description/proof of that for half Euclidean \(C^\infty\) manifold with boundary, open half ball is diffeomorphic to whole 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 knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
- The reader admits the proposition that for any Euclidean \(C^\infty\) manifold, any open ball is diffeomorphic to the whole space.
Target Context
- The reader will have a description and a proof of the proposition that for any half Euclidean \(C^\infty\) manifold with boundary, any open half ball is diffeomorphic to the whole 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:
\(\mathbb{H}^d\): \(= \text{ the Euclidean } C^\infty \text{ manifold with boundary }\)
\(p\): \(\in Bou (\mathbb{H}^d)\), where \(Bou (\mathbb{H}^d)\) denotes the manifold boundary of \(\mathbb{H}^d\)
\(H_{p, \epsilon}\): \(= \text{ the open half ball around } p \text{ on } \mathbb{H}^d\)
\(g\): \(: H_{p, \epsilon} \to \mathbb{H}^d, p' \mapsto (p' - p) / \sqrt{\epsilon^2 - \Vert p' - p \Vert^2}\)
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Statements:
\(H_{p, \epsilon} \cong_g \mathbb{H}^d\), where \(\cong_g\) denotes being diffeomorphic by \(g\)
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2: Natural Language Description
For the half Euclidean \(C^\infty\) manifold with boundary, \(\mathbb{H}^d\), any open half ball around any \(p \in Bou (\mathbb{H}^d)\), where \(Bou (\mathbb{H}^d)\) denotes the manifold boundary of \(\mathbb{H}^d\), \(H_{p, \epsilon} \subseteq \mathbb{H}^d\) is diffeomorphic to \(\mathbb{H}^d\) by \(g: H_{p, \epsilon} \to \mathbb{H}^d, p' \mapsto (p' - p) / \sqrt{\epsilon^2 - \Vert p' - p \Vert^2}\).
3: Proof
Whole Strategy: Step 1: see that \(g\) is a restriction of \(f\) cited in the proposition that for any Euclidean \(C^\infty\) manifold, any open ball is diffeomorphic to the whole space; Step 2: take the identity chart on \(\mathbb{H}^d\), \((\mathbb{H}^d \subseteq \mathbb{H}^d, id)\); Step 3: see that \(id \circ g \circ id^{-1} \vert_{id (H_{p, \epsilon})}: id (H_{p, \epsilon}) \to id (\mathbb{H}^d)\) has the \(C^\infty\) extension, \(f: B_{p, \epsilon} \to \mathbb{R}^d\); Step 4: see that \(id \circ g^{-1} id^{-1}: id (\mathbb{H}^d) \to id (\mathbb{H}^d)\) has the \(C^\infty\) extension, \(f^{-1}: \mathbb{R}^d \to \mathbb{R}^d\).
Step 1:
\(g\) is in fact a restriction of \(f\) cited in the proposition that for any Euclidean \(C^\infty\) manifold, any open ball is diffeomorphic to the whole space.
That implies that \(g\) is injective; the surjectiveness is obvious: while the \(d\)-th component of \(p\) is zero, for each \(p'' \in \mathbb{H}^d\), there is a \(p' \in B_{p, \epsilon}\) such that \(f (p') = p''\), but as the \(d\)-th component of \(p''\) is non-negative, the \(d\)-th component of \(p'\) is non-negative, which means that \(p' \in H_{p, \epsilon}\).
So, there is the inverse, \(g^{-1}: \mathbb{H}^d \to H_{p, \epsilon}\).
Step 2:
Let us take the identity chart, \((\mathbb{H}^d \subseteq \mathbb{H}^d, id)\).
Step 3:
\(C^\infty\)-ness of \(g\) is about whether \(id \circ g \circ id^{-1} \vert_{id (H_{p, \epsilon})}: id (H_{p, \epsilon}) \to id (\mathbb{H}^d)\) has a \(C^\infty\) extension, according to the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\). But \(f: B_{p, \epsilon} \to \mathbb{R}^d\) is indeed such an extension.
So, \(g\) is \(C^\infty\).
Step 4:
\(C^\infty\)-ness of \(g^{-1}\) is about whether \(id \circ g^{-1} \circ id^{-1}: id (\mathbb{H}^d) \to id (\mathbb{H}^d)\) has a \(C^\infty\) extension, according to the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\). But \(f^{-1}: \mathbb{R}^d \to \mathbb{R}^d\) is indeed such an extension: \(f^{-1}\) is really \(f^{-1}: \mathbb{R}^d \to B_{p, \epsilon}\), but the codomain can be expanded without any harm.
So, \(g^{-1}\) is \(C^\infty\).
