description/proof of that subset of \(C^\infty\) manifold with boundary that satisfies local-slice condition is embedded submanifold with boundary with subspace topology and adopting atlas
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of subset of \(C^\infty\) manifold with boundary that satisfies local-slice condition.
- The reader knows a definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of locally topologically closed upper half Euclidean topological space.
- The reader admits the proposition that any subspace of any Hausdorff topological space is Hausdorff.
- The reader admits the proposition that any subspace of any 2nd countable topological space is 2nd countable.
- 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 \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.
Target Context
- The reader will have a description and a proof of the proposition that any subset of any \(C^\infty\) manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.
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:
\(M\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\(S\): \(\subseteq M\)
\(d\): \(\in \mathbb{N}\) such that \(d \le d'\)
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Statements:
\(S \in \{\text{ the subsets of } M \text{ that satisfy local-slice condition with } d\}\)
\(\implies\)
\(S \text{ with the subspace topology and the adopting atlas } \in \{\text{ the embedded submanifolds with boundary of } M\}\)
//
2: Note
Although \(S\) is suppose to satisfy the local-slice condition (instead of the local-slice-or-half-slice condition), \(S\) may have a nonempty boundary, because a boundary point of \(M\) may become a boundary point of \(S\): for example, when \(M = \mathbb{H}^2\), \(S = \{s \in \mathbb{H}^2 \vert s^1 = 1\}\) has the nonempty boundary: \(\{(1, 0)\}\) is the boundary.
Even when \(M\) has a nonempty boundary, \(S\) may have the empty boundary: for example, when \(M = \mathbb{H}^2\), \(S = \{s \in \mathbb{H}^2 \vert s^2 = 1\}\) has the empty boundary.
3: Proof
Whole Strategy: Step 1: see that \(S\) is Hausdorff; Step 2: see that \(S\) is 2nd-countable; Step 3: see that \(S\) is locally topologically closed upper half Euclidean; Step 4: see that the adapting atlas is \(C^\infty\) compatible; Step 5: see that the inclusion, \(\iota: S \to M\), is a \(C^\infty\) embedding.
Step 1:
\(S\) is Hausdorff, by the proposition that any subspace of any Hausdorff topological space is Hausdorff.
Step 2:
\(S\) is 2nd-countable, by the proposition that any subspace of any 2nd countable topological space is 2nd countable.
Step 3:
Let us see that \(S\) is locally topologically closed upper half Euclidean.
Let \(s \in S\) be any.
There are an adopted chart around \(s\), \((U_s \subseteq M, \phi_s)\), a \(J \subseteq \{1, ..., d'\} = (j_1, ..., j_d)\), and a \(u \in U_s\) such that \(U_s \cap S = S_{J, u} (U)\).
When \((U_s \subseteq M, \phi_s)\) is an interior chart, \(\pi_J \circ \phi_s \vert_{U_s \cap S}: U_s \cap S \subseteq U \to \pi_J (\phi_s (U_s \cap S)) \subseteq \mathbb{R}^d\) is a homeomorphism from the open neighborhood of \(s\) on \(S\) into the open subset of \(\mathbb{R}^d\), as has been seen in Note for the definition of J-slice of chart domain with respect to point.
When \((U_s \subseteq M, \phi_s)\) is a boundary chart, \(\pi_J \circ \phi_s \vert_{U_s \cap S}: U_s \cap S \subseteq U \to \pi_J (\phi_s (U_s \cap S)) \subseteq \mathbb{H}^d \text{ or } \mathbb{R}^d\) (according to \(d' \in J\) or \(d' \notin J\)) is a homeomorphism from the open neighborhood of \(s\) on \(S\) into the open subset of \(\mathbb{H}^d\) or \(\mathbb{R}^d\), as has been seen in Note for the definition of J-slice of chart domain with respect to point.
That means that \(S\) is locally topologically closed upper half Euclidean: also any open subset of \(\mathbb{R}^d\) is allowed, as has been mentioned in Note for the definition of locally topologically closed upper half Euclidean topological space.
Step 4:
So, let \(\{(U_s \cap S \subseteq S, \pi_J \circ \phi_s \vert_{U_s \cap S}) \vert s \in S\}\) be an atlas for \(S\).
Let us see that the atlas is \(C^\infty\) compatible.
Let \((U_s \cap S \subseteq S, \pi_J \circ \phi_s \vert_{U_s \cap S})\) and \((U_{s'} \cap S \subseteq S, \pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S})\) be any charts such that \((U_s \cap S) \cap (U_{s'} \cap S) \neq \emptyset\).
Let us think of \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}: \pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S)) \to \pi_{J'} \circ \phi_{s'} ((U_{s'} \cap S) \cap (U_s \cap S))\).
We are going to apply 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, but the point is to carefully check that it is a legitimate chain of \(C^\infty\) maps.
Let us suppose that \((U_s \subseteq M, \phi_s)\) is an interior chart.
\(\pi_J \circ \phi_s \vert_{U_s \cap S} = \pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})} \circ \phi_s \vert_{U_s \cap S}\), where \(\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})}: S_{J, \phi (u)} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'} \to \mathbb{R}^d\) is a diffeomorphism.
\((\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
Let us suppose that \((U_s \subseteq M, \phi_s)\) is a boundary chart.
\(\pi_J \circ \phi_s \vert_{U_s \cap S} = \pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ \phi_s \vert_{U_s \cap S}\), where \(\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}: S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'} \to \mathbb{H}^d \text{ or } \mathbb{R}^d\) (according to \(d' \in J\) or \(d' \notin J\)) is a diffeomorphism.
\((\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
Let us suppose that \((U_{s'} \subseteq M, \phi_{s'})\) is an interior chart.
