description/proof of that restricted \(c^\infty\) vectors bundle w.r.t. embedded submanifold with boundary is embedded submanifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of restricted (C^\infty\) vectors bundle.
- The reader knows a definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and its any embedded submanifold with boundary, around each point on the submanifold with boundary, there is a trivializing open subset for the manifold with boundary whose intersection with the submanifold with boundary is a chart domain for the submanifold with boundary.
- 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 of the manifold with boundary.
- The reader admits the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, the restricted \(C^\infty\) vectors bundle w.r.t. any embedded submanifold with boundary is an embedded submanifold with boundary.
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:
\((E', M', \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles }\}\)
\(M\): \(\in \{\text{ the } d \text{ -dimensional embedded submanifolds with boundary of } M'\}\)
\((E, M, \pi)\): \(= \text{ the restricted } C^\infty \text{ vectors bundle }\)
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Statements:
\(E \in \{\text{ the embedded submanifolds with boundary of } E'\}\)
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2: Note
We already know that \(E\) is an immersed submanifold with boundary of \(E'\), by Note for the definition of restricted (C^\infty\) vectors bundle.
Definition-wise, \(E\) does not need to be an embedded submanifold with boundary of \(E'\) for an embedded submanifold with boundary \(M\), but in fact, \(E\) is an embedded submanifold with boundary of \(E'\), which we are going to prove.
3: Proof
Whole Strategy: Step 1: see that \(E\) has the subspace topology of \(E'\) by the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space; Step 2: see that the inclusion \(\iota: E \to E'\) is a \(C^\infty\) embedding.
Step 1:
Let us see that \(E\) has the subspace topology of \(E'\).
We are going to apply the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space.
Let \(m \in M\) be any.
There is a trivializing open subset around \(m\) for \(M'\), \(U'_m \subseteq M'\), such that \(U_m := U'_m \cap M\) is a chart domain for \(M\), by the proposition that for any \(C^\infty\) manifold with boundary and its any embedded submanifold with boundary, around each point on the submanifold with boundary, there is a trivializing open subset for the manifold with boundary whose intersection with the submanifold with boundary is a chart domain for the submanifold with boundary. Let the chart be \((U_m \subseteq M, \phi_m)\).
The corresponding \(U'_\beta\) s and \((U_\beta \subseteq M, \phi_\beta)\) s are legitimate as the ones used for constructing the topology and the atlas of \(E\) in the definition of restricted \(C^\infty\) vectors bundle: \(U_\beta\) is an embedded submanifold with boundary of \(M\), by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary and \(U_\beta\) is an embedded submanifold with boundary of \(M'\), 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 of the manifold with boundary. We take \(\{(U_\beta \subseteq M, \phi_\beta) \vert \beta \in B\}\) and \(\{U'_\beta \vert \beta \in B\}\) and \(\{\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^k \vert \beta \in B\}\), accordingly.
For each point on \(E\), let us take the open neighborhood of the point on \(E'\) for the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space as \(\pi'^{-1} (U'_\beta)\).
Let us see that \(\pi'^{-1} (U'_\beta)\) is indeed one the theorem requires.
\(\pi'^{-1} (U'_\beta)\) is indeed an open neighborhood of the point on \(E'\), because the point is contained in it and \(U'_\beta\) is open on \(M'\) and \(\pi'\) is continuous.
Is \(\pi'^{-1} (U'_\beta) \cap E\) an open subset of \(E\)?
Yes, because \(\pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi'^{-1} (M) = \pi'^{-1} (U'_\beta \cap M)\), by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets, \( = \pi^{-1} (U_\beta)\), which is open on \(E\), because \(U_\beta\) is open on \(M\) and \(\pi\) is continuous.
Does \(\pi'^{-1} (U'_\beta) \cap E \subseteq E \text{ as the subspace of } E\) equal \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta) \text{ as the subspace of } \pi'^{-1} (U'_\beta)\)?
