definition of \(C^\infty\) vectors subbundle of rank \(k\) of \(C^\infty\) vectors bundle of rank \(k'\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
Target Context
- The reader will have a definition of \(C^\infty\) vectors subbundle of rank \(k\) of \(C^\infty\) vectors bundle of rank \(k'\).
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:
\( (E', M, \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k'\}\)
\( E\): \(\subseteq E'\), with the subspace topology and any atlas that makes \(E\) an embedded submanifold with boundary of \(E'\)
\( \pi\): \(= \pi' \vert_E: E \to M\)
\( k\): \(\in \mathbb{N} \setminus \{0\}\) such that \(k \le k'\)
\(*(E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
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Conditions:
\(\forall m \in M (\pi^{-1} (m) \in \{\text{ the } k \text{ -dimensional vectors subspaces of } \pi'^{-1} (m)\})\)
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2: Note
This definition is not claiming that any \(E\) admits an atlas that makes \(E\) an embedded submanifold with boundary of \(E'\): it is saying that \(E\) is need to admit such an atlas in order for \((E, T, \pi)\) to be called "\(C^\infty\) vectors subbundle of \((E', T, \pi')\)".
This definition is not claiming that any \((E, M, \pi)\) that satisfies Conditions constitutes a \(C^\infty\) vectors bundle; it is saying that if \((E, M, \pi)\) constitutes a \(C^\infty\) vectors bundle with Conditions satisfied, it is called "\(C^\infty\) vectors subbundle of \((E', M, \pi')\)"
