definition of open submanifold with boundary of \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
Target Context
- The reader will have a definition of open submanifold with boundary of \(C^\infty\) manifold 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:
\( M'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( M\): \(\in \{\text{ the open subsets of } M'\}\) with the subspace topology and the atlas inherited from \(M'\)
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Conditions:
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2: Note
"the atlas inherited from \(M'\)" means this: some charts of \(M'\) cover \(M\); for each of such charts, \((U' \subseteq M', \phi')\), let \((U' \cap M \subseteq M, \phi' \vert_{U' \cap M})\) be a chart of \(M\).
For any open subset, \(M \subseteq M'\), it is indeed a \(C^\infty\) manifold with boundary: \((U' \cap M \subseteq M, \phi' \vert_{U' \cap M})\) is indeed a chart, because \(U' \cap M \subseteq M\) is open, \(\phi' \vert_{U' \cap M} (U' \cap M) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\) is open, and \(\phi' \vert_{U' \cap M}: U' \cap M \to \phi' \vert_{U' \cap M} (U' \cap M) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\) is homeomorphic, because \(\phi' \vert_{U' \cap M} (U' \cap M) = \phi' (U' \cap M)\) while \(\phi': U' \to \phi' (U') \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\) is homeomorphic; for any another chart, \((\widetilde{U'} \cap M \subseteq M, \widetilde{\phi'} \vert_{\widetilde{U'} \cap M})\), \(\widetilde{\phi'} \vert_{\widetilde{U'} \cap M} \circ {\phi' \vert_{U' \cap M}}^{-1} \vert_{\phi' \vert_{U' \cap M} (U \cap M \cap \widetilde{U'} \cap M)}: \phi' \vert_{U' \cap M} (U' \cap M \cap \widetilde{U'} \cap M) \to \widetilde{\phi'} \vert_{\widetilde{U'} \cap M} (U' \cap M \cap \widetilde{U'} \cap M)\) is diffeomorphic, because \(= \widetilde{\phi'} \circ {\phi}^{-1} \vert_{\phi' (U' \cap M \cap \widetilde{U'} \cap M)}\), which is diffeomorphic, because \(\widetilde{\phi'} \circ {\phi}^{-1}\) is a transition for \(M'\) and the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), 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 \(M'\).
\(M\) as the open submanifold with boundary is a codimension-0 embedded submanifold with boundary of \(M'\): for the inclusion, \(\iota: M \to M'\), \(\iota\) is a \(C^\infty\) embedding: it is obviously \(C^\infty\) (see the components function with respect to the charts, \((U' \cap M \subseteq M, \phi' \vert_{U' \cap M})\) and \((U' \subseteq M', \phi')\)); it is injective; for each \(m \in M\), \(d \iota_m\) is injective (in fact, bijective), 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; \(\tilde{\iota}: M \to \iota (M)\) is homeomorphic, because \(M\) has the subspace topology.
