1204: Rough q-Form over C∞ Manifold with Boundary

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definition of rough $q$-form over $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ $q$-covectors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ $(p,q)$-tensors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of rough section of continuous surjection.

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Target Context

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

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

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.

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References

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