description/proof of that for \(C^\infty\) vectors bundle, trivialization of chart trivializing open subset induces canonical chart map
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Note 1
- 3: Proof
- 4: Note 2
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism.
- The reader admits the proposition that the \(d\)-dimensional Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces whose (the product's) dimension equals \(d\).
- The reader admits the proposition that the \(d\)-dimensional closed upper half Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces and a closed upper half Euclidean space whose (the product's) dimension equals \(d\).
- The reader admits 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.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\((U \subseteq M, \phi)\): \(\in \{\text{ the charts of } M\}\), such that \(U \in \{\text{ the trivializing open subsets }\}\)
\(\Phi\): \(: \pi^{-1} (U) \to U \times \mathbb{R}^k\), \(\in \{\text{ the trivializations }\}\)
\(\pi_2\): \(: (U \times \mathbb{R}^k) \to \mathbb{R}^k\), \(= \text{ the projection }\)
\(\widetilde{\phi}\): \(: \pi^{-1} (U) \to U \times \mathbb{R}^k \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}, v \mapsto (\pi_2 (\Phi (v)), \phi (\pi (v)))\)
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Statements:
\((\pi^{-1} (U) \subseteq E, \widetilde{\phi}) \in \{\text{ the charts of } E\}\)
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2: Note 1
When the boundary of \(M\) is empty, \(\widetilde{\phi}\) is usually chosen to be such that \(v \mapsto (\phi (\pi (v)), \pi_2 (\Phi (v)))\) because \((\phi (\pi (v)), \pi_2 (\Phi (v))) \in \mathbb{R}^{d + k}\) anyway, but when the boundary of \(M\) is not empty, \((\phi (\pi (v)), \pi_2 (\Phi (v))) \notin \mathbb{H}^{d + k}\) in general, so, we have chosen \(\widetilde{\phi}\) as above, which is fine also for when the boundary of \(M\) is empty.
3: Proof
Whole Strategy: apply the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism; Step 1: see that \(\widetilde{\phi} (\pi^{-1} (U)) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is open; Step 2: see that \(\widetilde{\phi}\) is a diffeomorphism; Step 3: conclude the proposition.
Step 1:
Let us see that \(\widetilde{\phi} (\pi^{-1} (U)) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is open.
\(\widetilde{\phi} (\pi^{-1} (U)) = \mathbb{R}^k \times \phi (U)\).
\(\mathbb{R}^{d + k} \cong \mathbb{R}^k \times \mathbb{R}^d\) or \(\mathbb{H}^{d + k} \cong \mathbb{R}^k \times \mathbb{H}^d\), where \(\cong\) means being homeomorphic, by the proposition that the \(d\)-dimensional Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces whose (the product's) dimension equals \(d\) or the proposition that the \(d\)-dimensional closed upper half Euclidean topological space is homeomorphic to the product of any combination of some lower-dimensional Euclidean spaces and a closed upper half Euclidean space whose (the product's) dimension equals \(d\).
\(\mathbb{R}^k \times \phi (U) \subseteq \mathbb{R}^k \times \mathbb{R}^d \text{ or } \mathbb{R}^k \times \mathbb{H}^d\) is an open subset, by the definition of product topology, because \(\phi (U) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\) is an open subset.
So, \(\widetilde{\phi} (\pi^{-1} (U)) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is open.
Step 2:
Let us see that \(\widetilde{\phi}\) is diffeomorphic.
\(\widetilde{\phi} = \lambda \circ (\phi, id) \circ \Phi\), where \(\lambda: \mathbb{R}^{d + k} \to \mathbb{R}^{d + k}, (r^1, ..., r^d, r^{d + 1}, ..., r^{d + k}) \mapsto (r^{d + 1}, ..., r^{d + k}, r^1, ..., r^d)\).
\(\Phi: \pi^{-1} (U) \to U \times \mathbb{R}^k\) is diffeomorphic; \((\phi, id): U \times \mathbb{R}^k \to \phi (U) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k\) is obviously diffeomorphic; and \(\lambda \vert_{\phi (U) \times \mathbb{R}^k}: \phi (U) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k \to \mathbb{R}^k \times \phi (U) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is obviously diffeomorphic.
So, \(\widetilde{\phi}\) is diffeomorphic, 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.
As \(\widetilde{\phi}\) is a diffeomorphism, \((\pi^{-1} (U), \widetilde{\phi})\) is a chart of \(E\), by the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism.
4: Note 2
\(U\) has to be a chart trivializing open subset, not just a trivializing open subset, because otherwise, \(\phi\) would not exist.
By the proposition that for any \(C^\infty\) vectors bundle, there is a chart trivializing open cover, the chart trivializing open cover can be used to have a chart over any point, \(p \in M\).
