1128: q-Covectors Space at Point on C∞ Manifold with Boundary

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definition of $q$-covectors space at point on $C^{\mathrm{\infty }}$ 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 tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of antisymmetric tensors space with respect to field and $k$ same vectors spaces and vectors space over field.
• The reader knows a definition of $(p,q)$-Tensors Space at Point on $C^{\mathrm{\infty }}$ Manifold with Boundary.
• The reader knows a 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.

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Target Context

• The reader will have a definition of $q$-covectors space at point on $C^{\mathrm{\infty }}$ 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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$m$: $\in M$
$T_{m}M$: $=\text{ the tangent vectors space at }m$
$q$: $\in \mathbb{N}$
${\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$: $=\text{ the }q\text{ -covectors space }$
$f$: $:T_q^0(T_{m}M)\to L(T_{m}M,...,T_{m}M:\mathbb{R})$, $=\text{ the canonical 'vectors spaces - linear morphisms' isomorphism }$
$\ast {\mathrm{\Lambda }}_q(T_{m}M)$: $=f^{-1}({\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R}))$ when $q\ne 0$; $=\mathbb{R}$ when $q=0$, $\in \{\text{ the }\mathbb{R}\text{ vectors spaces }\}$
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Conditions:
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2: Note

Another definition defines "${\mathrm{\Lambda }}_q(T_{m}M):={\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$", but that would make ${\mathrm{\Lambda }}_q(T_{m}M)$ non-subspace of $T_q^0(T_{m}M)$, because we have defined $T_q^0(T_{m}M):=T_m{M}^{\ast }\otimes ...\otimes T_m{M}^{\ast }$, not $=L(T_{m}M,...,T_{m}M:\mathbb{R})$.

$T_{m}M$ is an $\mathbb{R}$ vectors space as is shown in Note for the definition of tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary and ${\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$ is indeed a vectors subspace of $L(T_{m}M,...,T_{m}M:\mathbb{R})$ as is shown in Note for the definition of antisymmetric tensors space with respect to field and $k$ same vectors spaces and vectors space over field. So, $f^{-1}({\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R}))$ is a vectors subspace of $T_q^0(T_{m}M)$.

${\mathrm{\Lambda }}_q(T_{m}M)$ is canonically 'vectors spaces - linear morphisms' isomorphic to ${\mathrm{\Lambda }}_q(T_{m}M:\mathbb{R})$, and quite often, the 2 spaces are implicitly identified by the isomorphism, which is the reason why ${\mathrm{\Lambda }}_q(T_{m}M)$ is called "$q$-covectors space".

${\mathrm{\Lambda }}_0(T_{m}M)=T_0^0(T_{m}M)$.

${\mathrm{\Lambda }}_1(T_{m}M)=T_1^0(T_{m}M)$, because each element of $T_1^0(T_{m}M)$ is antisymmetric.

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References

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