description/proof of that for \(C^\infty\) vectors bundle and section from subset of base space \(C^k\) at point where \(0 \lt k\), 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
- The reader knows a definition of \(C^\infty\) vectors bundle.
- The reader knows a definition of section of continuous surjection.
- 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 \(C^\infty\) vectors bundle, there is a chart trivializing open cover.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle and any section from any subset of the base space \(C^k\) at any point where \(0 \lt k\), there is a \(C^k\) extension on an open-neighborhood-of-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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\(S\): \(\subseteq M\)
\(p\): \(\in S\)
\(s\): \(: S \to \pi^{-1} (S) \subseteq E\), \(\in \{\text{ the sections of } \pi \vert_{\pi^{-1} (S)}\}\), \(C^l\) at \(p\), where \(0 \lt l\)
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Statements:
\(\exists V_p \subseteq M \in \{\text{ the open neighborhoods of } p\}, \exists s': V_p \to \pi^{-1} (V_p) \in \{\text{ the sections of } \pi \vert_{\pi^{-1} (V_p)}\} (s \vert_{S \cap V_p} = s' \vert_{S \cap V_p} \land s' \in \{\text{ the } C^l \text{ maps }\})\)
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2: Note
Compare with 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, which requires the codomain to be a subset of a \(C^\infty\) manifold without boundary. An issue because of which that proposition cannot be directly applied is that \(E\) may be with a nonempty boundary (when \(M\) is with a nonempty boundary), and another issue is that the extension is required to be a section, which that proposition does not guarantee. But the basic idea of this proposition is the same with that of that proposition: the rough reason for this proposition is that while \(E\) is locally \(U_p \times \mathbb{R}^k\), the boundary can exist only in the \(U_p\) part, but the extension needs to be the identity map with respect to the \(U_p\) part (because it is a section), and the concern is really about only the \(\mathbb{R}^k\) part, which has no boundary.
3: Proof
Whole Strategy: Step 1: take a chart trivializing subset of \(M\) around \(p\), with the chart, \((U_p \subseteq M, \phi_p)\), and the induced chart, \((\pi^{-1} (U_p) \subseteq E, \widetilde{\phi_p})\); Step 2: for the components function, \(f := \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)}: \phi_p (U_p \cap S) \to \widetilde{\phi_p} (\pi^{-1} (U_p))\), take an open neighborhood of \(\phi_p (p)\), \(U_{\phi_p (p)} \subseteq \mathbb{R}^d\), and an extension of \(f\), \(f': U_{\phi_p (p)} \to \mathbb{R}^{d + k}\); Step 3: tweak \(f'\) to have \(f'': U_{\phi_p (p)} \to \mathbb{R}^k \times \phi_p (U_p)\); Step 4: take \(V_p := {\phi_p}^{-1} (\phi_p (U_p) \cap U_{\phi_p (p)})\) and \(s' := {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p} \to \pi^{-1} (V_p)\); Step 5: see that \(s'\) satisfies the requirements.
Step 1:
Let us take a chart trivializing subset of \(M\) around \(p\), with the chart, \((U_p \subseteq M, \phi_p)\), which is possible by the proposition that for any \(C^\infty\) vectors bundle, there is a chart trivializing open cover.
Let us take the induced chart, \((\pi^{-1} (U_p) \subseteq E, \widetilde{\phi_p})\), which is possible by the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
\(s (U_p) \subseteq \pi^{-1} (U_p)\), because \(s\) is a section.
Step 2:
Let the components function be \(f := \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)}: \phi_p (U_p \cap S) \to \widetilde{\phi_p} (\pi^{-1} (U_p))\).
There is an open neighborhood of \(\phi_p (p)\), \(U_{\phi_p (p)} \subseteq \mathbb{R}^d\), and a \(C^l\) extension of \(f\), \(f': U_{\phi_p (p)} \to \mathbb{R}^{d + k}\), because that is what the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\) requires.
Step 3:
Denote \(f': (x^1, ..., x^d) \mapsto (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), f'^{k + 1} (x^1, ..., x^d), ..., , f'^{d + k} (x^1, ..., x^d))\). Each \(f'^j (x^1, ..., x^d)\) is \(C^l\).
Let us define \(f'': U_{\phi_p (p)} \to \mathbb{R}^k \times U_{\phi_p (p)}, (x^1, ..., x^d) \mapsto (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), x^1, ..., , x^d)\).
\(f''\) is obviously \(C^l\).
\(f \vert_{\phi_p (U_p \cap S)} = f' \vert_{\phi_p (U_p \cap S)} = f'' \vert_{\phi_p (U_p \cap S)}\), because \(f \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d) = f' \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d) = (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), f'^{k + 1} (x^1, ..., x^d), ..., , f'^{d + k} (x^1, ..., x^d)) = (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), x^1, ..., , x^d) = f'' \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d)\), because \(s\) is fiber-preserving and \(\widetilde{\phi_p}\) is induced from \(\phi_p\).
So, \(f''\) is a \(C^l\) extension of \(f\).
Step 4:
\(\phi_p (U_p) \cap U_{\phi_p (p)} \subseteq \phi_p (U_p)\) is an open neighborhood of \(\phi_p (p)\) on \(\phi_p (U_p)\).
Let us define \(V_p := {\phi_p}^{-1} (\phi_p (U_p) \cap U_{\phi_p (p)}) \subseteq M\) such that \(V_p \subseteq U_p\), which is an open neighborhood of \(p\).
Let us define \(s' := {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p}: V_p \to \pi^{-1} (V_p)\), which is possible because \(f'' \vert_{\phi_p \vert_{V_p} (V_p)}\) is into \(\mathbb{R}^k \times (U_{\phi_p (p)} \cap \phi_p (U_p)) \subseteq \mathbb{R}^k \times \phi_p (U_p) = \widetilde{\phi_p} (\pi^{-1} (U_p))\).
Step 5:
\(s'\) is indeed a section, obviously.
\(s'\) is \(C^l\), because while \(\phi_p \vert_{V_p}\) is \(C^\infty\) as \(: V_p \to \phi_p (V_p) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\), \(f''\) is \(C^l\) as \(: U_{\phi_p (p)} \subseteq \mathbb{R}^d \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\), and \(\widetilde{\phi_p}^{-1}\) is \(C^\infty\) as \(: \mathbb{R}^k \times \phi_p (U_p) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k} \to \pi^{-1} (U_p) \subseteq E\), the only concern for \(s'\) to be a legitimate chain of \(C^l\) maps is that \(\mathbb{R}^d \text{ or } \mathbb{H}^d\) for the codomain of \(\phi_p \vert_{V_p}\) is different from \(\mathbb{R}^d\) for the domain of \(f''\), but \(\phi_p \vert_{V_p}\) can be regarded to be \(\phi_p \vert_{V_p}: V_p \to \phi_p (V_p) \subseteq \mathbb{R}^d\), which obviously does not change \(C^\infty\)-ness, so, \(s'\) is indeed a legitimate chain of \(C^l\) maps, and is \(C^l\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
\(s' \vert_{V_p \cap S} = s \vert_{V_p \cap S}\), because \(s' \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ f \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)} \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = s \vert_{V_p} \vert_{V_p \cap S} = s \vert_{V_p \cap S}\).
