description/proof of that for diffeomorphism between \(C^\infty\) manifolds with boundary and \(C^\infty\) vectors field over domain, there is unique \(C^\infty\) vectors field over codomain map-related with vectors field over domain
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
- The reader knows a definition of map-related vectors fields pair for \(C^\infty\) map between \(C^\infty\) manifolds with boundary.
- The reader admits the proposition that for any \(C^\infty\) immersion between any \(C^\infty\) manifolds with boundary, its global differential is a \(C^\infty\) immersion.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Target Context
- The reader will have a description and a proof of 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.
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the diffeomorphisms }\}\)
\(V_1\): \(\in \{\text{ the } C^\infty \text{ vectors fields over } M_1\}\)
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Statements:
\(! \exists V_2 \in \{\text{ the } C^\infty \text{ vectors fields over } M_2\} ((V_1, V_2) \in \{\text{ the } f \text{ -related vectors fields pairs }\})\)
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2: Proof
Whole Strategy: Step 1: define \(V_2\) as \(: M_2 \to TM_2, m \mapsto d f_{f^{-1} (m)} V_1 (f^{-1} (m))\); Step 2: see that \(V_2\) is \(C^\infty\); Step 3: see that \(V_2\) is unique.
Step 1:
Let us define \(V_2: M_2 \to TM_2, m \mapsto d f_{f^{-1} (m)} V_1 (f^{-1} (m))\).
Step 2:
\(V_2 = d f \circ V_1 \circ f^{-1}\), where \(d f: TM_1 \to TM_2\) is the global differential of \(f\).
\(f^{-1}\) is \(C^\infty\), \(V_1\) is \(C^\infty\), and \(d f\) is \(C^\infty\) by the proposition that for any \(C^\infty\) immersion between any \(C^\infty\) manifolds with boundary, its global differential is a \(C^\infty\) immersion: any diffeomorphism is a \(C^\infty\) immersion.
So, \(V_2\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Step 3:
\(V_2\) is unique, because for each \(m \in M_2\), \(f (f^{-1} (m)) = m\), so, there is no other option but \(V_2 (m) = d f_{f^{-1} (m)} V_1 (f^{-1} (m))\).
