_p) = \mathbb{R}^d)\)
)
\(\land\)
(
\(p \in \{\text{ the boundary points of } M\}\)
\(\implies\)
\(\exists (H_p \subseteq M, \phi_p) \in \{\text{ the charts around } p \text{ on } M\} (\phi_p (H_p) = \mathbb{H}^d)\)
)
//
2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M\), and any \(p \in M\), if \(p\) is any interior point of \(M\), there is a chart around \(p\), \((B_p \subseteq M, \phi_p)\), such that \(\phi_p (B_p) = \mathbb{R}^d\), and if \(p\) is any boundary point of \(M\), there is a chart around \(p\), \((H_p \subseteq M, \phi_p)\), such that \(\phi_p (H_p) = \mathbb{H}^d\).
3: Proof
Whole Strategy: Step 1: suppose that \(p\) is any interior point and take any chart around \(p\), \((B_p \subseteq M, \phi_p)\), 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 \((B_p \subseteq M, f \circ \phi_p)\); Step 4: suppose that \(p\) is any boundary point and take any chart around \(p\), \((H_p \subseteq M, \phi_p)\), 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 \((H_p \subseteq M, g \circ \phi_p)\).
Step 1:
Let us suppose that \(p\) is any interior point.
Let us take any chart around \(p\), \((B_p \subseteq M, \phi_p)\), with \(B_p\) as a chart ball, which is possible, 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. \(\phi_p (B_p) \subseteq \mathbb{R}^d\) is an open ball around \(\phi_p (p)\).
Step 2:
Let us take the diffeomorphism, \(f: \phi_p (B_p) \to \mathbb{R}^d\), which is possible, by the proposition that for any Euclidean \(C^\infty\) manifold, any open ball is diffeomorphic to the whole space.
Step 3:
Let us take the chart, \((B_p \subseteq M, f \circ \phi_p)\), which is indeed a chart, because \(f \circ \phi_p (B_p) = \mathbb{R}^d\) is an open subset of \(\mathbb{R}^d\); \(f \circ \phi_p: B_p \to \mathbb{R}^d\) is homeomorphic; and the charts transition, \(f \circ \phi_p \circ {\phi_p}^{-1}: \phi_p (B_p) \to \mathbb{R}^d\) is \(f\), which is diffeomorphic.
Step 4:
Let us suppose that \(p\) is any boundary point.
Let us take any chart around \(p\), \((H_p \subseteq M, \phi_p)\), with \(H_p\) as a chart half ball, which is possible, 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. \(\phi_p (H_p) \subseteq \mathbb{H}^d\) is an open half ball around \(\phi_p (p)\).
Step 5:
Let us take the diffeomorphism, \(g: \phi_p (H_p) \to \mathbb{H}^d\), which is possible, by the proposition that for any half Euclidean \(C^\infty\) manifold with boundary, any open half ball is diffeomorphic to the whole space.
Step 6:
Let us take the chart, \((H_p \subseteq M, g \circ \phi_p)\), which is indeed a chart, because \(g \circ \phi_p (H_p) = \mathbb{H}^d\) is an open subset of \(\mathbb{H}^d\); \(g \circ \phi_p: H_p \to \mathbb{H}^d\) is homeomorphic; and the charts transition, \(g \circ \phi_p \circ {\phi_p}^{-1}: \phi_p (H_p) \to \mathbb{H}^d\) is \(g\), which is diffeomorphic.
