A description/proof of that for maps between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at corresponding points, where codomain of 1st map is open subset of domain of 2nd map, composition 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 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.
- 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.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Target Context
- The reader will have a description and a proof of the proposition that for any maps between arbitrary subsets of any \(C^\infty\) manifolds with boundary locally diffeomorphic at corresponding points, where the codomain of the 1st map is an open subset of the domain of the 2nd map, the composition 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, \(M_1, M_2, M_3\), any subsets, \(S_1 \subseteq M_1, S_2, S'_2 \subseteq M_2, S_3 \subseteq M_3\), such that \(S_2 \subseteq S'_2\) is an open subset of \(S'_2\), any point, \(p \in S_1\), and any maps, \(f_1: S_1 \to S_2, f'_2: S'_2 \to S_3\), such that \(f_1\) and \(f'_2\) are locally diffeomorphic at \(p\) and \(f_1 (p)\), \(f'_2 \circ f_1: S_1 \to S_3\) is locally diffeomorphic at \(p\).
2: Proof
\(f_2: S_2 \to S_3\) as the domain restriction of \(f'_2\) is locally diffeomorphic at \(f_1 (p)\), by 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.
As \(f'_2 \circ f_1 = f_2 \circ f_1\), we will prove that \(f_2 \circ f_1\) is locally diffeomorphic at \(p\), and then, \(f'_2 \circ f_1\) will be locally diffeomorphic at \(p\).
There are an open neighborhood, \(U_p \subseteq M_1\), of \(p\) and an open neighborhood, \(U_{f_1 (p)} \subseteq M_2\), of \(f_1 (p)\) such that \(f_1 \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f_1 (p)} \cap S_2\) is a diffeomorphism.
There are an open neighborhood, \(U'_{f_1 (p)} \subseteq M_2\), of \(f_1 (p)\) and an open neighborhood, \(U_{f_2 \circ f_1 (p)} \subseteq M_3\), of \(f_2 \circ f_1 (p)\) such that \(f_2 \vert_{U'_{f_1 (p)} \cap S_2}: U'_{f_1 (p)} \cap S_2 \to U_{f_2 \circ f_1 (p)} \cap S_3\) is a diffeomorphism.
\(U_{f_1 (p)} \cap U'_{f_1 (p)} \subseteq M_2\) is open on \(M_2\), and \(U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2\) is open on \(U_{f_1 (p)} \cap S_2\), because \(U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2 = U'_{f_1 (p)} \cap U_{f_1 (p)} \cap S_2\), and likewise, \(U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2\) is open on \(U'_{f_1 (p)} \cap S_2\).
So, \((f_1 \vert_{U_p \cap S_1})^{-1} (U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2)\) is open on \(U_p \cap S_1\), so, \(= U'_p \cap U_p \cap S_1\), where \(U'_p \subseteq M_1\) is an open subset of \(M_1\).
Likewise, \(f_2 \vert_{U'_{f_1 (p)} \cap S_2} (U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2)\) is open on \(U_{f_2 \circ f_1 (p)} \cap S_3\), so, \(= U'_{f_2 \circ f_1 (p)} \cap U_{f_2 \circ f_1 (p)} \cap S_3\), where \(U'_{f_2 \circ f_1 (p)} \subseteq M_3\) is an open subset of \(M_3\).
Then, there are an open neighborhood, \(U'_p \cap U_p \subseteq M_1\), of \(p\) and an open neighborhood, \(U'_{f_2 \circ f_1 (p)} \cap U_{f_2 \circ f_1 (p)}\), of \(f_2 \circ f_1 (p)\) such that \(f_2 \circ f_1 \vert_{U'_p \cap U_p \cap S_1}: U'_p \cap U_p \cap S_1 \to U'_{f_2 \circ f_1 (p)} \cap U_{f_2 \circ f_1 (p)} \cap S_3\) is a diffeomorphism, because \(f_1 \vert_{U_p \cap S_1} \vert_{U'_p \cap U_p \cap S_1}: U'_p \cap U_p \cap S_1 \to U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2\) is diffeomorphic and \(f_2 \vert_{U'_{f_1 (p)} \cap S_2} \vert_{U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2}: U_{f_1 (p)} \cap S_2 \cap U'_{f_1 (p)} \cap S_2 \to U'_{f_2 \circ f_1 (p)} \cap U_{f_2 \circ f_1 (p)} \cap S_3\) is diffeomorphic, 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, and the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point applies.
