A description/proof of that \(C^\infty\) function on \(C^\infty\) manifold is \(C^\infty\) on regular submanifold
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.
- The reader knows a definition of regular submanifold of \(C^\infty\) manifold.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold and its any regular submanifold, any \(C^\infty\) function on the super manifold is \(C^\infty\) on the regular submanifold.
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 \(C^\infty\) manifold, \(M'\), and any regular submanifold, \(M \subseteq M'\), any \(C^\infty\) function, \(f: M' \rightarrow \mathbb{R}\), is \(C^\infty\) on \(M\), which means that the restriction, \(f\vert_M: M \rightarrow \mathbb{R}\) is \(C^\infty\).
2: Proof
For any \(p \in M\), there is an adopted chart, \((U'_p \subseteq M', \phi'_p)\), and \(f \circ {\phi'_p}^{-1}\) is \(C^\infty\). There is the corresponding adopting chart, \((U_p, \phi_p)\), and \(f\vert_M \circ {\phi_p}^{-1}\) is \(C^\infty\), because \(f\vert_M \circ {\phi_p}^{-1} (r^1, r^2., ..., r^{d}) = f \circ {\phi'_p}^{-1} (r^1, r^2., ..., r^{d}, 0, ..., 0)\).
