A description/proof of that for 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
- The reader knows a definition of \(C^\infty\) vectors bundle.
- The reader admits the proposition that for any \(C^\infty\) manifold, any map from any open subset of the manifold onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold is a chart map if and only if it is a diffeomorphism.
Target Context
- The reader will have a description and a proof of the proposition that for any 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: Description
For any \(C^\infty\) manifold, \(M\), any vectors bundle, \(\pi: E \to M\), and any chart trivializing open subset, \(U\), where \((U, \phi)\) is the chart, the trivialization, \(\phi': \pi^{-1} (U) \to U \times \mathbb{R}^{d'}\), induces the canonical chart map, \(\phi'' = (\phi, id) \circ \phi': \pi^{-1} (U) \to U \times \mathbb{R}^{d'} \to \phi (U) \times \mathbb{R}^{d'}\), \(v \mapsto (\phi (\pi (v)), \lambda (\phi' (v)))\), where \(id\) is the identity map and \(\lambda\) is the projection from \((U \times \mathbb{R}^{d'})\) to \(\mathbb{R}^{d'}\).
2: Proof
\((\phi, id): U \times \mathbb{R}^{d'} \to \phi (U) \times \mathbb{R}^{d'}\) is obviously homeomorphic. It is obviously diffeomorphic.
\(\phi'' = (\phi, id) \circ \phi'\) is diffeomorphic as a composition of diffeomorphisms.
\(\phi (U) \subseteq \mathbb{R}^d\) and \(\phi (U) \times \mathbb{R}^{d'} \subseteq \mathbb{R}^d \times \mathbb{R}^{d'} = \mathbb{R}^{d + d'}\), open on \(\mathbb{R}^{d + d'}\). As \(\phi'': \pi^{-1} (U) \to \phi (U) \times \mathbb{R}^{d'}\) is a diffeomorphism, \((\pi^{-1} (U), \phi'')\) is a chart on \(E\), by the proposition that for any \(C^\infty\) manifold, any map from any open subset of the manifold onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold is a chart map if and only if it is a diffeomorphism.
3: Note
\(U\) 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 vectors bundle, there is a chart trivializing open cover, the chart trivializing open cover can be used to have a chart over any point, \(p \in M\).
