829: Open Submanifold with Boundary of C∞ Manifold with Boundary

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definition of open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.

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Target Context

• The reader will have a definition of open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M$: $\in \{\text{ the open subsets of }{M}^{\prime }\}$ with the subspace topology and the atlas inherited from $M^{\prime }$
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Conditions:
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2: Note

"the atlas inherited from $M^{\prime }$" means this: some charts of $M^{\prime }$ cover $M$; for each of such charts, $(U^{\prime }\subseteq M^{\prime },{\varphi }^{\prime })$, let $(U^{\prime }\cap M\subseteq M,{\varphi }^{\prime }{|}_{U^{\prime }\cap M})$ be a chart of $M$.

For any open subset, $M\subseteq M^{\prime }$, it is indeed a $C^{\mathrm{\infty }}$ manifold with boundary: $(U^{\prime }\cap M\subseteq M,{\varphi }^{\prime }{|}_{U^{\prime }\cap M})$ is indeed a chart, because $U^{\prime }\cap M\subseteq M$ is open, ${\varphi }^{\prime }{|}_{U^{\prime }\cap M}(U^{\prime }\cap M)\subseteq {\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$ is open, and ${\varphi }^{\prime }{|}_{U^{\prime }\cap M}:U^{\prime }\cap M\to {\varphi }^{\prime }{|}_{U^{\prime }\cap M}(U^{\prime }\cap M)\subseteq {\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$ is homeomorphic, because ${\varphi }^{\prime }{|}_{U^{\prime }\cap M}(U^{\prime }\cap M)={\varphi }^{\prime }(U^{\prime }\cap M)$ while ${\varphi }^{\prime }:U^{\prime }\to {\varphi }^{\prime }(U^{\prime })\subseteq {\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$ is homeomorphic; for any another chart, $(\widetilde{U^{\prime }}\cap M\subseteq M,\widetilde{{\varphi }^{\prime }}{|}_{\widetilde{U^{\prime }}\cap M})$, $\widetilde{{\varphi }^{\prime }}{|}_{\widetilde{U^{\prime }}\cap M}\circ {{\varphi }^{\prime }{|}_{U^{\prime }\cap M}}^{-1}{|}_{{\varphi }^{\prime }{|}_{U^{\prime }\cap M}(U\cap M\cap \widetilde{U^{\prime }}\cap M)}:{\varphi }^{\prime }{|}_{U^{\prime }\cap M}(U^{\prime }\cap M\cap \widetilde{U^{\prime }}\cap M)\to \widetilde{{\varphi }^{\prime }}{|}_{\widetilde{U^{\prime }}\cap M}(U^{\prime }\cap M\cap \widetilde{U^{\prime }}\cap M)$ is diffeomorphic, because $=\widetilde{{\varphi }^{\prime }}\circ {\varphi }^{-1}{|}_{{\varphi }^{\prime }(U^{\prime }\cap M\cap \widetilde{U^{\prime }}\cap M)}$, which is diffeomorphic, because $\widetilde{{\varphi }^{\prime }}\circ {\varphi }^{-1}$ is a transition for $M^{\prime }$ and the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point can be applied; $M$ is Hausdorff and 2nd-countable as a topological subspace of Hausdorff and 2nd-countable $M^{\prime }$.

$M$ as the open submanifold with boundary is a codimension-0 embedded submanifold with boundary of $M^{\prime }$: for the inclusion, $\iota :M\to M^{\prime }$, $\iota$ is a $C^{\mathrm{\infty }}$ embedding: it is obviously $C^{\mathrm{\infty }}$ (see the components function with respect to the charts, $(U^{\prime }\cap M\subseteq M,{\varphi }^{\prime }{|}_{U^{\prime }\cap M})$ and $(U^{\prime }\subseteq M^{\prime },{\varphi }^{\prime })$); it is injective; for each $m\in M$, $d{\iota }_m$ is injective (in fact, bijective), by the proposition that for any $C^{\mathrm{\infty }}$ 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; $\tilde{\iota }:M\to \iota (M)$ is homeomorphic, because $M$ has the subspace topology.

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References

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