description/proof of that for \(C^\infty\) vectors bundle and \(C^\infty\) local frame over open subset, around each point of open subset, there is possibly smaller chart for bundle that takes components w.r.t. frame
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader knows a definition of local \(C^\infty\) frame on \(C^\infty\) vectors bundle.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset.
- The reader admits the proposition that for any vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
- The reader admits the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle and any \(C^\infty\) local frame over any open subset, around each point of the open subset, there is a possibly smaller chart for the bundle that takes the components with respect to the frame.
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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\(U\): \(\in \{\text{ the open subsets of } M\}\)
\(\{s_1, ..., s_k\}\): \(\in \{\text{ the } C^\infty \text{ local frames over } U\}\)
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Statements:
\(\forall u \in U (\exists (U_u \subseteq M, \phi_u) \in \{\text{ the charts }\} \text{ such that } U_u \subseteq U ((\pi^{-1} (U_u) \subseteq E, \widetilde{\phi_u}) \in \{\text{ the charts }\} \text{ where } \widetilde{\phi_u}: \pi^{-1} (U_u) \to \mathbb{R}^k \times \phi_u (U_u), b^j s_j (p) \mapsto (b^1, ..., b^k, \phi_u \circ \pi (p))))\)
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2: Proof
Whole Strategy: Step 1: see that \(U\) is a trivializing open subset; Step 2: around each \(u \in U\), take any chart, \((U_u \subseteq M, \phi_u)\), such that \(U_u \subseteq U\), and see that \(U_u\) is a chart trivializing open subset: Step 3: take the trivialization over \(U_u\) cited in the proof of the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset; Step 4: take the chart canonically induced by the trivialization by the proposition that for any vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
Step 1:
\(U\) is a trivializing open subset, by the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset.
Step 2:
Around each \(u \in U\), take any chart, \((U_u \subseteq M, \phi_u)\), such that \(U_u \subseteq U\).
\(U_u\) is a trivializing open subset, by the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
So, \(U_u\) is a chart trivializing open subset.
Step 3:
Let us take the trivialization over \(U_u\) cited in the proof of the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset, \(\Phi: \pi^{-1} (U_u) \to U_u \times \mathbb{R}^k\), as \(b^j s_j (p) \mapsto (p, b^1, b^2, ..., b^k)\).
Step 4:
Let us take the chart canonically induced by the trivialization by the proposition that for any vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map, \(\widetilde{\phi_u}\): \(: \pi^{-1} (U_u) \to U_u \times \mathbb{R}^k \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}, v \mapsto (\pi_2 (\Phi_u (v)), \phi_u (\pi (v)))\), which is nothing but the chart claimed by this proposition.
