A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, any possible pair of domain chart and codomain chart satisfies condition of definition
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) 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 maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
- The reader admits the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets.
- The reader admits the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds, the map is \(C^k\) at any point if the restriction on any subspace open neighborhood of point domain 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 any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition.
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: Note
The definition requires directly only that there is a pair of a domain chart and a codomain chart that satisfies the conditions, not that any possible pair satisfies the condition, and this proposition confirms that the latter is indeed implied.
2: Description
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, \(p \in S_1\), any natural number (excluding 0) or \(\infty\) \(k\), and any map, \(f: S_1 \to S_2\), that is \(C^k\) at \(p\), the pair of any chart, \((U''_p \subseteq M_1, \phi''_p)\), around \(p\) and any chart, \((U'_{f (p)} \subseteq M_2, \phi'_{f (p)})\), around \(f (p)\) such that \(f (U''_p \cap S_1) \subseteq U'_{f (p)}\) satisfied the condition of the definition: \(\phi'_{f (p)} \circ f \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap S_1)}: \phi''_p (U''_p \cap S_1) \to \phi'_{f (p)} (U'_{f (p)})\) is \(C ^k\) at \(\phi''_p (p)\).
3: Proof
By the definition, 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 \(\phi'_p (p)\).
\(\phi'_{f (p)} \circ f \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap U'_p \cap S_1)} = \phi'_{f (p)} \circ {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \circ \phi'_p \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap U'_p \cap S_1)}\).
\(\phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap U''_p \cap S_1)}\) is \(C^k\) at \(\phi'_p (p)\), by the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point. \(\phi'_p \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap U'_p \cap S_1)}\) and \(\phi'_{f (p)} \circ {\phi_{f (p)}}^{-1}\) are \(C^k\) at \(\phi''_p (p)\) and \(\phi_{f (p)} (f (p))\), because the transition maps are diffeomorphic and by the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
So, the composition, \(\phi'_{f (p)} \circ {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi'_p}^{-1} \circ \phi'_p \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap U'_p \cap S_1)}\) is \(C^k\) at \(\phi''_p (p)\), by the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
\(\phi''_p (U''_p \cap U'_p \cap S_1) = \phi''_p (U''_p \cap S_1) \cap \phi''_p (U''_p \cap U'_p)\) (by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets) is an open neighborhood of \(\phi''_p (p)\) on \(\phi''_p (U''_p \cap S_1)\), because \(\phi''_p (U''_p \cap U'_p)\) is open on \(\mathbb{H}^{d_1}\) or \(\mathbb{R}^{d_1}\): when \((U''_p \subseteq M_1, \phi''_p)\) is a boundary chart, \(\phi''_p (U''_p \cap U'_p) = U \cap \mathbb{H}^{d_1}\) for an open \(U \subseteq \mathbb{R}^{d_1}\), and \(\phi''_p (U''_p \cap S_1) \cap \phi''_p (U''_p \cap U'_p) = \phi''_p (U''_p \cap S_1) \cap U \cap \mathbb{H}^{d_1} = \phi''_p (U''_p \cap S_1) \cap U\) because \(\phi''_p (U''_p \cap S_1) \subseteq \mathbb{H}^{d_1}\); otherwise, \(\phi''_p (U''_p \cap U'_p)\) itself is open on \(\mathbb{R}^{d_1}\).
So, \(\phi'_{f (p)} \circ f \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap S_1)}\) is \(C^k\) at \(\phi''_p (p)\), by the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds, the map is \(C^k\) at any point if the restriction on any subspace open neighborhood of point domain is \(C^k\) at the point.
