description/proof of that for \(C^\infty\) manifold with boundary and embedded submanifold with boundary, inverse of codomain restricted inclusion is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader knows a definition of double of \(C^\infty\) manifold with boundary.
- 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.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(M'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M\): \(\in \{\text{ the embedded submanifolds with boundary of } M'\}\)
\(\iota\): \(: M \to M'\), \(= \text{ the inclusion }\)
\(\iota'\): \(: M \to \iota (M) \subseteq M'\), \(= \text{ the codomain restriction of } \iota\)
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Statements:
\({\iota'}^{-1}: \iota (M) \subseteq M' \to M \in \{\text{ the } C^\infty \text{ maps }\}\)
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2: Note
This logic cannot be applied for immersed submanifold with boundary, which is not guaranteed to have any adopted chart or adopting chart.
\(\iota (M)\) equals \(M\) sets-wise, but is just a subset of \(M'\), so, \(\iota (M)\) and \(M\) need to be distinguished.
The \(C^\infty\)-ness of \({\iota}'^{-1}\) is according to the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
An immediate corollary is that when \(f: N \to M'\) such that \(f (N) \subseteq \iota (M)\) is \(C^\infty\), \(f': N \to M\), with the codomain of \(f\) replaced, is \(C^\infty\), because \(f' = {\iota'}^{-1} \circ f\) while \(f: N \to \iota (M) \subseteq M'\) and \({\iota'}^{-1}: \iota (M) \subseteq M' \to M\) are \(C^\infty\), by 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: \(f: N \to \iota (M) \subseteq M'\) is \(C^\infty\), 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 or expansion on any codomain that contains the range is \(C^k\) at the point.
As a caveat, when one tries to prove the \(C^\infty\)-ness of \(f: N_1 \to M \to N_2\) by proving the \(C^\infty\)-nesses of \(N_1 \to M\) and \(M \to N_2\), the \(C^\infty\)-ness of \(N_1 \to M'\) guarantees the \(C^\infty\)-ness of \(N_1 \to M\) by \(N_1 \to M' \to M\) if \(M\) is really an embedded submanifold with boundary of \(M'\) by this proposition, but if \(M\) is an immersed submanifold with boundary of \(M'\), the \(C^\infty\)-ness of \(N_1 \to M'\) does not guarantee the \(C^\infty\)-ness of \(N_1 \to M\), at least by this proposition.
3: Proof
Whole Strategy: do it in the 2 parts: the 1st part: suppose that \(M'\) has the empty boundary; the 2nd part: suppose that \(M'\) has a nonempty boundary; Step 1: suppose that \(M'\) has the empty boundary; Step 2: for each \(m' \in \iota (M)\), take an adopted chart, \((U'_{m'} \subseteq M', \phi'_{m'})\), and the corresponding adopting chart, \((U_{m'} \subseteq M, \phi_{m'})\), and see that \({\iota'}^{-1} (U'_{m'} \cap \iota (M)) \subseteq U_{m'}\); Step 3: see that \(\phi_{m'} \circ {\iota'}^{-1} \circ {\phi'_{m'}}^{-1} \vert_{\phi'_{m'} (U'_{m'} \cap \iota (M))}: \phi'_{m'} (U'_{m'} \cap \iota (M)) \to \phi_{m'} (U_{m'})\) is \(C^\infty\) at \(\phi'_{m'} (m')\); Step 4: suppose that \(m'\) has a nonempty boundary; Step 5: take the double of \(M'\), \(D (M')\), which has the empty boundary, and apply the 1st part conclusion to conclude the proposition.
Step 1:
For the 1st part, let us suppose that \(M'\) has the empty boundary.
The reason is that we use the slice charts criterion, which requires \(M'\) to be without boundary.
Step 2:
For each \(m' \in \iota (M)\), let us take an adopted chart, \((U'_{m'} \subseteq M', \phi'_{m'})\), and the corresponding adopting chart, \((U_{m'} \subseteq M, \phi_{m'})\), where \(U_{m'} = U'_{m'} \cap \iota (M)\) and \(\phi_{m'} = \pi_J \circ \phi'_{m'} \vert_{U_{m'}}\), where \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d\) is the projection.
\({\iota'}^{-1} (U'_{m'} \cap \iota (M)) \subseteq U_{m'}\), obviously.
