A description/proof of that restriction of \(C^\infty\) map on open domain and open codomain Is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) function on \(C^\infty\) manifold.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds, the restriction of the map on any open domain and any valid open codomain 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: Description
For any \(C^\infty\) manifolds, \(M_1, M_2\), any \(C^\infty\) map, \(f: M_1 \rightarrow M_2\), any open subset, \(U_1 \subseteq M_1\), and any open subset, \(U_2 \subseteq M_2\), such that \(f (U_1) \subseteq U_2\), \(f\vert_{U_1}: U_1 \rightarrow U_2\) is \(C^\infty\).
2: Proof
For any point, \(p \in U_1\), there are charts, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), and \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1}\) is \(C^\infty\) at \(\phi_p (p)\). \((U_p \cap U_1 \subseteq U_1, \phi_p\vert_{U_p \cap U_1})\) and \((U_{f (p)} \cap U_2 \subseteq U_2, \phi_{f (p)}\vert_{U_{f (p)} \cap U_2})\) are charts on \(U_1\) and \(U_2\), and \(\phi_{f (p)}\vert_{U_{f (p)} \cap U_2} \circ f\vert_{U_1} \circ {\phi_p\vert_{U_p \cap U_1}}^{-1}\) is \(C^\infty\) at \(\phi_p\vert_{U_p \cap U_1} (p)\), because it is a restriction of \(C^\infty\) \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1}\) on the open domain, \(\phi_p (U_p \cap U_1)\).
