2025-06-16

1169: \(q\)-Form over \(C^\infty\) Manifold with Boundary

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definition of \(q\)-form over \(C^\infty\) manifold with boundary

Topics


About: \(C^\infty\) manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of \(q\)-form over \(C^\infty\) 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\( q\): \(\in \mathbb{N}\)
\( (T^0_q (TM), M, \pi)\): \(= \text{ the } C^\infty (0, q) \text{ -tensors bundle over } M\)
\( (\Lambda_q (TM), M, \pi)\): \(= \text{ the } C^\infty q \text{ -covectors bundle over } M\)
\(*f\): \(: M \to T^0_q (TM)\) such that \(Ran (f) \subseteq \Lambda_q (TM)\) or \(: M \to \Lambda_q (TM)\), \(\in \{\text{ the sections }\}\)
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Conditions:
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2: Note


As Description says, "\(q\)-form" may mean \(: M \to T^0_q (TM)\) such that \(Ran (f) \subseteq \Lambda_q (TM)\) or mean \(: M \to \Lambda_q (TM)\), whose difference should not matter in most cases: \(\Lambda_q (TM)\) is an embedded submanifold with boundary of \(T^0_q (TM)\).

The set of all the \(C^\infty\) \(q\)-forms over \(M\) is denoted as \(\Omega_q (TM)\).


References


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