1168: C∞ q-Covectors Bundle over C∞ Manifold with Boundary

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definition of $C^{\mathrm{\infty }}$ $q$-covectors bundle 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 $q$-covectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ $q$-covectors bundle 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:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$q$: $\in \mathbb{N}$
${\mathrm{\Lambda }}_q(TM)$: $=\underset{m\in M}{⨄}{\mathrm{\Lambda }}_q(T_{m}M)$
$\pi$: $:{\mathrm{\Lambda }}_q(TM)\to M,v\mapsto m$, where $v\in {\mathrm{\Lambda }}_q(T_{m}M)$
$\ast ({\mathrm{\Lambda }}_q(TM),M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}{\text{ vectors bundle of rank }}_d{C}_q\}$
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Conditions:
$({\mathrm{\Lambda }}_q(TM),M,\pi )\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ subbundles of }(T_q^0(TM),M,\pi )\}$
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${\mathrm{\Lambda }}_q(TM)$ is very often implicitly identified with $\underset{m\in M}{⨄}{\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$ by the canonical bijection, $f^{\prime }:{\mathrm{\Lambda }}_q(TM)\to \underset{m\in M}{⨄}{\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R}),t\in {\mathrm{\Lambda }}_q(T_{m}M)\mapsto f(t)$ where $f$ is the canonical 'vectors spaces - linear morphisms' isomorphism mentioned in the definition of canonical 'vectors spaces - linear morphisms' isomorphism between tensors space at point on $C^{\mathrm{\infty }}$ manifold with boundary and tensors space with respect to real numbers field and $p$ cotangent vectors spaces and $q$ tangent vectors spaces and field.

So, although each $t\in {\mathrm{\Lambda }}_q(TM)$ is not really any multilinear map, $f^{\prime }(t)$ is very often implicitly used instead of $t$.

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2: Note

$({\mathrm{\Lambda }}_q(TM),M,\pi )$'s being a subbundle of $(T_q^0(TM),M,\pi )$ means that ${\mathrm{\Lambda }}_q(TM)$ has the subspace topology of $T_q^0(TM)$ and the adopting atlas of $T_q^0(TM)$ (which means that ${\mathrm{\Lambda }}_q(TM)$ is the embedding submanifold with boundary of $T_q^0(TM)$).

Let us see that $({\mathrm{\Lambda }}_q(TM),M,\pi )$ is indeed a subbundle of $(T_q^0(TM),M,\pi )$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the union of any $k$-dimensional vectors subspaces of the fibers that allows any local $C^{\mathrm{\infty }}$ frames is a $C^{\mathrm{\infty }}$ vectors subbundle with the subspace topology and the adopting atlas.

Note that in the following argument, we implicitly identify ${\mathrm{\Lambda }}_q(TM)$ with $\underset{m\in M}{⨄}{\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$: for example, $d{x}^{j_1}\wedge ...\wedge d{x}^{j_q}$ is really on $\underset{m\in M}{⨄}{\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$, and we really mean $f^{\prime -1}(d{x}^{j_1}\wedge ...\wedge d{x}^{j_q})$, but such exactitude is too hard to be maintained (and is almost never seen in the literature).

${\mathrm{\Lambda }}_q(T_{m}M)$ is a ${}_d{C}_q$-dimensional vectors subspace of $T_q^0(T_{m}M)$.

For each $m\in M$, let us take any chart, $(U_m\subseteq M,{\varphi }_m)$.

$d{x}^j:U_m\to T_1^0(TM)$ is a local $C^{\mathrm{\infty }}$ section for $(T_1^0(TM),M,\pi )$.

For any $j_1<...<j_q$, $d{x}^{j_1}\wedge ...\wedge d{x}^{j_q}$ is a local $C^{\mathrm{\infty }}$ section for $(T_q^0(TM),M,\pi )$, because $=\sum _{\sigma \in S_q}sgn\sigma d{x}^{j_{{\sigma }_1}}\otimes ...\otimes d{x}^{j_{{\sigma }_q}}$ , by the proposition that the wedge product of any 1-covectors is the sum of the signed reordered tensor products of the 1-covectors, and each $d{x}^{j_{{\sigma }_1}}\otimes ...\otimes d{x}^{j_{{\sigma }_q}}$ is $C^{\mathrm{\infty }}$.

$\{d{x}^{j_1}\wedge ...\wedge d{x}^{j_q}|j_1<...<j_q\}$ is a required set of ${}_d{C}_q$ local $C^{\mathrm{\infty }}$ sections: for each $m^{\prime }\in U_m$, $\{d{x}^{j_1}{|}_{m^{\prime }}\wedge ...\wedge d{x}^{j_q}{|}_{m^{\prime }}|j_1<...<j_q\}$ is a basis for ${\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$, by the proposition that the $q$-covectors space of any vectors space has the basis that consists of the wedge products of the increasing elements of the dual basis of the vectors space.

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References

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