1065: Orientation of C∞ Manifold with Boundary

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definition of orientation of $C^{\mathrm{\infty }}$ manifold with boundary

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of orientation of point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of local $C^{\mathrm{\infty }}$ frame on $C^{\mathrm{\infty }}$ vectors bundle.

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Target Context

• The reader will have a definition of orientation of $C^{\mathrm{\infty }}$ manifold with boundary.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$O$: $={\cup }_{m\in M}{O}_m$, where $O_m$ is the set of the orientations of $T_{m}M$
$\ast o$: $:M\to O$ such that $o(m)\in O_m$
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Conditions:
$\mathrm{\forall }m\in M(\mathrm{\exists }{U}_m\subseteq M\in \{\text{ the open neighborhoods of }m\},\mathrm{\exists }(v_1,...,v_d)\in \{\text{ the local }{C}^{\mathrm{\infty }}\text{ frames on }TM\text{ over }{U}_m\}(\mathrm{\forall }{m}^{\prime }\in U_m([(v_1(m^{\prime }),...,v_d(m^{\prime }))]=o(m^{\prime }))))$
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When Conditions are not required, $o$ is called "point-wise orientation".

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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 $M$".

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References

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