definition of \(C^\infty\) embedding
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader knows a definition of injection.
- The reader knows a definition of \(C^\infty\) 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(*f\): \(: M_1 \to M_2\)
\( f'\): \(: M_1 \to f (M_1) \subseteq M_2\), \(= \text{ the codomain restriction of } f\)
//
Conditions:
\(f \in \{\text{ the injections }\} \cap \{\text{ the } C^\infty \text{ immersions }\}\)
\(\land\)
\(f' \in \{\text{ the homeomorphisms }\}\)
//
2: Natural Language Description
For any \(C^\infty\) manifolds with boundary, \(M_1, M_2\), any map, \(f: M_1 \to M_2\), such that \(f\) is an injective \(C^\infty\) immersion and the codomain restriction, \(f': M_1 \to f (M_1) \subseteq M_2\) is a homeomorphism
3: 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.
Often just 'embedding' is used as it is often 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.
