description/proof of that map between \(C^\infty\) manifolds with boundary is \(C^k\) if and only if domain restriction of map to each element of open cover is \(C^k\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader knows a definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition.
Target Context
- The reader will have a description and a proof of the proposition that any map between any \(C^\infty\) manifolds with boundary is \(C^k\) if and only if the domain restriction of the map to each element of any open cover is \(C^k\).
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\)
\(B\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{U_\beta \vert \beta \in B\}\): \(\in \{\text{ the open covers of } M_1\}\)
//
Statements:
\(f \in \{\text{ the } C^k \text{ maps }\}\)
\(\iff\)
\(\forall U_\gamma \in \{U_\beta \vert \beta \in B\} (f \vert_{U_\gamma} \in \{\text{ the } C^k \text{ maps }\})\)
//
\(f \vert_{U_\gamma}\) can be regarded as a map from the open submanifold with boundary of \(M_1\) or a map from the subset of \(M_1\), by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary and the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
2: Proof
Whole Strategy: Step 1: when \(k = 0\), conclude the proposition; suppose that \(0 \lt k\) thereafter; Step 2: suppose that \(f\) is \(C^k\), and see that \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the subset of \(M_1\); Step 3: see that \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the open submanifold with boundary of \(M_1\); Step 4: see that when \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the open submanifold with boundary of \(M_1\), it is \(C^k\) with it regarded as the map from the subset of \(M_1\); Step 5: suppose that \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the subset of \(M_1\), and see that \(f\) is \(C^k\).
Step 1:
Let us suppose that \(k = 0\).
When \(f\) is continuous, each \(f \vert_{u_\gamma}\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
When each \(f \vert_{u_\gamma}\) is continuous, \(f\) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
Let us suppose that \(0 \lt k\) hereafter.
Step 2:
Let us suppose that \(f\) is \(C^k\).
Let us see that \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the subset of \(M_1\).
Around each \(m \in U_\gamma\), as \(m \in M_1\), there are a chart, \((U_m \subseteq M_1, \phi_m)\), and a chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), such that \(f (U_m) \subseteq \phi_{f (m)} (U_{f (m)})\), by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
Also \((U_m \cap U_\gamma \subseteq M_1, \phi_m \vert_{U_m \cap U_\gamma})\) is a chart, by the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart, and \(f (U_m \cap U_\gamma) \subseteq \phi_{f (m)} (U_{f (m)})\) is satisfied.
By the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition, \(\phi_{f (m)} \circ f \circ {\phi_m \vert_{U_m \cap U_\gamma}}^{-1}: \phi_m \vert_{U_m \cap U_\gamma} (U_m \cap U_\gamma) \to \phi_{f (m)} (U_{f (m)})\) is \(C^k\) at \(\phi_m (m)\).
\(= \phi_{f (m)} \circ f \vert_{U_\gamma} \circ {\phi_m \vert_{U_m \cap U_\gamma}}^{-1} \vert_{\phi_m \vert_{U_m \cap U_\gamma} (U_m \cap U_\gamma \cap U_\gamma)}: \phi_m \vert_{U_m \cap U_\gamma} (U_m \cap U_\gamma \cap U_\gamma) \to \phi_{f (m)} (U_{f (m)})\), which is \(C^k\) at \(\phi_m (m)\), which is exactly the condition for \(f \vert_{U_\gamma}\) to be \(C^k\) at \(m\), by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
So, \(f \vert_{U_\gamma}\) is \(C^k\) at \(m\), and as \(m \in U_\gamma\) is arbitrary, \(f \vert_{U_\gamma}\) is \(C^k\).
Step 3:
Then, \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the open submanifold with boundary of \(M_1\), by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary and the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
Step 4:
When \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the open submanifold with boundary of \(M_1\), it is \(C^k\) with it regarded as the map from the subset of \(M_1\), by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary and the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
So, if we prove the "if" direction of the proposition with \(f \vert_{U_\gamma}\) regarded as the map from the subset of \(M_1\), the direction will hold also with \(f \vert_{U_\gamma}\) regarded as the map from the open submanifold with boundary of \(M_1\).
Step 5:
Let us suppose that each \(f \vert_{U_\gamma}\) is \(C^k\) with it regarded as the map from the subset of \(M_1\).
Let us see that \(f\) is \(C^k\).
For each \(m \in M_1\), there is a \(\gamma \in B\) such that \(m \in U_\gamma\).
As \(f \vert_{U_\gamma}\) is \(C^k\) at \(m\), there are a chart, \((U_m \subseteq M_1, \phi_m)\) and a chart, \((U_{f (m)} \subseteq M_2, \phi_{f (m)})\), such that \(f \vert_{U_\gamma} (U_m \cap U_\gamma) \subseteq U_{f (m)}\) and \(\phi_{f (m)} \circ f \vert_{U_\gamma} \circ {\phi_m}^{-1} \vert_{\phi_m (U_m \cap U_\gamma)}: \phi_m (U_m \cap U_\gamma) \to \phi_{f (m)} (U_{f (m)})\) is \(C^k\) at \(\phi_m (m)\), by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
But also \((U_m \cap U_\gamma \subseteq M_1, \phi_m \vert_{U_m \cap U_\gamma})\) is a chart, by the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart, and \(f (U_m \cap U_\gamma) \subseteq U_{f (m)}\) is satisfied.
\(\phi_{f (m)} \circ f \circ {\phi_m \vert_{U_m \cap U_\gamma}}^{-1} = \phi_{f (m)} \circ f \vert_{U_\gamma} \circ {\phi_m}^{-1} \vert_{\phi_m (U_m \cap U_\gamma)}: \phi_m (U_m \cap U_\gamma) \to \phi_{f (m)} (U_{f (m)})\) is \(C^k\) at \(\phi_m (m)\), which is exactly the condition for \(f\) to be \(C^k\) at \(m\).
As \(m \in M_1\) is arbitrary, \(f\) is \(C^k\).