\(\pi'^{-1} (U'_\beta) \cap E \subseteq E\) is nothing but \(\pi^{-1} (U_\beta)\), and \(\Phi_\beta: \pi^{-1} (U_\beta) \to U_\beta \times \mathbb{R}^k\) is homeomorphic.
For each subset, \(S \subseteq \pi'^{-1} (U'_\beta) \cap E\), \(S \cap E = S \cap \pi^{-1} (U_\beta)\), because while \(S \cap \pi^{-1} (U_\beta) \subseteq S \cap E\) is obvious, for each \(p \in S \cap E\), \(p \in S \subseteq \pi'^{-1} (U'_\beta)\), which implies that \(\pi' (p) = \pi (p) \in U'_\beta\), but also \(\pi (p) \in M\), so, \(\pi (p) \in U'_\beta \cap M = U_\beta\), so, \(p \in \pi^{-1} (U_\beta)\), and \(p \in S \cap \pi^{-1} (U_\beta)\), and so, \(S \cap E \subseteq S \cap \pi^{-1} (U_\beta)\).
Especially, \(\pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta)\).
\(\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^k\) is homeomorphic, and \(\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}: \pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta) \to \Phi'_\beta (\pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta)) = \Phi'_\beta (\pi^{-1} (U_\beta)) = \Phi_\beta (\pi^{-1} (U_\beta)) = U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is homeomorphic, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous. Note that it is homeomorphic with the domain, \(\pi'^{-1} (U'_\beta) \cap E\), regarded as the subspace of \(\pi'^{-1} (U'_\beta)\) and the codomain, \(U_\beta \times \mathbb{R}^k\), regarded as the subspace of \(U'_\beta \times \mathbb{R}^k\).
Let us think of the identity map, \(id: U_\beta \times \mathbb{R}^k \to U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\). In fact, \(U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) regarded as the subspace of \(U'_\beta \times \mathbb{R}^k\) is nothing but \(U_\beta \times \mathbb{R}^k\), because \(M\) is a topological subspace of \(M'\).
Now, for each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq E = \pi^{-1} (U_\beta)\), \(U\), \(\Phi_\beta (U) \subseteq U_\beta \times \mathbb{R}^k\) is open; \(id (\Phi_\beta (U)) \subseteq U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is open; \({\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}}^{-1} (id (\Phi_\beta (U))) \subseteq \pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\) is open. In fact, that \({\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}}^{-1} \circ id \circ \Phi_\beta\) is the identity map, because \(\Phi_\beta\) is the restriction of \(\Phi'_\beta\). That shows that each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq E\) is open on \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\).
Likewise, for each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\), \(U\), \(\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U) \subseteq U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is open; \({id}^{-1} (\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U)) \subseteq U_\beta \times \mathbb{R}^k\) is open; \({\Phi_\beta}^{-1} ({id}^{-1} (\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U)) \subseteq \pi'^{-1} (U'_\beta) \cap E \subseteq E\) is open. In fact, that \({\Phi_\beta}^{-1} \circ {id}^{-1} \circ \Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}\) is the identity map, because \(\Phi_\beta\) is the restriction of \(\Phi'_\beta\). That shows that each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\) is open on \(\pi'^{-1} (U'_\beta) \cap E \subseteq E\).
So, yes, \(\pi'^{-1} (U'_\beta) \cap E \subseteq E \text{ as the subspace of } E\) equals \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta) \text{ as the subspace of } \pi'^{-1} (U'_\beta)\).
So, \(E\) is the topological subspace of \(E'\).
Step 2:
Let us see that the inclusion \(\iota: E \to E'\) is a \(C^\infty\) embedding.
We already know that \(\iota\) is a \(C^\infty\) immersion, because \(E\) is an immersed submanifold with boundary of \(E'\).
The codomain restriction, \(\iota': E \to \iota (E) \subseteq E'\), is homeomorphic, because \(E\) has the subspace topology of \(E'\) by Step 1, and \(\iota'\) is the identity map.
So, \(\iota\) is a \(C^\infty\) embedding.
