definition of \(C^\infty\) immersion
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
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 differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
- The reader knows a definition of injection.
Target Context
- The reader will have a definition of \(C^\infty\) immersion.
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:
\( M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(*f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
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Conditions:
\(\forall p \in M_1 (d f_p: T p M_1 \to T p M_2 \in \{\text{ the injections }\})\)
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2: Natural Language Description
For any \(C^\infty\) manifolds with boundary, \(M_1, M_2\), any \(C^\infty\) map, \(f: M_1 \to M_2\), such that its differential, \(d f_p: T p M_1 \to T p M_2\), at any point, \(p \in M_1\), is an injection
