description/proof of that for \(C^\infty\) manifold with boundary, wedge product of \(C^\infty\) forms is \(C^\infty\) form
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(q\)-form over \(C^\infty\) manifold with boundary.
- The reader knows a definition of wedge product of multicovectors.
- The reader admits the proposition that any \(q\)-form over any \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, the wedge product of any \(C^\infty\) forms is a \(C^\infty\) form.
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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(f_1\): \(: M \to \Lambda_{q_1} (TM)\), \(\in \{\text{ the } C^\infty q_1 \text{ -forms }\}\)
\(f_2\): \(: M \to \Lambda_{q_2} (TM)\), \(\in \{\text{ the } C^\infty q_2 \text{ -forms }\}\)
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Statements:
\(f_1 \wedge f_2 \in \{\text{ the } C^\infty (q_1 + q_2) \text{ -forms }\}\)
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2: Proof
Whole Strategy: Step 1: apply the proposition that any \(q\)-form over any \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\).
Step 1:
No matter whether \(f_1 \wedge f_2\) is regarded to be \(: M \to T^0_{q_1} (TM)\) or \(: M \to \Lambda_{q_1} (TM)\), the proposition that any \(q\)-form over any \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\) can be applied.
Let \(V_1: M \to TM, ..., V_{q_1 + q_2}: M \to TM\) be any \(C^\infty\) vectors fields.
\(f_1 \wedge f_2 (V_1, ..., V_{q_1 + q_2}) = (q_1 + q_2)! / ({q_1}! {q_2}!) Asym (f_1 \otimes f_2) (V_1, ..., V_{q_1 + q_2}) = (q_1 + q_2)! / ({q_1}! {q_2}!) 1 / (q_1 + q_2)! \sum_{\sigma} (f_1 \otimes f_2) (V_{\sigma_1}, ..., V_{\sigma_{q_1 + q_2}}) = 1 / ({q_1}! {q_2}!) \sum_{\sigma} (f_1 (V_{\sigma_1}, ..., V_{\sigma_{q_1}}) f_2 (V_{\sigma_{q_1 + 1}}, ..., V_{\sigma_{q_1 + q_2}})\), but each \(f_1 (V_{\sigma_1}, ..., V_{\sigma_{q_1}})\) and \(f_2 (V_{\sigma_{q_1 + 1}}, ..., V_{\sigma_{q_1 + q_2}})\) are \(C^\infty\), by the proposition that any \(q\)-form over any \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\).
So, \(f_1 \wedge f_2 (V_1, ..., V_{q_1 + q_2})\) is \(C^\infty\).
