definition of map-related vectors fields pair for \(C^\infty\) map between \(C^\infty\) manifolds with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of map-related vectors fields pair for \(C^\infty\) map between \(C^\infty\) manifolds with boundary.
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:
\( M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\(*(V_1, V_2)\): \(V_j \in \{\text{ the } C^\infty \text{ vectors fields over } M_j\}\)
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Conditions:
\(\forall m_1 \in M_1 (d f_{m_1} V_1 (m_1) = V_2 (f (m_1)))\)
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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 \(C^\infty\) manifolds with boundary and any \(C^\infty\) vectors field over the domain, there is the unique \(C^\infty\) vectors field over the codomain map-related with the vectors field over the domain.
