definition of immersed 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.
- The reader knows a definition of \(C^\infty\) immersion.
Target Context
- The reader will have a definition of immersed 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 a not-necessarily-subspace topology
\(\land\)
\(M\) has an atlas such that the inclusion, \(\iota: M \to M'\), is a \(C^\infty\) immersion
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2: Note
This definition does not mean that for each arbitrary subset, \(M \subseteq M'\), a topology and an atlas can be chosen to make \(M\) a \(C^\infty\) manifold with boundary; it means that if a topology and an atlas can be chosen for an \(M\), \(M\) (with the topology and the atlas) is an immersed submanifold with boundary of \(M'\).
