definition of rank of \(C^\infty\) map between \(C^\infty\) manifolds with boundary 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 rank of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
Orientation
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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:
\( M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary } \}\)
\( M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary } \}\)
\( f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\( m\): \(\in M_1\)
\( d f_m\): \(: T_mM_1 \to T_{f (m)}M_2\), \(= \text{ the differential of } f \text{ at } m\)
\(*Rank (f)_m\): \(= Rank (d f_m)\)
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Conditions:
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2: Note
In general, the rank of \(f\) changes point to point.
When the rank of \(f\) is the same at all the points of \(M_1\), \(f\) is called to have "constant rank".
When at each point of \(M_1\), there is a neighborhood of the point over which \(f\) has the same rank, \(f\) is called to have "locally constant rank".
