description/proof of that injective map between \(C^\infty\) manifolds with boundary is \(C^\infty\) embedding, if domain restriction of map on each element of open cover is \(C^\infty\) embedding onto open subset of range or codomain
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 open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any injective map between any topological spaces is a continuous embedding if the domain restriction of the map on each element of any open cover is any continuous embedding onto any open subset of the range or the codomain.
- The reader admits the proposition that any map between any \(C^\infty\) manifolds with boundary is \(C^k\) if and only if the domain restriction of the map to each element of any open cover is \(C^k\).
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and any open submanifold with boundary, the differential of the inclusion at each point on the open submanifold with boundary is a 'vectors spaces - linear morphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that any injective map between any \(C^\infty\) manifolds with boundary is a \(C^\infty\) embedding, if the domain restriction of the map on each element of any open cover is a \(C^\infty\) embedding onto an open subset of the range or the codomain.
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the injections }\}\)
\(B\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{U_\beta \vert \beta \in B\}\): \(\in \{\text{ the open covers of } M_1\}\)
//
Statements:
\(\forall U_\gamma \in \{U_\beta \vert \beta \in B\} (f (U_\gamma) \subseteq f (M_1) \subseteq M_2 \in \{\text{ the open subsets of } f (M_1) \text{ or } M_2\} \land f \vert_{U_\gamma} \in \{\text{ the } C^\infty \text{ embeddings }\})\)
\(\implies\)
\(f \in \{\text{ the } C^\infty \text{ embeddings }\}\)
//
\(f \vert_{U_\gamma}\) is regarded as a map from the open submanifold with boundary of \(M_1\).
2: Note
"\(f (U_\gamma) \subseteq f (M_1) \subseteq M_2 \in \{\text{ the open subsets of } f (M_1) \text{ or } M_2\}\)" is important.
3: Proof
Whole Strategy: Step 1: see that \(f\) is a continuous embedding; Step 2: see that \(f\) is \(C^\infty\); Step 3: see that \(f\) is an immersion.
Step 1:
\(f \vert_{U_\gamma}\) is a continuous embedding.
\(f\) is a continuous embedding, by the proposition that any injective map between any topological spaces is a continuous embedding if the domain restriction of the map on each element of any open cover is any continuous embedding onto any open subset of the range or the codomain.
That means that the codomain restriction, \(f': M_1 \to f (M_1)\), is a homeomorphism.
Step 2:
\(f\) is \(C^\infty\), by the proposition that any map between any \(C^\infty\) manifolds with boundary is \(C^k\) if and only if the domain restriction of the map to each element of any open cover is \(C^k\).
Step 3:
Let us see that \(f\) is an immersion.
For each \(m \in M_1\), there is a \(U_\gamma \in \{U_\beta \vert \beta \in B\}\) such that \(m \in U_\gamma\).
Let \(\iota: U_\gamma \to M_1\) be the inclusion.
\(f \vert_{U_\gamma} = f \circ \iota\).
\(d (f \vert_{U_\gamma})_m = d f_{\iota (m)} \circ d \iota_m\).
\(d \iota_m\) is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that for any \(C^\infty\) manifold with boundary and any open submanifold with boundary, the differential of the inclusion at each point on the open submanifold with boundary is a 'vectors spaces - linear morphisms' isomorphism. So, there is the inverse, \({d \iota_m}^{-1}\).
\(d (f \vert_{U_\gamma})_m \circ {d \iota_m}^{-1} = d f_{\iota (m)} \circ d \iota_m \circ {d \iota_m}^{-1} = d f_{\iota (m)}\).
As \(d (f \vert_{U_\gamma})_m\) is injective and \({d \iota_m}^{-1}\) is injective, \(d f_{\iota (m)}\) is injective.
So, \(f\) is an immersion.
Step 4:
So, \(f\) is an injective \(C^\infty\) immersion such that \(f': M_1 \to f (M_1)\) is a homeomorphism.
So, \(f\) is a \(C^\infty\) embedding.
