description/proof of that for \(C^\infty\) manifold with boundary, immersed submanifold with boundary of immersed submanifold with boundary is immersed submanifold with boundary of manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, any immersed submanifold with boundary of any immersed submanifold with boundary is an immersed submanifold with boundary of the 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:
\(M''\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M'\): \(\in \{\text{ the immersed submanifolds with boundary of } M''\}\)
\(M\): \(\in \{\text{ the immersed submanifolds with boundary of } M'\}\)
//
Statements:
\(M \in \{\text{ the immersed submanifolds with boundary of } M''\}\)
//
2: Proof
Whole Strategy: Step 1: take the inclusions, \(\iota': M' \to M''\) and \(\iota: M \to M'\), and see that the inclusion, \(\iota' \circ \iota: M \to M''\), is a \(C^\infty\) immersion.
Step 1:
Let \(\iota': M' \to M''\) be the inclusion, which is a \(C^\infty\) immersion, by the definition of immersed submanifold with boundary.
Let \(\iota: M \to M'\) be the inclusion, which is a \(C^\infty\) immersion, by the definition of immersed submanifold with boundary.
\(\iota' \circ \iota: M \to M''\) is the inclusion, and is a \(C^\infty\) immersion, by the proposition that the composition of any \(C^\infty\) immersions is a \(C^\infty\) immersion.
So, \(M\) is an immersed submanifold with boundary of \(M''\).
