definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of subspace topology of subset of topological space.
- The reader knows a definition of \(C^\infty\) embedding.
Target Context
- The reader will have a definition of embedded 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:
\( M'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(*M\): \(\subseteq M'\), \(\in \{\text{ the \(C^\infty\) manifolds with boundary }\}\)
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Conditions:
\(M\) has the subspace topology
\(\land\)
\(M\) has an atlas such that the inclusion, \(\iota: M \to M'\), is a \(C^\infty\) embedding
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2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M'\), any \(C^\infty\) manifold with boundary, \(M \subseteq M'\), with the subspace topology and any atlas such that the inclusion, \(\iota: M \to M'\), is a \(C^\infty\) embedding
3: Note
This definition does not mean that for each arbitrary subset, \(M \subseteq M'\), an atlas can be chosen to make \(M\) a \(C^\infty\) manifold with boundary; it means that if an atlas can be chosen for an \(M\), \(M\) (with the subspace topology and the atlas) is an embedded submanifold with boundary of \(M'\).
This definition does not immediately claim that the possible atlas is unique, but in fact, the atlas is unique, although no proof is shown here. In fact, the atlas contains the adopting (a.k.a. slice) charts that correspond to the adopted charts of \(M'\).
'embedded submanifold with boundary of \(M'\)' equals 'regular submanifold with boundary of \(M'\)', which may be defined as any submanifold that satisfies the slicing condition, although no proof is shown that the 2 definitions form the same entity.
