A definition of \(C^\infty\) embedding
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) map.
- The reader knows a definition of injection.
- The reader knows a definition of immersion.
- The reader knows a definition of homeomorphism.
Target Context
- The reader will have a definition of \(C^\infty\) embedding.
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, \(M_1, M_2\), any \(C^\infty\) map, \(f: M_1 \rightarrow M_2\), such that \(f\) is an injective immersion and the codomain restriction, \(f': M_1 \rightarrow f (M_1)\) is a homeomorphism
2: Note
'Continuous embedding' and '\(C^\infty\) embedding' are different, while any \(C^\infty\) embedding is a continuous embedding, a continuous embedding is not necessarily a \(C^\infty\) embedding.
Usually just 'embedding' is used as it is usually obvious which: non-\(C^\infty\) map cannot be a \(C^\infty\) embedding and (just) embedding-ness of a \(C^\infty\) map is customarily understood to be \(C^\infty\) embedding-ness.
