A description/proof of that map between arbitrary subsets of \(C^\infty\) manifolds with boundary bijective and locally diffeomorphic at each point is diffeomorphism
Topics
About: \(C^\infty\) manifold with boundary
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 locally diffeomorphic at point.
- The reader admits 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.
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 bijective and locally diffeomorphic at each point is a diffeomorphism.
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, \(M_1, M_2\), and any subsets, \(S_1 \subseteq M_1, S_2 \subseteq M_2\), any map, \(f: S_1 \to S_2\), that is bijective and locally diffeomorphic at each point is a diffeomorphism.
2: Proof
As \(f\) is bijective, there is the inverse, \(f^{-1}: S_2 \to S_1\).
\(f\) is \(C^\infty\), by 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.
For each \(f (p) \in S_2\), there are an open neighborhood, \(U_p \subseteq M_1\), of \(p \in S_1\) and an open neighborhood, \(U_{f (p)} \subseteq M_2\), of \(f (p)\) such that \(f \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f (p)} \cap S_2\) is a diffeomorphism. That means that there are an open neighborhood, \(U_{f (p)} \subseteq M_2\), of \(f (p)\) and an open neighborhood, \(U_p \subseteq M_1\), of \(p = f^{-1} (f (p)) \in S_1\) such that \(f^{-1} \vert_{U_{f (p)} \cap S_2}: U_{f (p)} \cap S_2 \to U_p \cap S_1\) is a diffeomorphism, so, \(f^{-1}\) is locally diffeomorphic at each \(f (p)\), and is \(C^\infty\), by 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.
So, \(f\) is a diffeomorphism.
