definition of orientation of \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of orientation of point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of local \(C^\infty\) frame on \(C^\infty\) vectors bundle.
Target Context
- The reader will have a definition of orientation of \(C^\infty\) manifold with boundary.
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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( O\): \(= \cup_{m \in M} O_m\), where \(O_m\) is the set of the orientations of \(T_mM\)
\(*o\): \(: M \to O\) such that \(o (m) \in O_m\)
//
Conditions:
\(\forall m \in M (\exists U_m \subseteq M \in \{\text{ the open neighborhoods of } m\}, \exists (v_1, ..., v_d) \in \{\text{ the local } C^\infty \text{ frames on } TM \text{ over } U_m\} (\forall m' \in U_m ([(v_1 (m'), ..., v_d (m'))] = o (m'))))\)
//
When Conditions are not required, \(o\) is called "point-wise orientation".
2: Note
In other words, any point-wise orientation is an assignment of an orientation for each point of \(M\).
Conditions requires that that assignment is "continuous".
While a noncontinuous assignment is often not useful by itself, a point-wise orientation is usually used as that 1st, a point-wise orientation is defined, and then, the point-wise orientation is proved to be an orientation.
This definition is not claiming that the continuous assignment is inevitably possible: it is just saying that when a continuous assignment is possible, the assignment is called "orientation of \(M\)".
