A description/proof of that for 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
- 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 vectors bundle, a chart open subset on the base space is not necessarily any trivializing open subset.
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, \(\pi: E \rightarrow M\), and any chart open subset, \(U\), \(U\) is not necessarily any trivializing open subset (probably). "probably" means that at least now I do not know any proof that every chart open subset is a trivializing open subset, so, I cannot assume so.
2: 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.
