A description/proof of that map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point is \(C^\infty\) at point
Topics
About: \(C^\infty\) manifold with boundary
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that any map between arbitrary subsets of any \(C^\infty\) manifolds with boundary locally diffeomorphic at any point is \(C^\infty\) 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, \(S_1 \subseteq M_1, S_2 \subseteq M_2\), and any point, \(p \in S_1\), any map, \(f: S_1 \to S_2\), that is locally diffeomorphic at \(p\) is \(C^\infty\) at \(p\).
2: Proof
There are an open neighborhood, \(U_p \subseteq M_1\), of \(p\) and an open neighborhood, \(U_{f (p)} \subseteq M_2\), of \(f (p) \in S_2\), such that \(f \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f (p)} \cap S_2\) is a diffeomorphism.
As \(f \vert_{U_p \cap S_1}\) is \(C^\infty\), there are a chart, \((U'_p \subseteq M_1, \phi'_p)\), and a chart, \((U'_{f (p)} \subseteq M_2, \phi'_{f (p)})\), such that \(f \vert_{U_p \cap S_1} (U'_p \cap U_p \cap S_1) \subseteq U'_{f (p)}\) and \(\phi'_{f (p)} \circ f \vert_{U_p \cap S_1} \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap U_p \cap S_1)}: \phi'_p (U'_p \cap U_p \cap S_1) \to \phi'_{f (p)} (U'_{f (p)})\) is \(C^\infty\) at \(\phi'_p (p)\).
Also \((U'_p \cap U_p \subseteq M_1, \phi'_p \vert_{U'_p \cap U_p})\) is a chart, and \(f (U'_p \cap U_p \cap S_1) = f \vert_{U_p \cap S_1} (U'_p \cap U_p \cap S_1) \subseteq U'_{f (p)}\) and \(\phi'_{f (p)} \circ f \circ {\phi'_p \vert_{U'_p \cap U_p}}^{-1} \vert_{\phi'_p (U'_p \cap U_p \cap S_1)}: \phi'_p (U'_p \cap U_p \cap S_1) \to \phi'_{f (p)} (U'_{f (p)}) = \phi'_{f (p)} \circ f \vert_{U_p \cap S_1} \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap U_p \cap S_1)}\) is \(C^\infty\) at \(\phi'_p (p)\).
So, \(f\) is \(C^\infty\) at \(p\).
3: Note
Typically, \(S_1 = M_1\) and \(S_2 = M_2\), and any map between any \(C^\infty\) manifolds with boundary locally diffeomorphic at \(p\) is \(C^\infty\) at \(p\).
