A definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds 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: Definition
For any \(C^\infty\) manifolds with (possibly empty) boundary, \(M_1, M_2\), and any subsets, \(S_1 \subseteq M_1, S_2 \subseteq M_2\), any \(C^\infty\) bijection, \(f: S_1 \to S_2\), such that also the inverse, \(f^{-1}: S_2 \to S_1\) is \(C^\infty\), where \(C^\infty\)-ness is by the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
2: Note
It is important to be aware that this definition is not by the existence of a diffeomorphic extension: there are a \(C^\infty\) extension for a direction and a \(C^\infty\) extension for the other direction, but the \(C^\infty\) extensions are not dictated to be the inverses to each other or each \(C^\infty\) extension is not dictated to be even injective. If you want to claim that there is a diffeomorphic extension, you will have to prove it.
