A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point, restriction on open subset of domain that contains point is locally diffeomorphic at point
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 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.
- The reader admits 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 or expansion on any codomain that contains the range 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 arbitrary subsets of any \(C^\infty\) manifolds with boundary locally diffeomorphic at any point, the restriction on any open subset of the domain that contains the point is locally diffeomorphic 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 \(S_1 \subseteq S'_1\) is open on \(S'_1\), any point, \(p \in S_1\), and any map, \(f': S'_1 \to S_2\), such that \(f'\) is locally diffeomorphic at \(p\), the domain restriction, \(f: S_1 \to S_2\) is locally diffeomorphic 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 \(f (p)\) such that \(f' \vert_{U'_p \cap S'_1}: U'_p \cap S'_1 \to U_{f (p)} \cap S_2\) is a diffeomorphism.
\(f \vert_{U'_p \cap S_1} = f' \vert_{U'_p \cap S'_1} \vert_{U'_p \cap S_1}: U'_p \cap S_1 \to f \vert_{U'_p \cap S_1} (U'_p \cap S_1)\) is a diffeomorphism, by 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 and 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 or expansion on any codomain that contains the range is \(C^k\) at the point.
But the issue is that the range has not been proved to be an open subset of \(S_2\) (the invariance of domain theorem does not apply here, because the diffeomorphism is not from any open subset of any \(C^\infty\) manifold without boundary).
\(U'_p \cap S_1\) is an open subset of \(U'_p \cap S'_1\), because as \(S_1\) is open on \(S'_1\), \(S_1 = U''_p \cap S'_1\), where \(U''_p \subset M_1\) is an open subset of \(M_1\), so, \(U'_p \cap S_1 = U'_p \cap U''_p \cap S'_1 = U''_p \cap U'_p \cap S'_1\). So, \(f' \vert_{U'_p \cap S'_1} (U'_p \cap S_1)\) is open on \(U_{f (p)} \cap S_2\), so, \(f' \vert_{U'_p \cap S'_1} (U'_p \cap S_1) = U'_{f (p)} \cap U_{f (p)} \cap S_2\), where \(U'_{f (p)}\) is an open subset of \(M_2\).
So, there are an open neighborhood, \(U'_p \subseteq M_1\), of \(p\) and an open neighborhood, \(U'_{f (p)} \cap U_{f (p)} \subseteq M_2\), of \(f (p)\) such that \(f \vert_{U'_p \cap S_1}: U'_p \cap S_1 \to U'_{f (p)} \cap U_{f (p)} \cap S_2\) is a diffeomorphism.
