definition of rough \(q\)-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 rough \(q\)-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 \(: M \to \Lambda_q (TM)\), \(\in \{\text{ the rough sections }\}\)
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Conditions:
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2: Note
As Description says, "rough \(q\)-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 \(T^0_q (TM)\).
Usually, we need only (non-rough) forms, but we sometimes need to talk about a rough form in order to 1st introduce a may-be-rough form and then prove that it is really a non-rough form.
