description/proof of that for \(C^\infty\) vectors bundle, chart open subset on base space is not necessarily trivializing open subset (probably)
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 (Not Complete)
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, a chart open subset on the base space is not necessarily any trivializing open subset (probably).
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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles }\}\)
\(U\): \(\in \{\text{ the chart domains of } M\}\)
//
Statements:
Not necessarily \(U \in \{\text{ the trivializing open subsets of } M\}\) (probably)
//
2: Natural Language Description
For any \(C^\infty\) vectors bundle, \((E, M, \pi)\), and any chart domain, \(U \subseteq M\), \(U\) is not necessarily any trivializing open subset (probably).
3: Note
The author is honestly not sure whether \(U\) needs to be a trivializing open subset of \(M\) or not.
The reason why we need this proposition even in this unsure state is that at least we need a reminder for not assuming what we have not proved.
4: Proof (Not Complete)
I know that I need to show a counter example, but I have not managed to.
At least, the definition of \(C^\infty\) vectors bundle does not directly require that every chart open subset on the base space is a trivializing open subset, and it seems unlikely that that every chart open subset on the base space is implied to be a trivializing open subset, at 1st glance.
For any tangent vectors bundle case, every chart open subset on the base space is a trivializing open subset (the atlas of the tangent vectors bundle is defined to make it so), but that is not so-immediately generalized to any general \(C^\infty\) vectors bundle case.