Step 3:
Whether \({\iota'}^{-1}\) is \(C^\infty\) at \(m'\) is about whether \(\phi_{m'} \circ {\iota'}^{-1} \circ {\phi'_{m'}}^{-1} \vert_{\phi'_{m'} (U'_{m'} \cap \iota (M))}: \phi'_{m'} (U'_{m'} \cap \iota (M)) \to \phi_{m'} (U_{m'})\) is \(C^\infty\) at \(\phi'_{m'} (m')\), by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
\(\phi_{m'} \circ {\iota'}^{-1} \circ {\phi'_{m'}}^{-1} \vert_{\phi'_{m'} (U'_{m'} \cap \iota (M))}\) is \((x^1, ..., x^d, 0, ..., 0) \mapsto (x^1, ..., x^d)\), by the natures of adopted chart and adopting chart.
It can be extended to the \(C^\infty\) map, \(f': \mathbb{R}^{d'} \to \mathbb{R}^d, (x^1, ..., x^d, 0, ..., 0) \mapsto (x^1, ..., x^d)\).
So, \(\phi_{m'} \circ {\iota'}^{-1} \circ {\phi'_{m'}}^{-1} \vert_{\phi'_{m'} (U'_{m'} \cap \iota (M))}\) is indeed \(C^\infty\) at \(\phi'_{m'}\), and so, \({\iota'}^{-1}\) is \(C^\infty\) at \(m'\).
As \(m' \in \iota (M)\) is arbitrary, \({\iota'}^{-1}\) is \(C^\infty\) all over \(\iota (M)\).
Step 4:
For the 2nd part, let us suppose that \(M'\) has a nonempty boundary.
Step 5:
Let us take the double of \(M'\), \(D (M')\).
\(D (M')\) is a \(C^\infty\) manifold without boundary that has a regular domain, \(\widetilde{M'}\), that is diffeomorphic to \(M'\).
Let a diffeomorphism be \(g: M' \to \widetilde{M'}\).
Let the restriction of \(g\) be \(g': M \to g (M) := \widetilde{M} \subseteq \widetilde{M'}\), where \(\widetilde{M}\) is the embedded submanifold with boundary of \(\widetilde{M'}\) that (\(\widetilde{M}\)) is diffeomorphic to \(M\): it is "restriction" map-between-sets-wise, but the domain and the codomain are replaced by the \(C^\infty\) manifolds with boundary, \(M\) and \(\widetilde{M}\). \(\widetilde{M}\) can be indeed construed so, because while \(\widetilde{M'}\) is diffeomorphic to \(M'\), \(\widetilde{M}\) is to \(\widetilde{M'}\) what \(M\) is to \(M'\).
\(\widetilde{M}\) is an embedded submanifold with boundary of \(D (M')\), because \(\widetilde{M'}\) is an embedded submanifold with boundary (in fact, a regular domain) of \(D (M')\), by the proposition that for any \(C^\infty\) manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary.
Let \(\widetilde{\iota}: \widetilde{M} \to D (M')\) be the inclusion.
Let \(\widetilde{\iota}': \widetilde{M} \to \widetilde{\iota} (\widetilde{M}) \subseteq D (M')\) be the codomain restriction of \(\widetilde{\iota}\).
By the 1st part conclusion, \({\widetilde{\iota}'}^{-1}: \widetilde{\iota} (\widetilde{M}) \subseteq D (M') \to \widetilde{M}\) is \(C^\infty\).
Let \(\widetilde{\tau}: \widetilde{M'} \to D (M')\) be the inclusion, which is \(C^\infty\), because \(\widetilde{M'}\) is an embedded submanifold with boundary of \(D (M')\).
Let \(\widetilde{\tau}': \widetilde{M'} \to \widetilde{\tau} (\widetilde{M'}) \subseteq D (M')\) be the codomain restriction of \(\widetilde{\tau}\), also which is \(C^\infty\).
\({\iota'}^{-1}: \iota (M) \subseteq M' \to M = g'^{-1} \circ {\widetilde{\iota}'}^{-1} \circ \widetilde{\tau}' \circ g \vert_{\iota (M)}: \iota (M) \subseteq M' \to g (\iota (M)) = \widetilde{\iota} (\widetilde{M}) \subseteq \widetilde{M'} \to \widetilde{\tau} (\widetilde{\iota} (\widetilde{M})) = \widetilde{\iota} (\widetilde{M}) \subseteq D (M') \to \widetilde{M} \to M, m' \mapsto g (m') \mapsto \widetilde{\tau} (g (m')) = g (m') \mapsto g (m') \to m'\).
As it is a composition of \(C^\infty\) maps, it is \(C^\infty\), by 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.
