description/proof of that for maps between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, composition is \(C^k\) at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
- 4: Proof
Starting Context
- The reader knows a 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 maps between any topological spaces continuous at corresponding points, the composition is continuous 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 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.
- 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.
Target Context
- The reader will have a description and a proof of 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.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(S_1\): \(\in \{\text{ the subsets of } M_1\}\)
\(S_2\): \(\in \{\text{ the subsets of } M_2\}\)
\(S'_2\): \(\in \{\text{ the subsets of } M_2\}\), such that \(S_2 \subseteq S'_2\)
\(S_3\): \(\in \{\text{ the subsets of } M_3\}\)
\(p\): \(\in S_1\)
\(k\): \(\in \mathbb{N} \cup \{\infty\}\)
\(f_1\): \(: S_1 \to S_2\), \(\in \{\text{ the maps } C^k \text{ at } p\}\)
\(f_2\): \(: S'_2 \to S_3\), \(\in \{\text{ the maps } C^k \text{ at } f_1 (p)\}\)
\(f_2 \circ f_1\): \(: S_1 \to S_3\)
//
Statements:
\(f_2 \circ f_1 \in \{\text{ the maps } C^k \text{ at } p\}\)
//
2: Natural Language 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 \(S_2 \subseteq S'_2\), any point, \(p \in S_1\), any natural number (including 0) or \(\infty\) \(k\), 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 \(C^k\) at \(p\) and \(f_1 (p)\), \(f_2 \circ f_1: S_1 \to S_3\) is \(C^k\) at \(p\).
3: Note
It is crucial that \(S_2\) and \(S'_2\) are regarded to be some subsets of the same \(M_2\): if \(f_2\) is \(C^k\) with \(S'_2\) regarded as a subset of another \(C^\infty\) manifold with boundary, \(M'_2\), with the same set but a different topology or a different atlas with \(M_2\), this proposition cannot be applied.
For an obvious example, for the \(k = 0\) case, if we were allowed to choose a different topology for \(M'_2\), I would choose the discrete topology for \(M'_2\), which would make any \(f_2\) be continuous (any map from any discrete topological space is continuous), and the proposition would imply that for whatever \(f_2\), \(f_2 \circ f_1\) would be continuous, which is of course not true.
We will sometimes call a composition of \(C^k\) maps "legitimate chain of \(C^k\) maps" when the requirements for this proposition are satisfied (when \(S_2\) and \(S'_2\) are regarded to be some subsets of the same \(M_2\) together with \(S_2 \subseteq S'_2\)). That is because we (or at least I) tend to slip in checking the requirements.
4: Proof
Let us suppose that \(k = 0\).
\(f_2 \circ f_1\) is continuous at \(p\), by the proposition that for any maps between any topological spaces continuous at corresponding points, the composition is continuous at the point.
Let us suppose that \(1 \le k\) including \(\infty\).
There are a chart, \((U'_{f_1 (p)} \subseteq M_2, \phi'_{f_1 (p)})\), around \(f_1 (p)\) and a chart, \((U_{f_2 \circ f_1 (p)} \subseteq M_3, \phi_{f_2 \circ f_1 (p)})\), around \(f_2 \circ f_1 (p)\) such that \(f_2 (U'_{f_1 (p)} \cap S'_2) \subseteq U_{f_2 \circ f_1 (p)}\) and \(\phi_{f_2 \circ f_1 (p)} \circ f_2 \circ {\phi'_{f_1 (p)}}^{-1} \vert_{\phi'_{f_1 (p)} (U'_{f_1 (p)} \cap S'_2)}\) is \(C^k\) at \(\phi'_{f_1 (p)} (f_1 (p))\).
As \(f_1\) is continuous at \(p\) (see Note in the definition), there is an open neighborhood, \(U_p \subseteq S_1\), of \(p\) on \(S_1\) such that \(f_1 (U_p) \subseteq U'_{f_1 (p)} \cap S_2\) where \(U_p = U' \cap S_1\) for an open subset, \(U' \subseteq M_1\). There is a chart, \((U'_p \subseteq M_1, \phi'_p)\), around \(p\) such that \(U'_p \subseteq U'\). \(f_1 (U'_p \cap S_1) \subseteq f_1 (U_p) \subseteq U'_{f_1 (p)} \cap S_2 \subseteq U'_{f_1 (p)}\). By the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at 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, \(\phi'_{f_1 (p)} \circ f_1 \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}\) is \(C^k\) at \(\phi'_p (p)\).
\(f_2 \circ f_1 (U'_p \cap S_1) \subseteq f_2 (U'_{f_1 (p)} \cap S_2) \subseteq f_2 (U'_{f_1 (p)} \cap S'_2) \subseteq U_{f_2 \circ f_1 (p)}\). \(\phi_{f_2 \circ f_1 (p)} \circ f_2 \circ f_1 \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)} = \phi_{f_2 \circ f_1 (p)} \circ f_2 \circ {\phi'_{f_1 (p)}}^{-1} \vert_{\phi'_{f_1 (p)} (U'_{f_1 (p)} \cap S'_2)} \circ \phi'_{f_1 (p)} \circ f_1 \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap S_1)}\), which is \(C^k\) at \(\phi'_p (p)\), by 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.
