definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point
Topics
About: \(C^\infty\) manifold
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: Structured Description
Here is the rules of Structured Description.
Entities:
\( M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary } \}\)
\( M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary } \}\)
\( S_1\): \(\subseteq M_1\)
\( S_2\): \(\subseteq M_2\)
\( s\): \(\in S_1\)
\(*f\): \(: S_1 \to S_2\)
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Conditions:
\(\exists U_s \subseteq M_1 \in \{\text{ the open neighborhoods of } s\}, \exists U_{f (s)} \subseteq M_2 \in \{\text{ the open neighborhoods of } f (s)\} (f \vert_{U_s \cap S_1}: U_s \cap S_1 \to U_{f (s)} \cap S_2 \in \{\text{ the diffeomorphisms }\})\)
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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 \(s\).
