description/proof of that for map from subset of \(C^\infty\) manifold with boundary into subset of \(C^\infty\) manifold \(C^k\) at point, there is \(C^k\) extension on-open-neighborhood-of-point domain
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
- 4: Proof
Starting Context
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits 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.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition.
Target Context
- The reader will have a description and a proof of the proposition that for any map from any subset of any \(C^\infty\) manifold with boundary into any subset of any \(C^\infty\) manifold \(C^k\) at any point, there is a \(C^k\) extension on an open-neighborhood-of-the-point domain.
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:
\(M_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } C^\infty \text{ manifolds }\}\)
\(S_1\): \(\subseteq M_1\)
\(S_2\): \(\subseteq M_2\)
\(p\): \(\in S_1\)
\(f\): \(: S_1 \to S_2\), \(\in \{\text{ the } C^k \text{ maps at } p\}\), where \(k \in (\mathbb{N} \setminus \{0\}) \cup \{\infty\}\)
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Statements:
\(\exists U_p \in \{\text{ the open neighborhoods of } p \text{ on } M_1\}, \exists U_{f (p)} \in \{\text{ the open neighborhoods of } f (p) \text{ on } M_2\}, \exists \tilde{f}: U_p \to U_{f (p)} \in \{\text{ the } C^k \text{ maps }\} (\tilde{f} \vert_{U_p \cap S_1} = f \vert_{U_p \cap S_1})\)
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2: Natural Language Description
For any \(d_1\)-dimensional \(C^\infty\) manifold with boundary, \(M_1\), any \(d_2\)-dimensional \(C^\infty\) manifold, \(M_2\), any subset, \(S_1 \subseteq M_1\), any subset, \(S_2 \subseteq M_2\), any point, \(p \in S_1\), and any map \(C^k\) at \(p\), where \(k \in (\mathbb{N} \setminus \{0\}) \cup \{\infty\}\), \(f: S_1 \to S_2\), there are an open neighborhood of \(p\), \(U_p \subseteq M_1\), an open neighborhood of \(f (p)\), \(U_{f (p)} \subseteq M_2\), and a \(C^k\) map, \(\tilde{f}: U_p \to U_{f (p)}\), such that \(\tilde{f} \vert_{U_p \cap S_1} = f \vert_{U_p \cap S_1}\).
3: Note
\(M_2\) is without boundary, which is necessary for this proposition, which is the reason why the conclusion of this proposition cannot be adopted as a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point.
4: Proof
Whole Strategy: while the definition of \(C^k\)-ness at point is about having a chart on \(M_1\) and a chart on \(M_2\) and a \(C^k\) extension of the coordinates function, \(f'\), \(\tilde{f}\) is constructed based on \(f'\), but for that, the charts need to be chosen specifically; Step 1: take a chart around \(p\), \((U'_p \subseteq M_1, \phi'_p)\), a chart around \(f (p)\), \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), and a \(C^k\) extension of the coordinates function, \(f': U_{\phi'_p (p)} \to \mathbb{R}^{d_2}\), according to a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), with the conditions that \(\phi_{f (p)} (U_{f (p)}) = \mathbb{R}^{d_2}\) and \(\phi'_p (U'_p) \subseteq U_{\phi'_p (p)}\); Step 2: define \(U_p := U'_p\) and \(\tilde{f}: U_p \to U_{f (p)} := {\phi_{f (p)}}^{-1} \circ f' \circ \phi'_p\); Step 3: see that \(\tilde{f}\) satisfies the necessary conditions.
Step 1:
There is a chart, \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), such that \(\phi_{f (p)} (U_{f (p)}) = \mathbb{R}^{d_2}\), by 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.
While \(U_{f (p)} \cap S_2\) is an open neighborhood of \(f (p)\) on \(S_2\), as \(f\) is continuous at \(p\) as is described in Note for the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), there is an open neighborhood of \(p\), \(U''_p \cap S_1 \subseteq S_1\), such that \(f (U''_p \cap S_1) \subseteq U_{f (p)} \cap S_2 \subseteq U_{f (p)}\). There is a chart, \((U'_p \subseteq M_1, \phi'_p)\), such that \(U'_p \subseteq U''_p\). \(f (U'_p \cap S_1) \subseteq f (U''_p \cap S_1) \subseteq U_{f (p)}\).
By the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition, the pair of \((U'_p \subseteq M_1, \phi'_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\) satisfies the condition of the definition.
Furthermore, let us take \(U'_p\) as such that \(\phi'_p (U'_p) \subseteq U_{\phi'_p (p)}\) as is described in the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
Now we have a \(C^k\) extension of the coordinates function, \(f': U_{\phi'_p (p)} \to \mathbb{R}^{d_2}\).
Step 2:
Let us define \(U_p := U'_p\) and \(\tilde{f}: U_p \to U_{f (p)} := {\phi_{f (p)}}^{-1} \circ f' \circ \phi'_p\), which is valid, because the domain of \(\phi'_p\) is \(U_p\), the codomain of \(\phi'_p\), \(\phi'_p (U'_p)\), is contained in the domain of \(f'\), \(U_{\phi'_p (p)}\), the codomain of \(f'\), \(\mathbb{R}^{d_2}\), is contained in the domain of \({\phi_{f (p)}}^{-1}\), \(\mathbb{R}^{d_2}\), and the codomain of \({\phi_{f (p)}}^{-1}\) is \(U_{f (p)}\).
Step 3:
Let us see that \(\tilde{f}\) is a \(C^k\) map.
For each \(p' \in U'_p\), let us take the charts, \((U'_p \subseteq M_1, \phi'_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\).
\(\tilde{f} (U'_p \cap U'_p) \subseteq U_{f (p)}\).
\(\phi_{f (p)} \circ \tilde{f} \circ {\phi'_p}^{-1}: \phi'_p (U'_p \cap U'_p) \to \mathbb{R}^{d_2}\) is \(= \phi_{f (p)} \circ {\phi_{f (p)}}^{-1} \circ f' \circ \phi'_p \circ {\phi'_p}^{-1} = f'\), which is \(C^k\).
Let us see that \(\tilde{f} \vert_{U_p \cap S_1} = f \vert_{U_p \cap S_1}\).
\(\tilde{f} \vert_{U_p \cap S_1} = {\phi_{f (p)}}^{-1} \circ f' \circ \phi'_p \vert_{U_p \cap S_1} = {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U_p \cap S_1)} \circ \phi'_p \vert_{U_p \cap S_1}\), because \(f'\) is chosen to satisfy \(f'\vert_{\phi'_p (U_p \cap S_1)} = \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U_p \cap S_1)}\), so, \(= f \vert_{U_p \cap S_1}\).