\(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})} \circ \phi_{s'} \vert_{U_{s'} \cap S}\), where \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})}: S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'} \to \mathbb{R}^d\) is a diffeomorphism.
Let us suppose that \((U_{s'} \subseteq M, \phi_{s'})\) is a boundary chart.
\(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ \phi_{s'} \vert_{U_{s'} \cap S}\), where \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}: S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'} \to \mathbb{H}^d \text{ or } \mathbb{R}^d\) (according to \(d' \in J\) or \(d' \notin J\)) is a diffeomorphism.
Now, when \((U_{s'} \subseteq M, \phi_{s'})\) is an interior chart and \((U_s \subseteq M, \phi_s)\) is an interior chart, \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})} \circ (\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}) \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
But \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})}\) is \(C^\infty\) from \(S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'}\) into \(\mathbb{R}^d\), \(\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}\) is \(C^\infty\) from \(\phi_s (U_{s'} \cap U_s) \subseteq \mathbb{R}^{d'}\) into \(\mathbb{R}^{d'}\), and \((\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\) is \(C^\infty\) from \(\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S)) \subseteq \mathbb{R}^d\) into \(\mathbb{R}^{d'}\), so, it is \(C^\infty\) as a legitimate chain of \(C^\infty\) maps, 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.
When \((U_{s'} \subseteq M, \phi_{s'})\) is a boundary chart and \((U_s \subseteq M, \phi_s)\) is an interior chart, \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ (\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}) \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
But \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\) is \(C^\infty\) from \(S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'}\) into \(\mathbb{H}^d\) or \(\mathbb{R}^d\), \(\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}\) is \(C^\infty\) from \(\phi_s (U_{s'} \cap U_s) \subseteq \mathbb{R}^{d'}\) into \(\mathbb{H}^{d'}\), and \((\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'})})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\) is \(C^\infty\) from \(\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S)) \subseteq \mathbb{R}^d\) into \(\mathbb{R}^{d'}\), so, it is \(C^\infty\) as a legitimate chain of \(C^\infty\) maps, likewise.
When \((U_{s'} \subseteq M, \phi_{s'})\) is an interior chart and \((U_s \subseteq M, \phi_s)\) is a boundary chart, \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})} \circ (\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}) \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
But \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'})}\) is \(C^\infty\) from \(S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \subseteq \mathbb{R}^{d'}\) into \(\mathbb{R}^d\), \(\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}\) is \(C^\infty\) from \(\phi_s (U_{s'} \cap U_s) \subseteq \mathbb{H}^{d'}\) into \(\mathbb{R}^{d'}\), and \((\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\) is \(C^\infty\) from \(\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S)) \subseteq \mathbb{H}^d \text{ or } \mathbb{R}^d\) into \(\mathbb{H}^{d'}\), so, it is \(C^\infty\) as a legitimate chain of \(C^\infty\) maps, likewise.
When \((U_{s'} \subseteq M, \phi_{s'})\) is a boundary chart and \((U_s \subseteq M, \phi_s)\) is a boundary chart, \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)} \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))} = \pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}} \circ (\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}) \circ (\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\).
But \(\pi_{J'} \vert_{S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}}\) is \(C^\infty\) from \(S_{J', \phi_{s'} (u')} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'} \subseteq \mathbb{H}^{d'}\) into \(\mathbb{H}^d\) or \(\mathbb{R}^d\), \(\phi_{s'} \circ \phi_s^{-1} \vert_{\phi_s (U_{s'} \cap U_s)}\) is \(C^\infty\) from \(\phi_s (U_{s'} \cap U_s) \subseteq \mathbb{H}^{d'}\) into \(\mathbb{H}^{d'}\), and \((\pi_J \vert_{S_{J, \phi (u)} (\mathbb{R}^{d'}) \cap \mathbb{H}^{d'}})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\) is \(C^\infty\) from \(\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S)) \subseteq \mathbb{H}^d \text{ or } \mathbb{R}^d\) into \(\mathbb{H}^{d'}\), so, it is \(C^\infty\) as a legitimate chain of \(C^\infty\) maps, likewise.
So, \(\pi_{J'} \circ \phi_{s'} \vert_{U_{s'} \cap S} \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1} \vert_{\pi_J \circ \phi_s ((U_{s'} \cap S) \cap (U_s \cap S))}\) is \(C^\infty\) in any case.
So, the atlas is \(C^\infty\) compatible.
So, \(S\) is a \(C^\infty\) manifold with boundary.
Step 5:
Let \(\iota: S \to M\) be the inclusion.
Let us see that \(\iota\) is a \(C^\infty\) immersion.
Let \(s \in S\) be any.
Let us choose the adopted chart, \((U_s \subseteq M, \phi_s)\), and the corresponding adopting char, \((U_s \cap S \subseteq S, \pi_J \circ \phi_s \vert_{U_s \cap S})\).
The components function, \(\phi_s \circ \iota \circ (\pi_J \circ \phi_s \vert_{U_s \cap S})^{-1}\) is like \(: (x^{j_1}, ..., x^{j_d}) \mapsto (\phi_s (u)^1, ..., x^{j_1}, ..., x^{j_d}, ..., \phi_s (u)^{d'})\), where whether the 1st component is really \(\phi_s (u)^1\) or \(x^{j_1}\) and whether the last component is really \(\phi_s (u)^{d'}\) or \(x^{j_d}\) really depend on \(J\), but anyway, it is \(C^\infty\), and its differential is injective, by the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this, so, \(\iota\) is a \(C^\infty\) immersion.
The codomain restriction of \(\iota\), \(\iota: S \to \iota (S) \subseteq M\), is a homeomorphism, because \(S\) has the subspace topology.
So, \(S\) is an embedded submanifold with boundary.
