description/proof of that for \(C^\infty\) embedding between \(C^\infty\) manifolds with boundary, restriction of embedding on embedded submanifold with boundary domain is \(C^\infty\) embedding
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) embedding.
- The reader knows a definition of embedded submanifold with boundary 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 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 \(C^\infty\) embedding between any \(C^\infty\) manifolds with boundary, the restriction of the embedding on any embedded submanifold with boundary domain is a \(C^\infty\) embedding.
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:
\(M'_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M'_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ embeddings }\}\)
\(M_1\): \(\in \{\text{ the embedded submanifolds with boundary of } M'_1\}\)
\(f'\): \(: M_1 \to M_2, m \mapsto f (m)\)
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Statements:
\(f' \in \{\text{ the } C^\infty \text{ embeddings }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f'\) is injective; Step 2: see that \(f'\) is \(C^\infty\); Step 3: see that \(f'\) is a \(C^\infty\) immersion; Step 4: see that the codomain restriction of \(f'\), \(f'': M_1 \to f' (M_1)\), is homeomorphic; Step 5: conclude the proposition.
Step 1:
\(f'\) is injective as \(f\) is so.
Step 2:
Let \(\iota: M_1 \to M'_1\) be the inclusion.
\(\iota\) is a \(C^\infty\) embedding, because \(M_1\) is an embedded submanifold with boundary of \(M'_1\).
\(f' = f \circ \iota\) 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.
Step: 3
The differential is, \(d f' = d f \circ d \iota\). \(d \iota\) is injective, because \(\iota\) is a \(C^\infty\) embedding. \(d f\) is injective, because \(f\) is a \(C^\infty\) embedding. So, \(d f'\) is injective.
So, \(f'\) is a \(C^\infty\) immersion.
Step 4:
Let \(f'': M_1 \to f' (M_1)\) be the codomain restriction of \(f'\).
Let us see that \(f''\) is homeomorphic.
Let \(\iota': M_1 \to \iota (M_1)\) be the codomain restriction of \(\iota\), which is homeomorphic because \(\iota\) is a \(C^\infty\) embedding.
Let \(f''': \iota (M_1) \to f (\iota (M_1))\) be the domain and the codomain restriction of \(f\), which is homeomorphic because \(f\) is a \(C^\infty\) embedding, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
\(f'' = f''' \circ \iota'\), which is homeomorphic as a composition of homeomorphisms.
Step 5:
So, \(f'\) is an injective \(C^\infty\) embedding whose codomain restriction is homeomorphic, which means that \(f'\) is a \(C^\infty\) embedding.
3: Note
As an immediate corollary, when \(M_1\) is an open submanifold with boundary, \(f'\) is a \(C^\infty\) embedding: any open submanifold with boundary is an embedded submanifold with boundary.
