A definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
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: Definition
For any \(C^\infty\) manifolds with (possibly empty) boundary, \(M_1, M_2\), any subsets, \(S_1 \subseteq M_1, S_2 \subseteq M_2\), any point, \(p \in S_1\), and any natural number (excluding 0) or \(\infty\) \(k\), any map, \(f: S_1 \to S_2\), such that there are a chart, \((U'_p \subseteq M_1, \phi'_p)\), around \(p\) and a chart, \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), around \(f (p)\) such that \(f (U'_p \cap S_1) \subseteq U_{f (p)}\) and \(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}: \phi'_p (U'_p \cap S_1) \to \phi_{f (p)} (U_{f (p)})\) is \(C ^k\) at \(\phi'_p (p)\) by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
When this definition is satisfied, there is the open subset, \(U_{\phi'_p (p)}\), cited in the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), on the whole of which \(f'\) is \(C^k\) and \(f' \vert_{U_{\phi'_p (p)} \cap \phi'_p (U'_p \cap S_1)} = \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{U_{\phi'_p (p)} \cap \phi'_p (U'_p \cap S_1)}\).
As \(U_{\phi'_p (p)} \cap \phi'_p (U'_p)\) is an open subset of \(\phi'_p (U'_p)\), \({\phi'_p}^{-1} (U_{\phi'_p (p)} \cap \phi'_p (U'_p))\) is open on \(U'_p\) and on \(M_1\), and so, \({\phi'_p}^{-1} (U_{\phi'_p (p)} \cap \phi'_p (U'_p))\) can be taken to be \(U'_p\) instead satisfying the conditions, and then, \(\phi'_p (U'_p) \subseteq U_{\phi'_p (p)}\).
When \(\phi'_p (U'_p)\) is open on \(\mathbb{R}^{d_1}\), \(U_{\phi'_p (p)}\) can be taken to be contained in \(\phi'_p (U'_p)\), and then, \(U_{\phi'_p (p)} \subseteq \phi'_p (U'_p)\). If furthermore \({\phi'_p}^{-1} (U_{\phi'_p (p)} \cap \phi'_p (U'_p))\) is taken to be \(U'_p\) instead as in the previous paragraph, \(\phi'_p (U'_p) = U_{\phi'_p (p)}\).
2: Note
\(k = 0\) is excluded because that case has been already defined as map continuous at point.
But when \(f\) is \(C^k\) at \(p\) where \(1 \le k\), \(f\) is \(C^0\) at \(p\): for any open neighborhood, \(U'_{f (p)} \subseteq S_2\), of \(f (p)\), we can think of only the case, \(U'_{f (p)} \subseteq U_{f (p)}\), because otherwise, we can think of \(U'_{f (p)} \cap U_{f (p)}\), which is open on \(S_2\), instead; \(U'_{f (p)} = U''_{f (p)} \cap S_2\) where \(U''_{f (p)} \subseteq M_2\) is open and we can take \(U''_{f (p)} \subseteq U_{f (p)}\); \(\phi_{f (p)} (U''_{f (p)}) \subseteq \phi_{f (p)} (U_{f (p)})\) is open on \(\mathbb{H}^{d_2}\) or \(\mathbb{R}^{d_2}\) and \(\phi_{f (p)} (U''_{f (p)}) = U_{\phi_{f (p)} (f (p))} \cap \mathbb{H}^{d_2} \text{ or } U_{\phi_{f (p)} (f (p))} \cap \mathbb{R}^{d_2}\) where \(U_{\phi_{f (p)} (f (p))} \subseteq \mathbb{R}^{d_2}\) is open; the extension of \(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}\), \(f': U'_{\phi_p (p)} \to \mathbb{R}^{d_2}\), is \(C^k\) and continuous at \(\phi_p (p)\); there is an open neighborhood, \(U''_{\phi_p (p)} \subseteq U'_{\phi_p (p)}\), of \(\phi_p (p)\) such that \(f' (U''_{\phi_p (p)}) \subseteq U_{\phi_{f (p)} (f (p))}\); \(U''_{\phi_p (p)} \cap \phi'_p (U'_p)\) is open on \(\phi'_p (U'_p)\) and \({\phi'_p}^{-1} (U''_{\phi_p (p)} \cap \phi'_p (U'_p)) \subseteq U'_p\) is open on \(U'_p\) and on \(M_1\); \(U''_p := {\phi'_p}^{-1} (U''_{\phi_p (p)} \cap \phi'_p (U'_p)) \cap S_1\) is open on \(S_1\); \(f (U''_p) = {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \circ \phi'_p (U''_p) = {\phi_{f (p)}}^{-1} \circ f' \circ \phi'_p (U''_p)\), but \(\phi'_p (U''_p) \subseteq U''_{\phi_p (p)}\), so, \(f' \circ \phi'_p (U''_p) \subseteq U_{\phi_{f (p)} (f (p))}\) and it is also contained in \(\mathbb{H}^{n'}\) or \(\mathbb{R}^{n'}\) because the coordinates function maps into there, so, it is contained in \(\phi_{f (p)} (U''_{f (p)})\); so, \(f (U''_p) \subseteq U''_{f (p)} \cap S_2 = U'_{f (p)}\).
When \(M_1\) and \(M_2\) are any Euclidean \(C^\infty\) manifolds, this definition coincides with the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), because in that case, if \(f\) satisfies the latter definition, we can take \((U'_p = M_1 \subseteq M_1, \phi'_p = id)\) and \((U_{f (p)} = M_2 \subseteq M_2, \phi_{f (p)} = id)\) for the former definition, and \(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)} = f\), which is \(C^k\) at \(\phi'_p (p) = p\) by the latter definition; if \(f\) satisfies the former definition, while \(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}\) is \(C^k\) at \(\phi'_p (p)\), \(id \circ {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)} \circ \phi'_p \circ {id}^{-1} \vert_{U'_p \cap S_1} = f \vert_{U'_p \cap S_1}\) is \(C^k\) at \(p\), because the transition maps are diffeomorphisms, by the proposition that for any map between arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), its restriction on any subset that contains the point is \(C^k\) at the point and the proposition that the composition of any maps between arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), is \(C^k\) at the point, and so, \(f\) is \(C^k\) at \(p\), by the proposition that any map between arbitrary subsets of any \(C^\infty\) manifolds with boundary is \(C^k\) at any point, where \(k\) includes \(\infty\), if its restriction on any open subspace domain that contains the point is \(C^k\) at the point.
