1129: Map-Related Vectors Fields Pair for C∞ Map Between C∞ Manifolds with Boundary

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definition of map-related vectors fields pair for $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary

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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 field on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at point.

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Target Context

• The reader will have a definition of map-related vectors fields pair for $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary.

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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:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f$: $:M_1\to M_2$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$\ast (V_1,V_2)$: $V_j\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors fields over }{M}_j\}$
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Conditions:
$\mathrm{\forall }{m}_1\in M_1(d{f}_{m_1}{V}_1(m_1)=V_2(f(m_1)))$
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2: Note

This definition does not claim that there is such a $V_2$ for each $V_1$.

Whether there is a $V_2$ for a $V_1$ depends on $V_1$.

For example, when $V_1=0$, $V_2=0$ will do.

When $f$ is any diffeomorphism, for $V_2:m_2\to d{f}_{f^{-1}(m_2)}{V}_1(f^{-1}(m_2))$, $(V_1,V_2)$ is $f$-related, by the proposition that for any diffeomorphism between any $C^{\mathrm{\infty }}$ manifolds with boundary and any $C^{\mathrm{\infty }}$ vectors field over the domain, there is the unique $C^{\mathrm{\infty }}$ vectors field over the codomain map-related with the vectors field over the domain.

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References

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