definition of local \(C^\infty\) frame on \(C^\infty\) vectors bundle
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 knows a definition of section of continuous map.
Target Context
- The reader will have a definition of local \(C^\infty\) frame on \(C^\infty\) vectors bundle.
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:
\( (E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\( U\): \(\in \{\text{ the open subsets of } M\}\)
\(*\{s_1, ..., s_k\}\): \(s_j: U \to \pi^{-1} (U) \in \{\text{ the } C^\infty \text{ sections of } \pi \vert_{\pi^{-1} (U)}\}\)
//
Conditions:
\(\forall m \in U (\{s_1 (m), ..., s_k (m)\} \in \{\text{ the bases of } \pi^{-1} (m)\})\)
//
2: Note
Whether the domain or the codomain of \(s_j: U \to \pi^{-1} (U)\) is regarded to be the subset of the ambient \(M\) or \(E\) or is regarded to be the embedded submanifold with boundary of \(M\) or the embedded submanifold with boundary of \(E\) as the restricted \(C^\infty\) vectors bundle does not really matter, by the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^\infty\)-ness does not change when the domain or the codomain is regarded to be the subset.
