A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, restriction on domain that contains point is \(C^k\) at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
Target Context
- The reader will have a description and a proof of the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
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: Description
For any \(C^\infty\) manifolds with (possibly empty) boundary, \(M_1, M_2\), any subsets, \(S_1, S'_1 \subseteq M_1, S_2 \subseteq M_2\), such that \(S'_1 \subseteq S_1\), any point, \(p \in S'_1\), any natural number (including 0) or \(\infty\) \(k\), and any map, \(f: S_1 \to S_2\), such that \(f\) is \(C^k\) at \(p\), \(f \vert_{S'_1}: S'_1 \to S_2\) is \(C^k\) at \(p\).
2: Proof
Let us suppose that \(k = 0\).
For any open neighborhood, \(U_{f (p)} \subseteq S_2\), of \(f (p)\), there is an open neighborhood, \(U_p \subseteq S_1\), of \(p\) such that \(f (U_p) \subseteq U_{f (p)}\). \(U_p \cap S'_1 \subseteq S'_1\) is an open neighborhood of \(p\) on \(S'_1\), and \(f \vert_{S'_1} (U_p \cap S'_1) \subseteq U_{f (p)}\).
Let us suppose that \(1 \le k\) including \(\infty\).
There are a chart, \((U'_p \subseteq M_1, \phi'_p)\), around \(p\) and a chart, \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), around \(f (p)\) such that \(f (U'_p \cap S_1) \subseteq U_{f (p)}\) and \(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}\) is \(C^k\) at \(\phi'_p (p)\).
\(f \vert_{S'_1} (U'_p \cap S'_1) \subseteq U_{f (p)}\) and \(\phi_{f (p)} \circ f \vert_{S'_1} \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S'_1)} = \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S'_1)}\) is \(C^k\) at \(\phi'_p (p)\), by the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
So, the same charts pair can be used for \(f \vert_{S'_1}\).
