A definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point
Topics
About: \(C^\infty\) manifold with boundary
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point.
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, \(M_1, M_2\), any subsets, \(S_1 \subseteq M_1, S_2 \subseteq M_2\), and any point, \(p \in S_1\), any map, \(f: S_1 \to S_2\), such that there are an open neighborhood, \(U_p \subseteq M_1\), of \(p\) and an open neighborhood, \(U_{f (p)} \subseteq M_2\), of \(f (p) \in S_2\), such that \(f \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f (p)} \cap S_2\) is a diffeomorphism
2: Note
Typically, \(S_1 = M_2\) and \(S_2 = M_2\), and then, \(f\) is a map between some \(C^\infty\) manifolds with boundary local diffeomorphic at \(p\).
