1165: C∞ Vectors Subbundle of Rank k of C∞ Vectors Bundle of Rank k′

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definition of $C^{\mathrm{\infty }}$ vectors subbundle of rank $k$ of $C^{\mathrm{\infty }}$ vectors bundle of rank $k^{\prime }$

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ vectors subbundle of rank $k$ of $C^{\mathrm{\infty }}$ vectors bundle of rank $k^{\prime }$.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$(E^{\prime },M,{\pi }^{\prime })$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }{k}^{\prime }\}$
$E$: $\subseteq E^{\prime }$, with the subspace topology and any atlas that makes $E$ an embedded submanifold with boundary of $E^{\prime }$
$\pi$: $={\pi }^{\prime }{|}_E:E\to M$
$k$: $\in \mathbb{N}\setminus \{0\}$ such that $k\le k^{\prime }$
$\ast (E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
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Conditions:
$\mathrm{\forall }m\in M({\pi }^{-1}(m)\in \{\text{ the }k\text{ -dimensional vectors subspaces of }{\pi }^{\prime -1}(m)\})$
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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 $E^{\prime }$: it is saying that $E$ is need to admit such an atlas in order for $(E,T,\pi )$ to be called "$C^{\mathrm{\infty }}$ vectors subbundle of $(E^{\prime },T,{\pi }^{\prime })$".

This definition is not claiming that any $(E,M,\pi )$ that satisfies Conditions constitutes a $C^{\mathrm{\infty }}$ vectors bundle; it is saying that if $(E,M,\pi )$ constitutes a $C^{\mathrm{\infty }}$ vectors bundle with Conditions satisfied, it is called "$C^{\mathrm{\infty }}$ vectors subbundle of $(E^{\prime },M,{\pi }^{\prime })$"

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References

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