A description/proof of that for \(C^\infty\) map between \(C^\infty\) manifolds, restriction of map on regular submanifold domain and regular submanifold codomain Is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) map.
- The reader knows a definition of regular submanifold.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map between \(C^\infty\) manifolds, the restriction of the map on any regular submanifold domain and any regular submanifold codomain is \(C^\infty\).
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, \(M'_1, M'_2\), any \(C^\infty\) map, \(f': M'_1 \rightarrow M'_2\), any regular submanifold, \(M_1 \subseteq M'_1\), and any regular submanifold, \(M_2 \subseteq M'_2\), such that \(f' (M_1) \subseteq M_2\), \(f = f'\vert_{M_1}: M_1 \rightarrow M_2\) is \(C^\infty\).
2: Proof
\(f'\) is continuous, and \(f\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Around any point, \(p \in M_1\), there are some adopted charts, \((U'_p \subseteq M'_1, \phi'_p)\) and \((U'_{f' (p)} \subseteq M'_2, \phi'_{f' (p)})\), such that \(f' (U'_p) \subseteq U'_{f' (p)}\), because as \(f'\) is continuous, there is an open neighborhood, \(N'_p \subseteq M'_1\), such that \(f' (N'_p) \subseteq U'_{f' (p)}\), and if \(\lnot f' (U'_p) \subseteq U'_{f' (p)}\), \(f' (U'_p \cap N'_p) \subseteq U'_{f' (p)}\), and \((U'_p \cap N'_p, \phi'_p\vert_{U'_p \cap N'_p})\) can be the adopted chart instead. There are the corresponding adopting charts, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\). \(f (U_p) \subseteq U_{f (p)}\), because \(f (U_p) = f' (U_p) = f' (U'_p \cap M_1) \subseteq U'_{f' (p)} \cap M_2 = U_{f (p)}\). \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1}, (x^1, x^2 , . . ., x^{d_1}) \mapsto (y^1, y^2, . . ., y^{d_2})\) is the restriction of \(\phi'_{f' (p)} \circ f' \circ {\phi'_p}^{-1}, (x^1, x^2 , . . ., x^{d_1}, x^{d_1 + 1}, x^{d_1 + 2}, . . ., x^{d'_1}) \mapsto (y^1, y^2, . . ., y^{d_2}, y^{d_2 + 1}, y^{d_2 + 2}, . . ., y^{d'_2})\) on the domain such that \((x^{d_1 + 1}, x^{d_1 + 2}, . . ., x^{d'_1}) = (0, 0, . . ., 0)\) and on the codomain such that \((y^{d_2 + 1}, y^{d_2 + 2}, . . ., y^{d'_2}) = (0., 0, . . ., 0)\), and is \(C^\infty\).
