description/proof of that for \(C^\infty\) manifold, regular domain, \(C^\infty\) manifold with boundary, and \(C^\infty\) map from regular domain into \(C^\infty\) manifold with boundary, corresponding map with domain regarded as subset of manifold is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
- 4: Proof
Starting Context
- The reader knows a definition of regular domain of \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader admits the proposition that for any \(C^\infty\) manifold, any subset, and any point on the subset, if a chart satisfies the local slice condition for embedded submanifold or the local slice condition for embedded submanifold with boundary, its any sub-open-neighborhood does so.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold, its any regular domain, any \(C^\infty\) manifold with boundary, and any \(C^\infty\) map from the regular domain into the \(C^\infty\) manifold with boundary, the corresponding map with the domain regarded as the subset of the manifold is \(C^\infty\).
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:
\(M'_1\): \(\in \{ \text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds } \}\)
\(M_1\): \(\in \{\text{ the regular domains of } M'_1\}\)
\(M_2\): \(\in \{ \text{ the } C^\infty \text{ manifolds with boundary } \}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\(S\): \(= M_1\), as the subset of \(M'_1\)
\(f'\): \(: S \to M_2, s \mapsto f (s)\)
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Statements:
\(f' \in \{\text{ the } C^\infty \text{ maps }\}\)
//
2: Natural Language Description
For any \(d'\)-dimensional \(C^\infty\) manifold, \(M'_1\), any regular domain of \(M'_1\), \(M_1\), any \(C^\infty\) manifold with boundary, \(M_2\), any \(C^\infty\) map, \(f: M_1 \to M_2\), the subset of \(M'_1\), \(S = M_1\), and the map, \(f': S \to M_2, s \mapsto f (s)\), \(f'\) is \(C^\infty\).
3: Note
Although \(f\) and \(f'\) map each point to the same point, they are not the same map: whether \(f\) is \(C^\infty\) or not depends on the atlas of \(M_1\), while whether \(f'\) is \(C^\infty\) or not depends on the atlas of \(M'_1\), so, the proposition is not trivial.
4: Proof
Whole Strategy: Step 1: for each \(m \in M_1\), take an adopted chart, \((U'_m \subseteq M'_1, \phi'_m)\), for \(M_1\), take any chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), take an open neighborhood of \(m\), \(V_m = V'_m \cap M_1 \subseteq M_1\), such that \(f (V_m) \subseteq U_{f (m)}\), and take the adopted chart, \((U'_m \cap V'_m \subseteq M'_1, \phi'_m \vert_{U'_m \cap V'_m})\), and the corresponding adopting chart, \((U'_m \cap V'_m \cap M_1 \subseteq M_1, \pi_J \circ \phi'_m \vert_{U'_m \cap V'_m \cap M_1})\); Step 2: see what \(f\)'s being \(C^\infty\) at \(m\) means with respect to \((U_m := U'_m \cap V'_m \cap M \subseteq M_1, \phi_m := \pi_J \circ \phi'_m \vert_{U_m})\); Step 3: see what \(f'\)' being \(C^\infty\) at \(m\) means with respect to \((U'_m \cap V'_m \subseteq M'_1, \phi'_m \vert_{U'_m \cap V'_m})\); Step 4: conclude the proposition.
Step 1:
Let \(m \in M_1\) be any.
As \(M_1\) is an embedded submanifold with boundary of \(M'_1\), there is an adopted chart, \((U'_m \subseteq M'_1, \phi'_m)\), for \(M_1\).
Let us take any chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\).
As \(f\) is continuous, there is an open neighborhood of \(m\), \(V_m \subseteq M_1\), such that \(f (V_m) \subseteq U_{f (m)}\). As \(M_1\) is a topological subspace of \(M'_1\), \(V_m = V'_m \cap M_1\), where \(V'_m \subseteq M'_1\) is an open neighborhood of \(m\) on \(M'_1\).
\((U'_m \cap V'_m \subseteq M'_1, \phi'_m \vert_{U'_m \cap V'_m})\) is an adopted chart, by the proposition that for any \(C^\infty\) manifold, any subset, and any point on the subset, if a chart satisfies the local slice condition for embedded submanifold or the local slice condition for embedded submanifold with boundary, its any sub-open-neighborhood does so.
The corresponding adopting chart is \((U_m := U'_m \cap V'_m \cap M_1 \subseteq M_1, \phi_m := \pi_J \circ \phi'_m \vert_{U_m})\).
Step 2:
\(f (U'_m \cap V'_m \cap M_1) \subseteq f (V'_m \cap M_1) = f (V_m) \subseteq U_{f (m)}\).
So, by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\), \(\phi_{f (m)} \circ f \circ {\phi_m}^{-1}: \phi_m (U_m) \to \phi_{f (m)} (U_{f (m)})\) is \(C^\infty\) at \(\phi_m (m)\).
Step 3:
\(f' (U'_m \cap V'_m \cap S) = f (U'_m \cap V'_m \cap M_1) \subseteq U_{f (m)}\).
By the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\), \(f'\) is \(C^\infty\), if \(\phi_{f (m)} \circ f' \circ {\phi'_m \vert_{U'_m \cap V'_m}}^{-1} \vert_{\phi'_m \vert_{U'_m \cap V'_m} (U'_m \cap V'_m \cap S)}: \phi'_m \vert_{U'_m \cap V'_m} (U'_m \cap V'_m \cap S) \to \phi_{f (m)} (U_{f (m)})\) is \(C^\infty\) at \(\phi'_m \vert_{U'_m \cap V'_m} (m) = \phi'_m (m)\).
Step 4:
\(\phi_m (U_m) = \pi_J \circ \phi'_m \vert_{U'_m \cap V'_m \cap M_1} (U'_m \cap V'_m \cap M_1) = \phi'_m \vert_{U'_m \cap V'_m \cap M_1} (U'_m \cap V'_m \cap M_1)\), because as \(M_1\) is a regular domain, which means that the codimension of \(M_1\) is \(0\), \(\pi_J\) does nothing.
On the other hand, \(\phi'_m \vert_{U'_m \cap V'_m} (U'_m \cap V'_m \cap S) = \phi'_m \vert_{U'_m \cap V'_m \cap M_1} (U'_m \cap V'_m \cap M_1)\).
So, in fact, the domain of \(\phi_{f (m)} \circ f \circ {\phi_m}^{-1}\) and the domain of \(\phi_{f (m)} \circ f' \circ {\phi'_m \vert_{U'_m \cap V'_m}}^{-1} \vert_{\phi'_m \vert_{U'_m \cap V'_m} (U'_m \cap V'_m \cap S)}\) are the same, and the 2 maps are the same.
So, \(\phi_{f (m)} \circ f' \circ {\phi'_m \vert_{U'_m \cap V'_m}}^{-1} \vert_{\phi'_m \vert_{U'_m \cap V'_m} (U'_m \cap V'_m \cap S)}\) is \(C^\infty\) at \(\phi'_m (m)\), as \(\phi_{f (m)} \circ f \circ {\phi_m}^{-1}\) is \(C^\infty\) at \(\phi_m (m) = \phi'_m (m)\).
So, \(f'\) is \(C^\infty\).
